Fairing Methods of Planar and Space Curves Under Design Constraints - Applications in Computer-Aided Ship Design

NATIONAL TECHNICAL UNIVERSITY OF ATHENSDEPARTMENT OF NAVAL ARCHITECTURE AND MARINE ENGINEERINGFairing Methodsof Planar and Space Curvesunder Design Constraints -Applications in Computer-Aided Ship DesignbyKonstantinos G. Pigounakis

Submitted in Partial Ful�lmentof theRequirements for the Degree:Doctor of EngineeringThesis Supervisor:Panagiotis D. Kaklis, Associate Professor

Athens, February 1997

Fairing Methods of Planar and Space Curvesunder Design Conditions -Applications in Computer-Aided Ship DesignbyKonstantinos G. PigounakisSubmitted in Partial Ful�lment of the Requirements for the Degree:Doctor of EngineeringAdvisory CommitteePanagiotis Kaklis, Associate Professor (Supervisor)Apostolos Papanikolaou, ProfessorGerasimos Politis, Assistant ProfessorExamination CommitteeVasilios Dougalis, Professor { National and Capodistrean University of AthensPanagiotis Kaklis, Associate Professor { National Technical University of AthensTheodoros Loukakis, Professor { National Technical University of AthensApostolos Papanikolaou, Professor { National Technical University of AthensGerasimos Politis, Assistant Professor { National Technical University of AthensPanagiotis Sakkalis, Assistant Professor { Agricultural University of AthensStylianos Stamatakis, Assistant Professor { Aristotle University of ThessalonikiAthens, February 1997

To my parents,George and E�e.

Abbriviations2D / 3D Two-/Three-DimensionalCAD Computer-Aided DesignCAGD Computer-Aided Geometric DesignCAM Computer-Aided ManufacturingDW Design WaterlineFEM Finite Element MethodFf Frenet frameFOB / FOs Flat of Bottom / SideGM Metacentre, Metacentric heightLCB / LCF Longitudinal Centre of Buoyancy /FlotationNLP Non-Linear ProgrammingNTUA National Technical University of AthensNURBS Non-Uniform Rational B-SplinesOK O�ending KnotPDE Partial Di�erential EquationsQP Quadratic ProgrammingSAC Stational Area CurveSLC Stational Leverarm CurveSQL Sequential Quadratic ProgrammingSt StationWl Waterline

List of Symbols6 (a;b) Smaller angle between two vectors, a and ba?b a is vertical to ba � b Inner product of a and ba� b Outer or cross product of a and ba0 or da=ds Di�erentiation of a with respect to arc length_a or da=du (da=dt) Di�erentiation with respect to the parameterjaj Absolute value of aja b cj Determinant of three vectors, a, b, and c(a b c) Plane determined by the point vectors, a, b, and c(a b) Plane determined by the free vectors, a and bkak Euclidean norm of vector ar DisplacementA Area de�ned by a curve and the x�axisAnj;k Parameter-dependent terms of L for a B�spline of degree nAW Waterplane areaAX Maximum sectional area�j Parameter-dependent coe�cients of _Q within a segmentB=BW Breadth of ship (moulted) / Breadth at the waterlineb Unit binormal vector (or binormal vector)Cr Order of continuity rCB Block coe�cientCP Prismatic coe�cientCW Waterplane coe�cientCX Maximum section coe�cientD Control polygon of a B�spline curve

D Depth of ship� Di�erence symbol (e.g., �dj�1 = dj � dj�1)di = (dxi ; dyi ; dzi )T Vertex i of the control polygon� Global tolerancehi Absolute torsion discontinuity at ui� Sum of hi for a cubic spline curve�� Maximum permissible value for �Ixx; Iyy Inertias of area with respect to the x� and y�axisJ , J0, Jr Fairing functionalsK Curvature vectork Signed curvaturek; kmax k�th iteration, maximum number of iterations� Curvature (absolute value)L Numerator of the signed curvatureLOA=LW Overall length of ship / Length at the waterlineLCB Longitudinal centre of buoyancyLCF Longitudinal centre of otation�j Parameter-dependent coe�cients of L for a cubic B�splinelimx!x0 Limit of variable x going to the value x0Mxx; Myy Moments of area with respect to the x� and y�axisMj(t) j�th transformed basis function of degree 3Nnj (u) j�th B�spline basis function of degree nNnj (r)(u) r�th parametric derivative of Nnj (u)n Unit principal normal vector (or normal vector)� Sum of nodal sign changes of torsion, wio(�) Lower-case Landau order symbolP Problem referenceQ(t) Spline (B�spline) curve with components Qx(t); Qy(t); Qz(t)Q0(t) Initial spline curve

Q�(t) Changed (faired) spline curveIR The set of real numbersR Ratio of discontinuities and fairness metricsri Tolerance radius, corresponding to the i�th control vertice�i = (�1; 0; 0) Desired direction of signed curvature in i�th segmentS(t) Parametric curve in IR3 with components x(t); y(t); z(t)s(t) Arc length of a parametric curvePk1j=k0 Sum of the index j between the values k0 and k1R x1x0 Integral between the values x0 and x1T Local parametrizationT Draught of shipt Unit tangent vector� Torsion�max Absolute maximum value of torsionU = fu0; u1; :::; ung Parametrization or knot vectorui i�th knotwi Torsion sign changes at nodal points, uiz�i ; zki ; zKi Magnitude of discontinuity at ui for �0, or k0, or K0, respectively� Sum of z�i�� Maximum permissible value for �

AbstractThe present work can be classi�ed in the areas of Computer Aided Geometric and ShipDesign (CAGD - CAShipD). New automatic methods for fairing planar and spatial curves,which can be subjected to geometric and/or general design constraints, are examined. Themethods handle low degree polynomial spline curves of at least second order derivativecontinuity (C2): The curves are given through the B�spline representation, which hasbeen chosen for its exibility in geometric applications. Along with the fairing methodsa process is proposed for developing ship lines when integrated and shape characteristicsof a ship-hull are available.The dissertation starts with a general introduction to the Di�erential Geometry of para-metric curves and the B�spline representation, so that the reader becomes acquaintedwith the notions, the mathematical tools and the notation of the work.A local and iterative fairing method for three dimensional cubic B�splines follows, whichcan be considered an extension and generalization of [Sapidis & Farin '90]. A new fairnesscriterion for spatial C2 cubic splines is introduced, focusing on the quality of the curvatureand torsion plots as well as on the shape characteristics of the curve. New su�cientconditions derive for satisfying the new criterion and the proposed algorithm improvesthe curve locally at every step by raising di�erent kinds of nodal discontinuities.The second method is also automatic and handles spatial cubic B�splines. Shape con-straints are set for the control polygon of every segment of the curve, which secure thei

iismooth movement of the Frenet frame vectors along the curve. The solution of the result-ing Non-Linear Programming problem exhibits substantial improvement of the torsionplot of the curve, while the curvature plot is less a�ected.The third method is appropriate for fairing planar curves under geometric constraints,such as end conditions, shape, and deviation of existing data poins, and/or integral con-straints, e.g. area, �rst and second moments of area. The method is e�ciently used forsolving a number of problems, and it is mainly applicable to Ship Design.Based on the latter method, a process is proposed for developing ship lines, when hydro-static characteristics of the desirable hull are available. Although the process is similarto the traditional method that naval architects follow, it assures that the produced linessatisfy all requirements within a small tolerance throughout the process.All fairing methods incorporate tolerance constraints and bound the di�erence betweenthe initial and the faired curve. This fact secures that the resulting faired curve standsclose to the initial one, which is important for design applications.

AcknowledgementsThe present thesis consummates my postgraduate studies at the National Technical Uni-versity of Athens, which have been partially funded by the Greek Scholarship Foundation,the Commission of the European Union and the Committee of Research of NTUA, towhich I owe special thanks.First, I would like to express my sincere gratitude to my supervisor, Professor PanagiotisKaklis, not only for his continual guidance and help, but also for his friendship and hismoral support all these years I have known him. It has been a great fortune for me towork under his supervision and be taught by his way of thinking and acting. He has givenme the opportunity to start materializing my scienti�c ambitions, though some times thismay have turned out to be an èncumbrance' for his personal life.I would like to thank Professor Apostolos Papanikolaou for his valuable and always willingassistance and support. His deep knowledge and ideas for Ship Design helped me manytimes to keep in touch with the essence of my thesis, and his remarks were always to thetarget.I am indebted to Dr. Nickolas Sapidis for his constant and valuable support, his helpfulcomments, his ideas and his friendship. Thanks to him I got in touch with the CAGDworld.I must thank Assistant Professor Gerasimos Politis for his interest in my work and hispractical advice, which saved me from aimless loss of time.iii

ivI am also indebted to the members of the examination committee, who honoured me byaccepting participating in that.I was also fortunate to collaborate and be a friend with Melkon Isirikian, Spyridon Kap-niaris and Polyvios Kehagioglou, who had worked for their diploma theses under thesupervision of Prof. Kaklis. Especially to Melkon and Spyridon, I am indebted for theirhelp.It would have been a great omission, had I not mention my trusted friends and colleagues,Elias Karyampas and Christodoulos Koskinas, who encouraged me in the �rst steps ofpostgraduate studies and have been of great help throughout the years.Special thanks go to all my colleagues of the Ship Design Laboratory, who have alwaysstood by me and honoured me with their trust and friendship. Along with them, Iwould like to thank the members of the laboratories of Ship Design and of ShipbuildingTechnology, and especially Professor Vassilios Papazoglou for his care and support andDr. Nicolas Tsouvalis for his advice and friendship. Also, I feel that the accomplishmentof this dissertation is undoubtly due to the Department of Naval Architecture and MarineEngineering, to which I am grateful.I am grateful to Professor Joseph Hoschek of the Technische Hochschule Darmstadt(THD), Germany, for his support and experienced advice all the time I spent with him.I also thank the postgraduate students and the personnel of the Zentrum f�ur PraktisheMathematik for their spontaneous help. They made my stay there an exceptional expe-rience.I feel obliged to Dr. D. Papageorgiou (General Motors/USA) and Dr. Kaufmann (Mercedes-Benz/BRD) who both provided me data from their design departments and allowed meto use them and publish the results of my work.Last but not least, I would like to express my deep gratitude to my father, George, my

vmother, Eustathia, and my sister Garyfalia. This work would not have even startedwithout their consent and constant support, moral and �nancial. Along with them, Iwould like to thank Despena for her love and patience.

ContentsChapter 1. Introduction 11.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Organisation of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Fairing methods for spatial curves . . . . . . . . . . . . . . . . . . 41.3.2 Planar fairing under constrains - Ship lines development . . . . . . 6Chapter 2. Basic concepts and de�nitions 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Elements from Di�erential Geometry . . . . . . . . . . . . . . . . . . . . . 102.2.1 Arc length of a regular curve . . . . . . . . . . . . . . . . . . . . . 102.2.2 Frenet frame (Ff) - Curvature and torsion . . . . . . . . . . . . . . 102.2.3 Signed curvature - In ections . . . . . . . . . . . . . . . . . . . . . 132.3 B�spline curves: de�nitions and properties . . . . . . . . . . . . . . . . . 152.3.1 De�nition of integral B�splines . . . . . . . . . . . . . . . . . . . . 162.3.2 Di�erentiation and integration formulae . . . . . . . . . . . . . . . 17vi

Contents viiChapter 3. Local fairing for 3D curves 183.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.1 Mathematical descriptions for fairness . . . . . . . . . . . . . . . . 183.2 Fairness indicators and discrete fairness metrics for cubic curves . . . . . . 203.3 Local continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.1 Torsion continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.2 Continuity related to curvature . . . . . . . . . . . . . . . . . . . . 243.3.3 Continuity of third order (C3) . . . . . . . . . . . . . . . . . . . . 253.4 Automatic method for local fairing . . . . . . . . . . . . . . . . . . . . . . 263.4.1 Determination of the o�ending knot . . . . . . . . . . . . . . . . . 273.4.2 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.6 Conclusions - Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Chapter 4. Fairing of 3D curves under shape constraints 414.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Frenet frame and in ections . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 More local properties of C2 cubic B�splines . . . . . . . . . . . . . . . . . 434.4 Acuteness and local convexity . . . . . . . . . . . . . . . . . . . . . . . . . 454.5 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 504.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

viii Contents4.7 Conclusions - Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Chapter 5. Fairing of 2D curves under design constraints 625.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2.1 End conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.2 Calculation of integrated measurs . . . . . . . . . . . . . . . . . . . 635.2.3 Local convexity of planar cubic B�splines . . . . . . . . . . . . . . 665.3 Fairing of B�spline curves under constraints . . . . . . . . . . . . . . . . 675.3.1 The problem (Pdesign) . . . . . . . . . . . . . . . . . . . . . . . . . 675.3.2 Solution of the problem (Pdesign) - An example . . . . . . . . . . . 695.4 Conclusions - Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Chapter 6. Applications in Computer Aided Ship Design 776.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2 Ship design and hull geometry . . . . . . . . . . . . . . . . . . . . . . . . 786.3 Development of ship lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.3.1 Main characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3.2 Process for solving the problem (Pmesh) . . . . . . . . . . . . . . . 826.4 A CAShipD example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.5 Conclusions - Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

List of Figures2.1 The Frenet frame vectors when a planar curve turns left (right). . . . . . . 133.1 The conditions for �; K0; and C3 continuity, expressed as geometric locii . 243.2 Orthogonal projections of the initial (dashed) and �nal (solid) chine curve. 333.3 Curvature (upper) and torsion (lower) plots of the initial (dashed) and �nal(solid) chine curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Binormal vector distribution of the initial (upper) and �nal (lower) chinecurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Curvature vector distribution of the initial (upper) and �nal (lower) chinecurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6 Orthogonal projections of the initial (dashed) and �nal (solid) GM curve. 373.7 Curvature (upper) and torsion (lower) plots of the initial (dashed) and �nal(solid) GM curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.8 Binormal vector distribution of the initial (upper) and �nal (lower) GMcurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.9 Curvature vector distribution of the initial (upper) and �nal (lower) GMcurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40ix

x List of Figures4.1 Comparison of two curves with the same sign of torsion, but di�erent move-ment of Frenet frame. The lower curve exhibits a rapid change of shapebut no in ection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Acute polygons. Left: Planar. Right: Spatial. . . . . . . . . . . . . . . . . 464.3 Local coordinate system. Left: The initial position of the coordinate sys-tem, P1xy. Right: The relative position of the two coordinate systems,P1xy and P2x0y0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Subspaces that the two coordinate systems can create, and possible posi-tions of the components of _Q(u) and L(u). . . . . . . . . . . . . . . . . . . 494.5 Orthogonal projections of the initial (dashed) and �nal (solid) chine curve. 544.6 Curvature (upper) and torsion (lower) plots of the initial (dashed) and �nal(solid) chine curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.7 Binormal vector distribution of the initial (upper) and �nal (lower) chinecurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.8 Curvature vector distribution of the initial (upper) and �nal (lower) chinecurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.9 Orthogonal projections of the initial (dashed) and �nal (solid) GM curve. 584.10 Curvature (upper) and torsion (lower) plots of the initial (dashed) and �nal(solid) GM curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.11 Binormal vector distribution of the initial (upper) and �nal (lower) GMcurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.12 Curvature vector distribution of the initial (upper) and �nal (lower) GMcurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.1 Planar symmetrical section of a symmetric body. Axis of symmetry: Ox. . 70

List of Figures xi5.2 Control polygons: Interpolant of the initial data (long dashed), A�nelytransformed curve (short dashed), Solution 1 (thin solid), Solution 2 (thicksolid). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Curves: Interpolant of the initial data (long dashed), A�nely transformedcurve (short dashed), Solution 1 (thin solid), Solution 2 (thick solid). . . . 755.4 Curvature Plot: Interpolant of the initial data (long dashed), A�nelytransformed curve (short dashed), Solution 1 (thin solid), Solution 2 (thicksolid). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.1 Initial Hull: Body plan St0.1 of Hull0.1. . . . . . . . . . . . . . . . . . . . 886.2 Initial Hull: Body plan St0.4 of Hull0.2. . . . . . . . . . . . . . . . . . . . 886.3 Sets of waterlines of the initial hull: Wl0.1 (upper) and Wl0.2 (lower). . . 896.4 Aft (left) and fore (right) of Wl0.1 (upper) and Wl0.2 (lower). . . . . . . 896.5 Initial (dashed-I) and modi�ed (solid-II) distributions, SAC and SLC, ofthe initial hull. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.6 Comparison between SAC-1/SLC-1 (dashed) and SAC-F/SLC-F (solid). . 906.7 Initial Hull: Body plan St1.1 of Hull1.1. . . . . . . . . . . . . . . . . . . . 916.8 Final Hull: Body plan St1.5 of Hull1.3. . . . . . . . . . . . . . . . . . . . 916.9 Distributions of Aw, LCF , Ixx and Iyy with respect to the draught of theship (G-distributions). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.10 Sets of waterlines: Wl1.1 (upper) and Wl1.2 (lower). . . . . . . . . . . . . 936.11 Waterlines: Wl1.4 (upper and lower). . . . . . . . . . . . . . . . . . . . . 93

List of Tables3.1 The fairness metrics for the initial and �nal chine curve. . . . . . . . . . . 303.2 The fairness metrics for the initial and �nal GM curve. . . . . . . . . . . . 315.1 Characteristics of the boundary curves of the example section. . . . . . . 726.1 Initial and required characteristics of the hull. . . . . . . . . . . . . . . . . 856.2 The characteristics of hulls calculated through the development process. . 85

xii

Chapter 1Introduction1.1 PrefaceThe recent developments in hardware and software tools have stimulated the transition ofdesign from traditional methods to integrated computer systems. The increasing comput-ing power that has been achieved, leads even the most reluctant to bene�t of ComputerAided Design (CAD) and Manufacturing (CAM), which o�er improvement of productiv-ity, consistently high quality and reduced development time for new products. All theseadvantages enable a �rm to keep up with the modern, customer-driven market conditions,which require capability to design and release frequently new and improved products, sothat this �rm gains strategic bene�ts [Roy '95]. That explains the apparent increase ofCAD companies with innovated systems, targeting even not-so- ourishing markets andindustry sectors, like the marine market and Shipbuilding, which are relatively �nite[Wake '93] { if not shrinking.The need for a wide variety of types, morphology and size of products has boosted Com-puter Aided Geometric Design (CAGD), the branch of CAD that focuses on free formcurve-, surface- and solid-modelling. Since the appearance of a product is important forpotential buyers [Burchard et al. '94], the construction of visually pleasing objects hasbecome vital for industrial design and styling, although this boost was initially not due to1

2 Chapter 1. Introductionmarket reasons, but to practical de�ciencies. Pierre B�ezier, one of the pioneers in CAGD,explains why R�enault adopted his ideas in the early sixties by pointing out the necessity of\a complete, distortion-free and unquestionable de�nition" for their designs, particularlywhen they had to \exchange information" [B�ezier '90].From 1960 until today, research and development have resulted in quite a few methodsfor constructing curves, surfaces and solids. Now industry research is focused in shapequality and fairness [Burchard et al. '94], while vendors aim at an even cooperation be-tween solid and surface modellers [Smith '96]. For a non-connoisseur of the principlesof CAGD it may seem oxymoron that, in a state-of-the-art system { like those basedon Non-Uniform Rational B-Splines (NURBS) { one often faces undesirable shape fea-tures (in ections/oscillations). Unfortunately, this occurs when any spline representation{ NURBS included { is used to interpolate/approximate lower dimension entities, or to�ll a gap between entities, or even when two or more entities must be joined together[Jensen et al. '91], [Massabo '96]. Also, there is a certain need for tools to repair im-ported geometry from `foreign' systems [Smith '96].In order to develop a CAGD system able to produce entities as good as a draftsman candraw, a `complete set' of aesthetic constraints must be available, like the ones used bydraftsmen in judging the quality of a curve or a surface, which should be followed by theirtranslation into mathematical conditions [Burchard et al. '94]. Though that may soundcomplicated, the principle of the simplest shape is widely accepted { originating mainlyin the ideas of [Birkho� '33] { and has been followed by the majority of researchers. Theprocess of eliminating undesirable features of geometric entities is called fairing, and itis by no means in uenced only by styling. A number of practical constraints, like man-ufacturability, functionality and performance according to hydrodynamics/aerodynamicsshould be also taken into account, so that such a process can be successful.Materialising a CAGD system with robust techniques for automatic fairing is not an easy

1.2. Organisation of the dissertation 3task and is still a challenge [Burchard et al. '94], [Jensen et al. '91]. That is justi�ed bythe plethora of existing fairing tools [Burchard et al. '94], [Catley '89], [CETENA '85],[Hohenberger & Reuding '95], [Hottel et al. '91], [Klass et al. '91], [KCS '94], which, how-ever, often stand far from satisfying the end-users [Hays '92] and lack of `simple' utilities,like fairing of truly spatial curves, or automatic fairing under design requirements alongwith the styling ones.Even in the area of Computer Aided Ship Design (CAShipD), spatial curves are ùnavoid-able evils', since they can be found on every vessel, from a high-speed craft to an ultra-largetanker [Clement '63], [Clement & Blount '63], [Comstock-PNA '67], [Serter '92], [Serter'94], so fairing tools are quite necessary for ship hull generation. Nevertheless, for thatprocess, spatial curve fairing is not the target. As in most engineering applications, ge-ometry should satisfy a number of requirements and consist the optimal solution. Thisapproach starts to become a trend in CAD, and the incorporation of constraints in mod-elling processes is a fact [Anderl & Mendgen '96], [Feng & Kusiak '95], [Zou et al. '96].For these reasons, the dissertation investigates possible ways of automating local fairing for3D curves, based on two di�erent approaches; one that aims at àberrations' of curvatureand torsion, and one that controls the shape through the Frenet frame. Additionally, itgives attention to a fairing method for planar curves, which incorporates several designconstraints, derived mainly from the area of Ship Design. Based on that method, a processfor the development of hull lines is proposed, which can be automated and consist a toolfor CAShipD.1.2 Organisation of the dissertationThe �rst chapter of the thesis continues with a general literature review related to thework areas, while the second chapter focuses on the mathematical background needed

4 Chapter 1. Introductionfor comprehending the contents of the following chapters. It gives a brief review of thenotions of Di�erential Geometry related with planar and spatial parametric curves, andsome elements for the B�spline representation.Chapters 3 and 4 deal with fairing methods for three-dimensional (3D) curves. New fairingcriteria are introduced and algorithms are developed in order to ful�l them. In Chapter 3di�erent kinds of continuity are discussed, and an iterative local fairing algorithm isproposed. In Chapter 4 the behaviour of the Frenet frame is controlled in any segment,and this results to the quality improvement of the curve.Chapter 5 proposes a method for fairing planar curves under constraints. Some constraintsare set for the curve itself (end conditions, curvature sign, tolerances), while others are setfor integral characteristics (area, �rst and second order moments of area) of the planardomain bounded by the curve and one of the cartesian axes, assuming that the curvecan be described by a function of one variable. The well known energy functional is thefairness criterion, consisting of two integrated derivative magnitudes of the curve. Mainlybased on the method of Chapter 5, a design process is proposed in Chapter 6, whichdevelops ship lines under a number of design requirements and can be automated.1.3 Literature Review1.3.1 Fairing methods for spatial curvesThough a number of fairness criteria for 3D curves have been proposed (see [Farin '93],ch. 23, [Hoschek & Lasser '93], ch. 13, and references therein), the majority of themrefer to integral functionals, which, in most cases, involve one or more squared parametricderivatives up to third order, and in few cases are directly related to curvature and torsion.Using integral functionals of the former kind, many algorithms have been proposed using

1.3. Literature Review 5polynomial splines [Eck & Hadenfeld '95a], [Fang & Gossard '95], [Hohenberger & Reud-ing '95], [Hagen & Bonneau '91], [Huanzong et al. '92], [Hu & Schumaker '86], [Reinsch'67], [Reinsch '71], most of which solve the interpolation/ approximation (or fairing)problem analytically and quickly, but the results are not always satisfactory, partiallybecause of the assumption that the parametrization of the curve is close to the naturalone (see [Pigounakis et al. '95]). Proposed improvements, such as in [Liu et al. '92] and[Vassilev '96], have not been tested for truly 3D curves, so this kind of algorithms shouldbe considered supplements of the planar case.The second category of integral functional criteria are mainly covered by [Roulier et al. '91],[Roulier & Rando '94], and by [Moreton & S�equin '93], [Moreton & S�equin '95]. Thoughmore sophisticated, these criteria cannot be implemented with analytic methods and theresulting algorithms are either simple and iterative or materialized by Non-Linear Pro-gramming (NLP) techniques1. Also, in Moreton's work the employed criteria deal onlywith curvature and the variation of curvature, while they ignore torsion. On the otherhand, [Roulier & Rando '94] seems to focus more on the integrated study of integral func-tional methods, since it explains the e�ect of a variety of fairness criteria and proposesiterative algorithms for implementing them.The criteria mentioned above exhibit two inherent disadvantages, namely, (i) they assumethat the fairness of curve is almost constant along its length, so they act globally, and(ii) they cannot cope up with the shape of the curve, given by the sign of torsion. Anexception for (i) is [Eck & Hadenfeld '95a], where a global fairness measure is used fora local criterion. A more detailed analysis for global and local criteria is given in theintroduction of Chapter 3.A completely di�erent approach is found in [Wagner et al. '95], where a new kind ofpolynomial splines, called spring splines, is introduced, which incorporate constraints for1NLP techniques are also iterative. Nonetheless, this distinction aims to underline that NPL techniquesare much more time consuming and their iterations may require the solution of large linear systems.

6 Chapter 1. Introductioncurvature and torsion, based on mechanics. Nevertheless, the whole problem is set in arather simpli�ed way and no results for 3D curves are given.The discussion above points out the necessity of local fairing criteria and methods whichare mainly intended for spatial curves, i.e., they take into consideration both curvature andtorsion. Such criteria and a method for C2 cubic B�splines are presented in Chapter 3.Experience has shown that fairing of 3D curves means, among others, endeavour for`right' shape, which is not always evident. Presumably, that is why the �rst papers onshape preservation in IR3 with polynomial splines are very recent [Goodman & Ong '91],[Kaklis & Karavelas '96], [Labenski & Piper '96]. Along with them, a number of methodsfor detecting shape failures, and particularly in ections can be used for or extented to 3Dspace [Goodman '91], [Hoschek '84], [Manocha & Canny '91], [Hansen & Nielsen '95a],[Hansen & Nielsen '95b]. However, the problem of hailing any kind of shape failures, likesign-changes of torsion, has not been faced yet. The only exception is met in [Jones '87],where a method is proposed for shape manipulation of spatial curves, though the condi-tions are rather application oriented and not tested for 3D curves. Also, a constructivemethod has been introduced in [Higashi & Kaneko '88] for B�ezier curves, but the imposedcondition seems too stringent, since it guarantees monotonic curvature variation.In the fourth chapter of the dissertation, a method for elimination of shape failures isproposed, which controls the Frenet frame vectors along a cubic C2 B�spline. Thecontrol is achieved by setting constraints on the control polygon of the curve.1.3.2 Planar fairing under constrains - Ship lines developmentIndustrial design lacks of methods that incorporate design requirements along with curve/surface generation, and that gives birth to design spirals until geometry comes in accor-dance with functionality. Especially in ship design, such conditions are in the every-day

1.3. Literature Review 7practice, because a hull must be a compromise of numerous and sometimes controversialconstraints. As a result, the elimination of such spirals is still considered unavoidable,while CAD systems simply \speed the trip around the classic design spiral" [Hays '92].So, the majority of ship designers change the geometry of a ship in a more or less arbitraryway in order to achieve the appropriate characteristics [Firth '95], while the more pro-gressive take advantage of the mathematical representation of their system and estimatethe result of the modi�cations [Birmingham & Smith '95].At the antipode of this trend, there exist a few methods for generating/ fairing curvesand surfaces under general constraints. The Institut f�ur Schi�s- und Meerstechnik of theTechnische Universit�at Berlin have proposed procedures for developing fair curves andsurfaces under constraints [Nowacki & L�u '94], [Nowacki et al. '90], [Nowacki et al. '95],[Standerski '88]. More speci�cally, [Nowacki et al. '90] and [Nowacki & L�u '94] solve theproblem of fairing a planar curve under end and area constraints. The former refers to aninterpolation scheme using a single-segment B�ezier curve, while the latter approximatesgiven data with quintic Hermite polynomial splines. In [Standerski '88] a ship hull ismodelled by a B�spline surface with uniform parametrization and rectangular grid forthe control points. For speci�c isoparametric lines (sections) area moments of �rst andsecond order are required, but the restrictive setting of the problem leads to results,which can be used only as guidelines and not for lines development. In none of the aforementioned works exist constraints for shape, and all of them employ an energy functionalminimization as a fairness criterion.Along with them, [Ganos & Papanikolaou '95] proposes a method for generation of hullforms of traditional greek �shing boats, which employs non-parametric cubic form splinesand a rectangular grid. Though the framework seems rather restrictive, the results arequite satisfactory, at least for the particular, simple-shaped hulls. Another related work is[Lowe et al. '94], where hydrodynamic criteria are incorporated in a Partial Di�erential

8 Chapter 1. IntroductionEquations (PDE)2 method for yacht hull generation.For di�erent purposes, [Bercovirer & Jacobi '94] suggests a constrained minimization al-gorithm for o�set curve construction with composite B�ezier curves employing a FiniteElement Method (FEM). Also, [Gopalsamy & Reddy '93] presents an algorithm for deter-mining the shape of a curve along with the minimization of a combined functional forerror and fairness.The �fth chapter of this thesis develops a method for fairing planar B�spline curves ofarbitrary �practically up to seventh� degree, by minimizing a combined error and energyfunctional under constraints for the ends of the curve, the sign of curvature, the area, the�rst and second moment of area, as well as tolerance constraints for the nodal points.In the last chapter of the dissertation, a process is proposed for the development of shiphull lines under constraints, taking advantage of the method of Chapter 5. Though de-sign constraints are employed only in [Standerski '88] and [Ganos & Papanikolaou '95],the hull generation problem has been faced by a few researchers [Bedi & Vickers '89],[Kouh & Chau '93], [Liu et al. '92], [Rogers & Fog '89], [Rossier '90], and nowadays nu-merous systems provide such methods [AUTOSHIP '96], [KCS '94], [Wake '93]. The nov-elty of the proposed process is that the design constraints are present throughout thecalculations and the designer can de�ne the desirable shape of any line or local area of thehull. Furthermore, the variety of constraints is the widest among the existing methods,and the process can be automated.2PDE methods are introduced and widely used by M.I.G. Bloor, M. Wilson and their collaboratorsat the Department of Mathematical Studies of the University of Leeds.

Chapter 2Basic concepts and de�nitions2.1 IntroductionDi�erential Geometry provides the necessary mathematical background for interroga-tion and study of the behaviour of a curve, which can be seen as a geometric set ofpoints, i.e. a locus, and, at the same time, as a path traced out by a particle in IR3[Millman & Parker '77], [Kreyszig '91], [Stoker '88], [Struik '88] etc. The latter consider-ation is compatible with the parametric expression of a curve, which o�ers informationnot only for the successive positions of the particle, but also for the time that this particlepasses through a speci�c position. One can understand that changing the correspondencebetween positions and times, the kinematic condition of the particle is altered while thepath remains the same. Though this piece of information is valuable, there are certain un-changed geometrical properties due to the fact that the curve is the same, no matter howthe particle moves along the path. These properties enable one to estimate the quality ofthe curve.Apart from notions of Di�erential Geometry, this chapter presents properties and intrin-sic characteristics of polynomial spline parametric curves expressed with the so-calledB�spline representation. The choice of B�splines is based on the CAD/CAM practice,where NURBS have become a standard, since their properties o�er many advantages9

10 Chapter 2. Basic concepts and de�nitionsfor changing and manipulating such curves e�ciently [Farin '91], [Farin '93], [Farin '94],[Hoschek & Lasser '93], [Piegl & Tiller '95], [Rogers & Adams '90].2.2 Elements from Di�erential Geometry2.2.1 Arc length of a regular curveLet S(t) = (x(t); y(t); z(t))T be a real vector function of a real variable t, representinga parametric curve in IR3. As stated in many books (e.g., [Stoker '88], pp. 12-13) andin order to avoid ambiquities, it is practical to work with a speci�c family of parametriccurves, of which all three components are at least one time continuously di�erentiableand, for any t the �rst derivative of at least one of the components is non-zero. Suchcurves are called regular.Since IR3 is a Euclidean space, the length, s(t), of the curve S(t) from a �xed point (say,from S(t = t0)) to a variable point equals tos(t) = Z tt0 jj _S(l)jjdl; (2.1)and is called the arc length of the curve. Dot denotes di�erentiation with respect to theparameter t and "jj � jj" is the Euclidean norm. It can be shown that the arc length is aninvariant property of the curve (see, e.g., [Willmore '93]). If a curve S(t) is de�ned as afunction of its arc length, its parametrization is called natural and holds jjS0jj = jj _S(s)jj =1. From now on, di�erentiation with respect to arc length, s, will be denoted by primeand the resulting derivatives will be also called natural.2.2.2 Frenet frame (Ff) - Curvature and torsionNo matter what the parametrization of the curve is, the �rst derivative of a parametriccurve is a vector, tangent to the curve, which points in the direction that the parameter

2.2. Elements from Di�erential Geometry 11increases. In other words, this vector gives the velocity of a moving particle for theparticular path. If such a vector, t, is of magnitude 1, namelyt = _S(t)jj _S(t)jj = S0(s); (2.2)is called the unit tangent vector or, simply, the tangent vector of the curve.Assuming that the curve is of continuity order Cr, r � 2, we can di�erentiate the relationt � t = 1 with respect to arc length and obtain t � t0 = 0. Hence, if the vector t0 := S00 isnot the null vector, it is orthogonal to t. The vectorsK = t0 = S00 and n = S00(s)jjS00(s)jj (2.3)are the curvature vector and the unit principal normal vector or, simply, the normalvector, respectively. The direction of these vectors does not depend on the orientation ofthe curve, i.e., their direction does not change if the parameter s is replaced by s� = �sor any other allowable parameter ([Kreyszig '91], p.34). For an arbitrary parametrizationof the curve, the expression of the curvature vector isK(t) = 1jj _S(t)jj2 h�S(t)� t(t)�t(t) � �S(t)�i; (2.4)where dot " � " denotes inner product.The magnitude of the curvature vector, which is known as curvature, �, measures therate of change of the tangent vector [Millman & Parker '77], [Stoker '88] [Kreyszig '91],[Porteous '94]. It can be shown (see, e.g., [Kreyszig '91], [Struik '88]) that� = jjS0(s)� S00(s)jj = jjL(t)jjjj _S(t)jj3 ; (2.5)where "� " denotes outer (or cross) product,andL(t) = _S(t)� �S(t): (2.6)The equation (2.5) is commonly used as a de�nition for curvature, which is assumedalways positive.

12 Chapter 2. Basic concepts and de�nitionsThe normal vector and the tangent vector are orthogonal and de�ne the osculating plane.For a planar curve the osculating plane is identical to the plane of the curve, while a spatialcurve belongs locally to its osculating plane. For di�erent ways of de�ning the osculatingplane, as well as its properties, one can refer [Kreyszig '91], [Stoker '88], [Willmore '93].Another vector, orthogonal to t and to n, can be introduced for any regular curve withnon-vanishing second natural derivative:b = t� n = L(t)jjL(t)jj : (2.7)This vector is called the unit binormal vector or, simply, the binormal vector. From theabove de�nition, one can observe that the vectors t; n; b; in this order, have the right-handed direction. The triplet they form is called the moving trihedron or the Frenetframe (Ff). Apart from the osculating plane, there are two more planes de�ned by theFf vectors: t de�nes the rectifying plane, and n de�nes the normal plane [Struik '88],[Kreyszig '91], [Willmore '93].Apart from measuring the change of tangent through curvature, it is of high interest tomeasure the vector of deviation of a curve from the osculating plane, or, in other words,the rate and direction of change of the osculating plane. Since b is the vector whichde�nes the plane, we study the derivative b0 for a Cr curve S, r � 3, and non-vanishingcurvature. It can be shown (see, e.g. [Kreyszig '91], [Stoker '88], [Willmore '93]) thatb0 = ��n, where � = �(s) is the torsion of the curve at S(s):� = �n � b0 = � � � = jS0 S00 S000jjjS00jj2 = j _S �S ...Sjjj _S� �Sjj2 ; (2.8)where "jabcj" denotes the determinant of the three vectors a, b and c.Unlike curvature, the sign of torsion has a geometric signi�cance: it shows whether a curve'curves' right (positive) or left (negative). The sign of torsion is orientation invariant, whilefor planar curves torsion vanishes identically (see [Kreyszig '91], p. 39).

2.2. Elements from Di�erential Geometry 13t

n

b

��

b

n

t

Figure 2.1: The Frenet frame vectors when a planar curve turns left (right).The relation between the Ff vectors and their natural derivatives are given through theSerret-Frenet formulae: 264 t0n0b0 375 = 264 0 � 0�� 0 �0 �� 0 375 264 tnb 375 : (2.9)2.2.3 Signed curvature - In ectionsIf � � 0 holds along an entire curve, then it follows from (2.5) that the curve is a straightline. In case that � 6= 0 and the curve is planar, b is constantly orthogonal to the planeof the curve, whereas n lies on the plane. If the planar curve turns to the left (right) withrespect to the tangent vector, then at any position along the curve the normal vectorpoints to the left (right) of the tangent vector, and the binormal vector points up (down);see Figure 2.1. In that particular case we can de�ne a positive/negative direction withreference to the osculating plane and introduce the signed curvature:k = �b = _S(t)� �S(t)jj _S(t)jj3 : (2.10)(A detailed analysis for planar curvature can be found in [Stoker '88], pp. 23{27.)Until now, it has been assumed that neither S00 nor S0�S00 equal to null vector, but this isnot an unusual case, especially in IR2. These two conditions have the same interpretation,

14 Chapter 2. Basic concepts and de�nitionsnamely the curve does not change in direction, so its tangent is constant. For an arbitraryparametrization the tangent vector could remain constant (�S = 0) or change only in mag-nitude ( _S� �S = 0), but both result to vanishing curvature. If (�(s�) = �(t�) = 0) holds,then this position, S(s�) = S(t�), is called a point of in ection. Excluding the case thatthe curve is a straight line itself, at the neighbourhood of an in ection the curve exhibits astraight line behaviour, since its tangent does not change. Then the osculating plane andthe Ff vectors can be de�ned using the �rst non-vanishing derivative of the curve, thoughat a point of in ection, even a C1-curve need not possess a unique osculating plane (anexample is given in [Willmore '93]). Apart from the tangent vector, which is determinedfor any regular curve, the other two vectors can be expressed through Taylor expansionusing higher derivatives of the curve at the neighbourhood of s�. Assuming that onlythe second natural derivative is zero at s� and the curve is at least C4�continuous in theneighbourhood of s�, while its �fth-order natural derivative exhists ahd is �nite there, weget: S(s� � h) = S0(s�)� hS00(s�) + h22 S000(s�)� h36 S0000(s�) + o(h4); (2.11)S00(s� � h) = S00(s�)� hS000(s�) + h22 S0000(s�) + o(h3); (2.12)S000(s� � h) = S000(s�)� hS0000(s�) + o(h2): (2.13)where o(�) is the lower-case Landau order symbol. From the de�nitions of the Ff vectorsand the equations (2.11)-(2.13) we have:limh!0n(s� � h) = � S000(s�)jjS000(s�)jj ; limh!0b(s� � h) = �S0(s�)� S000(s�)jjS000(s�)jj : (2.14)Following the same procedure as above, one can calculate the torsion at an in ection,which is non-vanishing and continuous:limh!0 �(s� � h) = 12 jS0(s�)S000(s�)S0000(s�)jjjS000(s�)jj2 : (2.15)For the special case that the curve is a cubic polynomial, the value of torsion at thein ection equals to zero according to (2.15).

2.3. B�spline curves: de�nitions and properties 15If the curve is determined through an arbitrary parametrization, the following expressionsfor the Ff vectors can be used:t(t) = _S(t)jj _S(t)jj ; b(t) = L(r)(t)jjL(r)(t)jj ; n(t) = b(t)� t(t); (2.16)where L(r)(t) = _S(t) � S(r)(t); and r � 2 is the order of the �rst derivative that L(r) isnon-vanishing at t.Looking at (2.14) one can realize that n and b are discontinuous at an in ection point.Nevertheless the osculating plane remains the same, though its orientation changes. Inorder that this discontinuity is raised, the following convention is usually adopted inDi�erential Geometry: at the neighbourhood of an in ection b is considered constantin direction and � is taken to be positive or negative according to the sign of the innerproduct t0 � n ([Brand '84], p. 310, [Struik '88], p. 15). This is in accordance with theformulae of (2.9) and o�ers an explanation for the signed curvature in plane. For a deeperanalysis on the Serret{Frenet equations, one can trace back in [Nomizu '59]. Nevertheless,in this work the convention for the sign of � at an in ection will not be followed, since Ffis going to be the principal mathematical tool for interrogating 3D curves.2.3 B�spline curves: de�nitions and propertiesThe most widely used method for de�ning curves, in computer-aided hull design andother CAD applications, is the B�spline method. Integral B�splines de�ne parametricpiecewise-polynomial curves, while rational B�splines de�ne parametric curves that con-sist of rational-polynomial segments. The fundamental properties of integral B�splinesare brie y reviewed below; detailed expositions for both representations are available in,e.g., [Farin '91], [Farin '93], [Hoschek & Lasser '93], [Piegl & Tiller '95].

16 Chapter 2. Basic concepts and de�nitions2.3.1 De�nition of integral B�splinesIntegral B�splines (in short, B�splines) are classical polynomial splines represented aslinear combinations of B�spline basis functions. The coe�cients of this family of splineshave a geometrical interpretetion, as they form a polygonal line, referred to as the con-trol polygon, which reveals the shape of the corresponding curve. The fundamentals ofB�splines are included in most of CAGD textbooks; see, for instance, [de Boor '78],[Farin '93], [Hoschek & Lasser '93] e.t.c.LetD = fdi = (dxi ; dyi ; dzi )T ; i = 0; 1; : : : ;Mg be a set of points in IR3, and U = fu0; u1; : : : ;uNg; N = M + n+1, a non-decreasing sequence of real numbers (knots). The B�spline,Q(u) = (Qx(u); Qy(u); Qz(u))T , of order n corresponding to the control polygon D andthe knot sequence (parametrization) U is given by:Q(u) = MXi=0diNni (u); u 2 [un; uM+1]; (2.17)where Nnj (u); j = 0; 1; : : : ;M; are the B�spline basis functions [de Boor '78], [Farin '93],[Schumaker '81], which are de�ned by the recursive formulaNni (u) = u� uiui+n � uiNn�1i (u) + ui+n+1 � uui+n+1 � ui+1Nn�1i+1 (u); u 2 (�1;+1); (2.18)with N0i (u) = ( 1 if u 2 [ui; ui+1);0 elsewhere:Assuming that all control points are distinct, the degree of continuity between the poly-nomial segments of the spline is dictated by the knot vector: if ui has multiplicity r (i.e.,ui = ui+1 = ui+2 = : : : = ui+r�1), then Q(u) is at least Cn�r at that knot. Finally,making the multiplicity of the �rst and the last knot equal to the degree of the splineplus one (: n + 1) forces the spline to interpolate the �rst and the last control point, aproperty highly desirable to designers. The particular structure of these basis functionsproduces a plethora of properties for the curve Q(u), making it a powerful tool for CAD

2.3. B�spline curves: de�nitions and properties 17applications. Among others, a B�spline has the `variation diminishing' and the `convexhull' properties, it can be e�ciently evaluated using the de Boor algorithm, and it pro-vides the ability of `local control'. Furthermore, the derivatives of a B�spline can be alsoexpressed as B�splines of lower degree; see [Hoschek & Lasser '93], [Farin '93].2.3.2 Di�erentiation and integration formulaeThe restriction of a B�spline curve, Q(u), on [ui; ui+1], i = n(1)M , with multiplicity(n + 1) at both ends, can be written as:Q(u) = iXj=i�ndjNnj (u); (2.19)and its derivative of order r equals to:Q(r)(u) = iXj=i�ndjNnj (r); (2.20)where the derivatives of the basis functions are calculated recursively:Nni (r)(u) = nui+n � uiNn�1i (r�1)(u) nui+n+1 � ui+1Nn�1i+1 (r�1)(u): (2.21)Especially the �rst and second derivatives are written in the form:_Q(u) = n iXj=i�n+1 �dj�1uj+n � ujNn�1j (u) (2.22)�Q(u) = n(n� 1) iXj=i�n+1" �dj�1uj+n � uj � �dj�n+1uj+2 � uj�1# Nn�1j (u)uj+2 � uj ; (2.23)where �dj�1 = dj � dj�1.If the knots at the ends of a curve, Q(u), uk1 and uk2, exhibit multiplicity n + 1 (opencurve), then it holds that:I = Z uk2uk1 Q(u)du = 1n + 1 k2�k1�n+2Xi=k1 di(ui+n+1 � ui): (2.24)

Chapter 3Local fairing for 3D curves3.1 Introduction[Farin et al. '87] introduced the concept of knot removal { knot reinsertion1 for a pla-nar cubic B�spline, which was subsequently applied in the development of local fairingalgorithms for curves [Eck & Hadenfeld '95a], [Farin & Sapidis '89], [Hottel et al. '91],[Pigounakis & Kaklis '94], [Poliako� '96], [Sapidis & Farin '90], and for point sets [Eck &Jaspert '94] (discrete version). Starting from the same idea, in this chapter criteria forevaluating the fairness of C2 spatial curves are introduced and methods for local fairingof C2 cubic B�splines are proposed. The basic tools for interrogation are the curvaturevector, the curvature plot and the torsion plot of the curve, i.e. the magnitudes of cur-vature and torsion versus parameter, since the proposed solutions aim to correct locallyundesirable behaviours in those plots.3.1.1 Mathematical descriptions for fairnessCurvature plots are extensively used by CAD researchers, developers and users for in-specting curves (and surfaces) on a computer screen [Catley '89], [Sapidis & Farin '90],1More about knot removal can be found in [Hoschek & Lasser '93], [Farin '93], as well as in[Eck & Hadenfeld '95b], while for knot insertion in [Boehm '80] among others.18

3.1. Introduction 19[Farin & Sapidis '89], [Eck & Hadenfeld '95a]. The basic idea can be found already in[Birkho� '33], and its applications have been evaluated in related studies on planar curves,which have concluded that a fair curvature plot is free of any unnecessary variation, i.e.,the distribution of curvature on a fair curve must be as uniform as possible [Burchard etal. '94], [Farin '94], [Sapidis '94]. A number of researchers have proposed mathematicalcriteria aiming at materializing the above idea, and these criteria can be divided into twocategories:Criterion A. (Direct Criterion for C2 Planar Curves) A curve is characterized as fairif the corresponding curvature plot is comprised of as few as possible monotone (strictlier:almost linear) segments [Farin et al. '87], [Farin & Sapidis '89], [Sapidis '94].Criterion B. (Indirect or Ènergy' Criterion for C2 Planar Curves) A curve is consideredfair if it minimizes the integral of the squared curvature or the squared slope of curvature(or an approximation to one those) with respect to arc length (see review of papers in[Roulier & Rando '94]).Indirect criteria, and the associated curve design and curve fairing methods, have beeninvestigated since the early days of CAD, with mixed results. Their major disadvan-tages are: (i) their e�ectiveness, in terms of consistently producing fair curves, has beenmarginal, (ii) they disallow user involvement as they always function as a `numericalblack box', and (iii) they are global, i.e., they a�ect the whole curve, while in practicedesigners are interested in removing local imperfections from a curve without altering thecorrect parts of it. On the other hand, the direct criterion A and the corresponding fair-ing methods [Farin et al. '87], [Farin & Sapidis '89], [Sapidis & Farin '90], are free of thelimitations (ii) and (iii). This fact, combined with the empirical observation that thesefairing algorithms do remove local undulations from curves, has created a considerableinterest in them among researchers and practitioners [Hottel et al. '91], [Klass et al. '91],[Eck & Jaspert '94], [Eck & Hadenfeld '95a].

20 Chapter 3. Local fairing for 3D curvesIn order to state a direct fairness criterion for spatial curves, analogous to the one statedabove for planar curves, the elimination of unnecessary variations in curvature and torsionis required, since this is the reason that causes a viewer's eye to stop when inspecting thecurve. Especially for a C2 curve, e.g., a cubic B�spline with simple knots, the torsion andthe derivative of curvature are discontinuous functions, namely they exhibit a variationat least at the knots. Based on the above, the following can be proposed:Criterion C. (Direct Fairness Criterion for C2 Spatial Curves) A C2 curve is character-ized as fair if (a) its curvature plot is comprised of as few as possible monotone (strictlier:almost linear) segments, (b) its torsion plot is as close as possible to being continuousalso with the fewest possible number of monotone (strictlier: almost linear) pieces, (c) thesign changes in the torsion plot are as few as possible, and (d) the value of the torsion,at each point of the curve, is as small as possible.Trying to adopt the proper indicators in order to judge the smoothness of a cubic spline,the notion of the curvature vector can be very helpful. [Manning '74] proposed that ameasure of smoothness for curves is the curvature vector, and this idea has been exploitedin a number of works dealing with geometric continuity; see, e.g., [Boehm '88].3.2 Fairness indicators and discrete fairness metricsfor cubic curvesThe third derivative of a C2 cubic spline curve, Q(u), is, in general, discontinuous at thenodal points, and this fact a�ects the derivative of curvature as well as the torsion of thecurve. For curvature, one can investigate the discontinuity of the slope of curvature, orof the slope of the signed curvature, or of the curvature vector with respect to arc length.The derivative of curvature with respect to arc length equals to�0 = d�ds = n � ...Qk _Qk3 � 3� t � �Qk _Qk2 ; (3.1)

3.2. Fairness indicators and discrete fairness metrics for cubic curves 21while the derivative of the signed curvature, as de�ned by (2.10), equals tok0 = dkds = t� ...Qk _Qk3 � 3k t � �Qk _Qk2 ; (3.2)and the derivative of the curvature vector with respect to arc length, is equal to:K0 = dKds = ��2t+ �0n + ��b = n(n � ...Q) + b(b � ...Q)k _Qk3 � 3K t � �Qk _Qk2 � �2t : (3.3)The absolute discontinuity of �0 at a nodal point (knot), ui, equals toz�i = j�0(u+i )� �0(u�i )j = kn(ui) ��...Q(ui)kk _Q(ui)k3 ; (3.4)where �...Q(ui) = ...Q(u+i ) � ...Q(u�i ), and, analogously, for the discontinuity of k0 at ui wehave the expression for the magnitude:zki = kk0(u+i )� k0(u�i )k = kt(ui)��...Q(ui)kk _Q(ui)k3 : (3.5)The discontinuity of the curvature vector at ui equals tozKi = kK0(u+i )�K0(u�i )k = kn(ui)�n(ui) ��...Q(ui)�+ b(ui)�b(ui) ��...Q(ui)�kk _Q(ui)k3 : (3.6)All these types of discontinuity that are presented above will be analysed in the nextsection, where conditions for di�erent kinds of continuity at ui are set.On the other hand, torsion is discontinuous at any nodal point, ui, and its absolutediscontinuity is hi = j�(u+i )� �(u�i )j = b(ui) ��...Q(ui)k _Q(ui)� �Q(ui)k : (3.7)Part (a) of Criterion C (x3.1.1) implies the requirement that the derivative of curvaturewith respect to arc length is continuous, which is not true, in general, at the knotsof a cubic spline. Similarly, from part (b) of Criterion C derives the requirement for

22 Chapter 3. Local fairing for 3D curvestorsion continuity. The corresponding discrete fairness metrics for the whole curve are,respectively: � := MXi=4 z�i and � := MXi=4 hi: (3.8)The third part of Criterion C is expressed through the sum� = MXi=4wi; where wi = ( 1 if �(u+i ) � �(u�i ) < 00 otherwise (3.9)while part (d) is checked by introducing the absolute maximum value of torsion, �max =maxfj�(u)jg 2, as another metric.3.3 Local continuityLet [ui�1; ui] and [ui; ui+1] be two consecutive parameter intervals of a cubic B�spline.These two segments are a�ected by the control points dj; j = i � 4; :::; i. We will seehow a speci�c kind of continuity can be achieved by moving anyone of those controlpoints. Since we are interested in the movement of one point only, we denote that pointby D = (Dx; Dy; Dz)T := dj, and we split the curve in two parts, namely the one thatis a�ected by D and the residual of the curve, Qres. The corresponding basis function isdenoted by N := N3j (ui) and the curve at ui can be written asQ(ui) = Qres(ui) +DN = (Qxres; Qyres; Qzres)T + (Dx; Dy; Dz)T �N; (3.10)For reasons of simplicity, the variable u is omitted and all functions are calculated atu = ui, unless it is stated otherwise.The derivatives of the curve are written, respectively,_Q = _Qres +D _N = ( _Qxres; _Qyres; _Qzres)T + (Dx; Dy; Dz)T � _N; (3.11)�Q = �Qres +D �N = ( �Qxres; �Qyres; �Qzres)T + (Dx; Dy; Dz)T � �N; (3.12)2For a cubic polynomial the absolute maximum value of torsion can be found analytically, since thenumerator of torsion equals to a constant value ([Kaklis & Karavelas '96], x3), and the denominator is apolynomial of fourth degree.

3.3. Local continuity 23whereas for the third derivative we have...Q(u�i ) = ...Q(u�i ) +D ...N(u�i ) = ...Q�res +D ...N�; (3.13)...Q(u+i ) = ...Q(u+i ) +D ...N(u+i ) = ...Q+res +D ...N+; (3.14)and the di�erence of the third derivative from the left and from the right is:�...Q = ...Q(u+)� ...Q(u�) = (�...Qxres;�...Qyres;�...Qzres)T + (Dx; Dy; Dz)T ��...N: (3.15)3.3.1 Torsion continuityIn order to achieve torsion continuity at ui moving only one point, we set the followingequation (see also (3.7)), assuming that �...Q(ui) 6= 0 and �(ui) 6= 0:hi = 0() b(ui) ��...Q(ui) = 0() �� _Q �Q �...Q��u=ui = 0: (3.16)The locus of the points,D, that satisfy (3.16) is a plane (see Fig. 3.1) and can be expressedas A1Dx + A2Dy + A3Dz + A4 = 0; (3.17)where A1 = �x ��...N � z ��...Qyres + y ��...Qzres;A2 = �y ��...N � x ��...Qzres + z ��...Qxres;A3 = �z ��...N � y ��...Qxres + x ��...Qyres;A4 = �x ��...Qxres +�y ��...Qyres +�z ��...Qzres; (3.18)and�x = _Qyres �Qzres� �Qyres _Qzres; �y = _Qzres �Qxres� �Qzres _Qxres; �z = _Qxres �Qyres� �Qxres _Qyres; (3.19) x = _N �Qxres � �N _Qxres; y = _N �Qyres � �N _Qyres; z = _N �Qzres � �N _Qzres: (3.20)

24 Chapter 3. Local fairing for 3D curves��

��

b

n

tK continuity

τ continuity

(u )

continuity3C

QiFigure 3.1: The conditions for �; K0; and C3 continuity, expressed as geometric locii3.3.2 Continuity related to curvatureAssuming again that �...Q(ui) 6= 0 and �(ui) 6= 0, then, according to (3.6), the continuityof the derivative of the curvature vector with respect to arc length holds whenzKi = 0() kn�n ��...Q�+ b�b ��...Q�ku=ui = 0: (3.21)The two vectors of the sum in (3.21) lie along n and b respectively, thus they are linearlyindependent and belong to the normal plane. Therefore, the condition holds when bothare of zero magnitude, namely �...Q(ui) ? (n;b) () t(ui) � �...Q(ui) = 0. Looking at(3.5), one observes thatzKi = 0() zki = 0() t(ui)��...Q(ui) = 0() _Q(ui)��...Q(ui) = 0: (3.22)With respect to the variable point, D, equation (3.22) is written as a system of threelinear equations of rank at most two, which represents a straight line (see Fig. 3.1):0B@ 0 �z ��y��z 0 �x�y ��x 0 1CA � (Dx; Dy; Dz)T + (�x; �y; �z)T = 0; where (3.23)�x = _N�...Qxres ��...N _Qxres; �y = _N�...Qyres ��...N _Qyres; �z = _N�...Qzres ��...N _Qzres; (3.24)

3.3. Local continuity 25�x = _Qyres�...Qzres � _Qzres�...Qyres; �y = _Qzres�...Qxres � _Qxres�...Qzres; �z = _Qxres�...Qyres � _Qyres�...Qxres:(3.25)For planar curves, the condition for �0 continuity coincides with the condition (3.22).Nevertheless, (3.22) is su�cient but not necessary for �0 continuity in 3D space. In thatcase the condition is set as follows:z�i = 0() n(ui) ��...Q(ui) = 0() ��b t�...Q��u=ui = 0: (3.26)The same condition with respect to D is:( y�y + z�z)(Dx)2 + ( z�z + x�x)(Dy)2 + ( x�x + y�y)(Dz)2� ( x�y + y�x)DxDy � ( y�z + z�y)DyDz � ( x�z + z�x)DzDx+ ( �z�y � �y�z + y�z � z�y)Dx+ ( �x�z � �z�x + z�x � x�z)Dy+ ( �y�x � �x�y + x�y � y�x)Dz+ ( �x�x + �y�y + �z�z) = 0 (3.27)Eq. (3.27) represents a quadric surface. The classi�cation of that surface is not obviousand depends on the relative positions of the other four control points that a�ect the curveat ui.3.3.3 Continuity of third order (C3)The continuity of the third derivative is achieved when�...Q(ui) = 0() (Dx; Dy; Dz)T = �(�...Qxres;�...Qxres;�...Qzres)T�...N : (3.28)It is obvious that the position of the control point which satis�es the C3 continuityat ui, satis�es also continuity of torsion and continuity of the derivative of curvature[Boehm '88], [Lee et al. '92]; see also Fig. 3.1. In case that D = di�2; then the positionfor C3 continuity coincides with the result of the 2�2 algorithm in [Sapidis & Farin '90].

26 Chapter 3. Local fairing for 3D curves3.4 Automatic method for local fairingAiming at minimizing the various previously mentioned discrete fairness metrics, di�erentmethods can be developed. The herein local fairing method is iterative, i.e., in every stepthe position of a speci�c control point is changed so that a particular kind of discontinuityis eliminated. After a number of iterations both curvature and torsion plots becomesmoother, so the corresponding curve is fairer than the initial one. Such methods are ofheuristic nature, since convergence cannot be guaranteed. Nevertheless, testing a similarone with practical examples has shown that they are most reliable [Pigounakis et al. '95],[Ives-Smith '96].According to the ideas of [Sapidis & Farin '90] and the pertinent bibliography for 2Dcurves, the obvious solution for reducing all kinds of discontinuity is to achieve C3 conti-nuity in every step. Practical experience has shown that C3 continuity is a rather stringentcondition and after a few iterations the resulting curve deviates from the initial one. Thusin every step the òptimal' continuity should be established.Two questions arise about the location that a discontinuity should be removed, and thecontrol point that should be changed for that purpose. For the �rst question the answeris based on the idea of the o�ending knot [Sapidis & Farin '90], though with an extendedsense (see below). For the second one, the answer is the same with [Sapidis & Farin '90],i.e., the control point that a�ects most the o�ending knot is chosen. This decision isbased on the fact that, for uniform parametrization, the basis function N3i�2(u) exhibitsthe highest value among all basis functions at the knot ui, therefore the movement of thecontrol point di�2 a�ects the curve at Q(ui) more than any other control point. It is notfar from true to say that in most practical cases the choice of di�2 is the right one. Evenif this were not exact, the theoretical background would not fail, since any control pointa�ecting ui can be chosen for erasing a discontinuity.

3.4. Automatic method for local fairing 273.4.1 Determination of the o�ending knotLet Q(k)(u) be a cubic B�spline curve at the k�th iteration1 of a local fairing procedure,which does not change the initial knot vector, U . For Q(k)(u) the following knots can befound:� up; so that zp = maxfz�i (k); i = 4(1)Mg;� uq; so that hq = maxfh(k)i ; i = 4(1)Mg;� ur; so that � (k)max 2 [ur�1; ur]; or � (k)max 2 [ur; ur+1];where � (k)max = maxfj� (k)(u)jg and � (k)(u) is the torsion of Q(k)(u). Since � (k)max appears ingeneral in the interior of a segment, two knots can be chosen. So, ur is determined as theknot where the curve also exhibits greater discontinuity of torsion.Next, the following ratios are calculated with respect to the discontinuities and metricsof the current iteration, k:Rz = zp� ; Rh = hq� ; and R� = �max� (0)max : (3.29)De�nition 3.1 The o�ending knot (OK) is that knot among fup; uq; urg, where the max-imum value of the corresponding ratios Rz; Rh; or R� is found3, unless wh = 1, when theo�ending knot is uq.De�nition 3.1 can be interpreted as follows: in every iteration, we look for any kind ofdiscontinuity, which, if eliminated, improves most the corresponding fairness metric ofthe curve. In order to incorporate part (c) of Criterion C, the de�nition `gives bounty' toknots sign changes of torsion.1The initial curve is denoted as the curve of zero iteration, Q(0).3It is not expected that two or more of the ratios exhibit exactly the same value, so the maximumcan be determined

28 Chapter 3. Local fairing for 3D curves3.4.2 The algorithmThe input data for the fairing procedure are, apart from the curve, Q(0)(u), the followingvalues, which are used as termination criteria:� a global tolerance, �, which should be an upper bound for the sum�:= PMi=4 jjQ(k)(ui)�Q(0)(ui)jj;� the maximum number of iterations, kmax;� values �� and �� corresponding to � and �; respectively, which determine whether thecurve can be considered fair enough1.The proposed algorithm can be described with the following steps:0 [ Calculate the fairness metrics, �(0); �(0); �(0) and � (0)max ]1 IF [ k � kmax ]THEN [ find the knots up; uq; ur; and u(k)OK ]ELSE STOP2 IF [ up = uq = ur ]THEN [ find the point, d(k)OK�2, which guarantees C3 continuity ]3 IF [ u(k)OK � up ]THEN [ find the closest point2, d(k)OK�2; for which zK(k) = 0 ]4 IF [ u(k)OK � uq ] OR [ u(k)OK � ur ]THEN [ find the closest point2, d(k)OK�2; for which h(k) = 0 ]5 IF [ �� ] AND [ [ �(k) � �� ] OR [ �(k) � �� ] ]THEN [replace Q(k�1)(u) with Q(k)(u) ] AND [ k = k + 1 ]ELSE STOP6 GOTO 11It is quite di�cult for the designer to set values for �� and ��, since the sense of fairness cannot beeasily interpreted in terms of discontinuity sums. In most cases these values are set to zero.2Closest control point to the existing control point, d(k�1)OK�2.

3.5. Examples 29In Step 2 of the algorithm, the position for C3 continuity is unique, whereas in Steps3 and 4 one should calculate the orthogonal projection of the existing control point,corresponding to the o�ending knot, to a geometric locus (plane for torsion continuity,straight line for curvature vector slope continuity) in order to �nd the new position. Thechoice of the closest point guarantees that the curve changes as less as possible.3.5 ExamplesThe performance of the algorithm is tested in two practical examples. The �rst oneoriginates in the Parent Model 4667-I of the TMB Series 62 for hard-chine planning hulls[Clement '63], [Clement & Blount '63]. The second data have been provided by GeneralMotors Design Center, (Warren, MI). In all tests the initial curve is depicted with dashedline, while the �nal one with solid line. The same convention stands for curvature andtorsion plots. In both examples, the �rst and the last control points remain unchanged.First the three projections on the cartesian planes are given (Figs. 3.2, 3.6), whereone can see the deviation between the initial and the �nal curve. Then the plots ofcurvature and of torsion vs. parameter are presented (Figs. 3.3, 3.7). Since the values oftorsion vary strongly 1, these �gures also depict the torsion plots in two di�erent scalings.Finally, 3D-views of the scaled distributions of the binormal vector { Figs. 3.4, 3.8 { andthe curvature vector { Figs. 3.5, 3.9 { along the initial and the �nal curve are given,respectively. An e�ort is made so that the best view is chosen for every 3D plot, thoughthat leads sometimes to confusion of the orientation of the curve; thus, an `S' is printednear the starting point of the curves in 3D plots.Example 1: The initial curve is an interpolant of thirteen (13) points, obtained bydigitizing the chine curve of the model 4667-I. The control polygon consists of �fteen1The torsion plot may not show the exact value of the local maxima of torsion, especially if thesemaxima correspond to peaks; nonetheless, the local maxima are analytically found, so the algorithm isaccurate.

30 Chapter 3. Local fairing for 3D curves(15) points and the parametrization of the curve corresponds to the chord-length of theinterpolated data. Because of digitizing, the quality of the initial curve is rather poor. Onecan notice three local maxima of torsion in the intervals [u11; u12]; [u12; u13]; and [u14; u15],which decrease to one, in [u13; u14]; after fairing; see Fig. 3.3. Also, the algorithm improvesthe curvature and torsion plot, the latter of which becomes of constant sign (negative).This improvement is obvious in Figs. 3.4 and 3.5: the strong variation of the binormalvector along the initial curve almost disappears in the �nal curve, while the distributionof the curvature vector becomes very smooth.The global tolerance, �, is set to 0:200m; (reference length: 10:000m), and the maximumnumber of iterations kmax = 100. The values of �� and �� are set to zero. After 81iterations, the maximum total deviation between the initial and the �nal nodal points isfound � = 0:197m and the maximum nodal deviation 0:027m. The fairness metrics ofthe initial and �nal curve are presented in Table 3.1.Fairness Metrics Q0(u) Q(u) Reductionmaxfzig14i=4 0.502139 0.059400 88.17 %maxfhig14i=4 2.577463 0.222267 91.37 %� 1.694275 0.157899 90.68 %� 14.407156 0.567479 96.06 %� 8. 0. 100.00 %�max 136.741117 6.726018 95.08 %Table 3.1: The fairness metrics for the initial and �nal chine curve.Example 2: The initial curve is a feature line of the hood of a car. The control polygoncontains twenty-two (22) points, and the parametrization is equidistant (uniform). Thequality of the curve is good. No serious problems can be detected from the curvatureplot, except for the intervals [u20; u21]; [u21; u22]; where the values of curvature vary. Onthe other hand, in the torsion plot there exist peaks near the two ends (Fig. 3.7). Theseproblematic areas are quite obvious in the distributions of binormal and the curvaturevector (Figs. 3.8, 3.9). The faired curve stays close to the initial one, while its curvature

3.6. Conclusions - Future work 31and torsion plots are quite smooth and the maximum values are signi�cantly reduced (Fig.3.7), which can be also noticed in the �gures of vector distributions along the curves.Here, the global tolerance is � = 10mm; (reference length: 700mm), and the maximumnumber of iterations kmax = 500. The values of �� and �� are also set to zero. The maximumtotal deviation is found � = 9:879mm, the maximum nodal deviation 2:624mm and thenumber of iterations k = 114. The fairness metrics of the initial and �nal curve arepresented in Table 3.2.Fairness Metrics Q0(u) Q(u) Reductionmaxfzig22i=4 5.072063�10�4 2.161102�10�5 95.74 %maxfhig22i=4 7.281034�10�2 9.207439�10�4 98.74 %� 1.397465�10�3 1.370568�10�4 90.19 %� 1.400674�10�1 7.087902�10�3 94.95 %� 11. 3. 72.72 %�max 23.157029 0.117412 99.49 %Table 3.2: The fairness metrics for the initial and �nal GM curve.3.6 Conclusions - Future workThe fairness criteria of this chapter are adequate for improving curves with local problems.Furthermore, the iterative nature of the presented method, as well as other possiblemethods based on these criteria, leads to an overall correction of curvature and torsion,so they can be compared with global methods that minimize integral functional (see[Ives-Smith '96], [Pigounakis et al. '95]). Such methods exhibit comparative advantages,namely they are easily implemented, automatic, fast, with low required information ininput, which make them considerably important for industrial use.Apart from developing variations of the algorithm presented here in order to eliminatespeci�c kinds of discontinuity, the interest should be focused on the investigation of thenecessary and su�cient condition for continuity of the derivative of curvature. The clas-

32 Chapter 3. Local fairing for 3D curvessi�cation of the quadric surface of Eq. (3.27) and the determination of the closest pointto the control point to be moved could result to more e�cient algorithms.

Figuresofexamples33

0

1.6

0 1.60

1.6

0 10

0

1.6

0 10

Figure 3.2: Orthogonal projections of the initial (dashed) and �nal (solid) chine curve.

34 Chapter 3. Local fairing for 3D curves

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15

-120

-100

-80

-60

-40

-20

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u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15

-8

-6

-4

-2

0

2

u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15Figure 3.3: Curvature (upper) and torsion (lower) plots of the initial (dashed) and �nal(solid) chine curve.

Figures of examples 35S

010

0

6

-2

0

2

S

010

0

6

-2

0

2

Figure 3.4: Binormal vector distribution of the initial (upper) and �nal (lower) chinecurve.

36 Chapter 3. Local fairing for 3D curvesS

10

0

6

-2

0

2

S

10

0

6

-2

0

2

Figure 3.5: Curvature vector distribution of the initial (upper) and �nal (lower) chinecurve.

Figuresofexamples374650

5050

950 10504650

5050

0 700

950

1050

0 700

Figure 3.6: Orthogonal projections of the initial (dashed) and �nal (solid) GM curve.

38 Chapter 3. Local fairing for 3D curves

0

0.002

0.004

0.006

0.008

0.01

0.012

u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15 u16 u17 u18 u19 u20 u21 u22

-1

-0.5

0

0.5

1

1.5

2


-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15 u16 u17 u18 u19 u20 u21 u22Figure 3.7: Curvature (upper) and torsion (lower) plots of the initial (dashed) and �nal(solid) GM curve.


4600

5100

0

700

650

1050

S

4600

5100

0

700

650

1050

Figure 3.8: Binormal vector distribution of the initial (upper) and �nal (lower) GMcurve.

40 Chapter 3. Local fairing for 3D curvesS

4600

5100

0

700

650

1050

S

4600

5100

0

700

650

1050

Figure 3.9: Curvature vector distribution of the initial (upper) and �nal (lower) GMcurve.

Chapter 4Fairing of 3D curvesunder shape constraints4.1 IntroductionAccording to Di�erential Geometry, a spatial curve is {up to a rigid motion{ fully deter-mined by its curvature and torsion [Kreyszig '91], [Stoker '88], [Struik '88], [Willmore '93].Since curvature is sign indi�erent, it is not suitable for shape interrogation, while torsionis useful for extracting basic shape information from its sign (see Chapter 2). For a moredetailed analysis, a good background in Di�eretial Geometry is needed in order to properlyinterpret the behaviour of the curve and propose methods for shape improvement.On the other hand, the Ff can also be employed for studying spatial curves [Stoker '88],[Struik '88], and that approach is becoming popular in the CAGD community; see, forexample, [Moreton '95], [Greiner et al. '95]. As shown in Chapter 2, the notion of in ec-tion can be introduced in exactly the same way for 2D and 3D curves, because at such apoint two of the Ff vectors change their direction; see Eqs. (2.14).In this chapter the combined use of the Ff and the sign of torsion is examined for the caseof a C2 cubic B�spline, so that the curve can exhibit the proper shape and its behaviourbecome predictable. The proposed method is based on a new notion, the latent in ection41

42 Chapter 4. Fairing of 3D curves under shape constraints(Def. 4.2), which is a stronger notion of in ection. The choice of cubic splines, whichexhibit discontinuous torsion at the nodal points, does not a�ect the principal shapefeatures of the method.4.2 Frenet frame and in ectionsWhen the curvature of a regular 3D curve vanishes at u�, namely L(u�) = 0, the curveexhibits a straight-line behaviour there. In such cases, if the tangent straight-line at u�intersects the curve, then L(u��) �L(u�+)=jjL(u)jj2 = �1. Since planar curves do not di�erfrom spatial ones in that, the direction of the in ection point can be uni�ed for both 2Dand 3D curves:De�nition 4.1 A curve Q(u) exhibits an in ection at u�, i� the binormal vector is dis-continuous there, i.e., b(u��) � b(u�+) = �1 or, equivalently, 6 (b(u��);b(u�+)) = �:The symbol 6 (�; �) denotes the smaller angle between two vectors. The requirement for anangle of � always holds for in ections in plane and in space, because the osculating planeof the curve must change orientation therein, though the osculating plane itself remainsthe same.In 3D space the binormal vector of a curve can change its direction enormously, but notso abruptly as it does at an in ection. That causes a substantial change of the osculatingplane, though the sign of torsion may remain constant (Fig. 4.1). In order to classify thisundesirable case, which is similar to an in ection, we give the following de�nition:De�nition 4.2 The curve Q(u) exhibits a latent in ection, i� there exists an interval(ua; ub), so that b(ua) � b(ub) � 0 or, equivalently, 6 (b(ua);b(ub)) � �=2.

4.3. More local properties of C2 cubic B�splines 43

n

n

t

b

b

n

n

nb

b

b

t

t

tt

nb

t

Figure 4.1: Comparison of two curves with the same sign of torsion, but di�erent move-ment of Frenet frame. The lower curve exhibits a rapid change of shape but no in ection.De�nition 4.2 di�ers from De�nition 4.1, in two ways. First, the positionsQ(ua) andQ(ub)can be close to, but not necessarily in the same neighbourhood of the curve. Second, theangle 6 (b(ua);b(ub)); though greater than �=2, does not necessarily equal to �: As aresult, De�nition 4.2 is more general than De�nition 4.1, and one can easily show that:Lemma 4.1 If a curve exhibits no latent in ections in an interval, then there exist noin ections therein.4.3 More local properties of C2 cubic B�splinesExpanding Eq. (2.22), the �rst derivative of a cubic B�spline, Q(u), in [ui; ui+1] can bewritten as:_Q(u) = 3 iXj=i�2�dj�1 N2j (u)uj+3 � uj = �1(u)�di�3+�2(u)�di�2+�3(u)�di�1witha1;2;3(u) � 0:(4.1)

44 Chapter 4. Fairing of 3D curves under shape constraintsTaking into consideration (4.1) and (2.23), L(u) equals to:L(u) = 18f�1(u)(�di�3��di�2)+�2(u)(�di�3��di�1)+�3(u)(�di�2��di�1)g (4.2)where�1(u) = 1(ui+1 � ui�2)(ui+2 � ui�1)"(ui+2 � ui+1)N1i (u)N1i�1(u)(ui+2 � ui)(ui+1 � ui�1) + (N1i�1(u))2(ui+1 � ui�1)#;�2(u) = 1(ui+1 � ui�2)(ui+3 � ui) (ui+1 � ui)N1i (u)N1i�1(u)(ui+2 � ui)(ui+1 � ui�1) ; (4.3)�3(u) = 1(ui+2 � ui�1)(ui+3 � ui)"(ui � ui�1)N1i (u)N1i�1(u)(ui+2 � ui)(ui+1 � ui�1) + (N1i (u))2(ui+2 � ui)#;One can easily realize that the coe�cients �j(u); j = 1; 2; 3 are non-negative, and foruniform parametrization �1(u) and �3(u) are always greater than �2(u). (An extendeddiscussion can be found in [Pigounakis & Kaklis '94]).The normal vector lies in the direction of the quantity L(u)� _Q(u), which, with the aidof (4.1) and (4.2), is written as a sum of nine factors:i�2Xj=i�3 i�1Xk=j+1 i�1Xr=i�1�6+j+k�2i(u) � �4+r�i(u)[(�dj ��dk)��dr]: (4.4)For u 2 [ui; ui+1], the above sum is a linear combination of the double cross products ofthe vectors �dj; j = i� 2(1)i, and its coe�cients are non-negative.Another quantity of high interest is the numerator of torsion, j _Q(u)�Q(u) ...Q(u)j, since itcontrols its sign. After some algebra the numerator can be expressed as:Ci � j�dj�3�dj�2�dj�1j; (4.5)where Ci is a positive constant number, depending on the the knots uj; j = i� 2(1)i+3.Remark: It is well known that each cross product of the type �dk�1 ��dk, divided bythe lengths of the two vectors involved, expresses a vector, the magnitude of which equals

4.4. Acuteness and local convexity 45to the sine of the angle between �dk�1 and �dk. This sine is also the numerator of thediscrete curvature of the points dk�1dkdk+1; see [Sauer '70], x2.3. Additionally, the scalarproduct in (4.5), divided by the lengths of the involved vectors, expresses the sine of thebihedral angle between the planes (dj�3dj�2dj�1) and (dj�2dj�1dj). That sine is thenumerator of the discrete torsion{ again in the sense that [Sauer '70] introduced{ of thepolygon dj�3dj�2dj�1dj. So, one can observe how the discrete properties of the controlpolygon a�ect the continuous ones of the corresponding curve.4.4 Acuteness and local convexityTaking advantage of the simple formulae derived in the previous section, we can predictthe local behaviour of the Ff of a spatial C2 cubic B�spline, when its control polygonpossesses some attributes. But �rst, some new de�nitions are necessary.De�nition 4.3 A planar polygon p0p1p2 is called acute, i� it is convex and 0 � !1; !2 <�=2, where !1 = 6 (p0p1); and !2 = 6 (p1p2); see Fig. 4.2.A planar acute polygon is also regular according to [Goodman '91].De�nition 4.4 A spatial polygon p0p1p2 is called acute, i� the orthogonal projection ofp2 on the plane of (p0;p1) forms an acute polygon with three legs or, equivalently, i�0 � !1; !2; � < �=2, where � is the smaller bihedral angle of the planes (p0;p1) and(p1;p2); see Fig. 4.2.It can be shown that the condition of De�nition 4.4 implies that the orthogonal projectionof p0 on the plane of (p1;p2) forms an acute polygon, too. So, the requirement for!1;2; � 2 [0; �=2) is equivalent to that for `three-legs acute polygon'.

46 Chapter 4. Fairing of 3D curves under shape constraints3

p

p

PP

P

2

1

P

0

1

2

2

0

1

p

1

2

p1

3

��

��

pp

P

P

P

P0

0

1

2

2

Figure 4.2: Acute polygons. Left: Planar. Right: Spatial.(1,0,0)

x

y

z

P

P P

P

0

1 2

3

-a

-b-d

e

c12

(0,0,0)

xy’-plane

y

z

P

P

0

2

3P

P1 x

y’

z’

Figure 4.3: Local coordinate system. Left: The initial position of the coordinate system,P1xy. Right: The relative position of the two coordinate systems, P1xy and P2x0y0.Using the above de�nitions, we can show how the change of the vectors of the Ff can bebounded and, thus, predictable.Proposition 4.1 If �di�3�di�2�di�1 is a spatial acute polygon, corresponding to a seg-ment of Q(u); u 2 [ui; ui+1], then the change of direction of the binormal vector is alwaysless than �=2 therein, i.e., there is no latent in ection in [ui; ui+1].Proof: Without loss of generality, we assume that the knots that a�ect the segmentQ(u);u 2 [ui; ui+1], de�ne the set T = ftj = (ui+j�2�ui)=(ui+1�ui); j = 0(1)5g; and the curveis expressed as bfQ = Q(t) through the transformation t = (u� ui)=(ui+1 � ui) 2 [0; 1]:

4.4. Acuteness and local convexity 47Furthermore it is assumed kp1k = 1:The coordinate system is set as shown in Figure 4.3, namely the xy-plane is identicalto the plane of (p0;p1); with x-axis starting at the second vertex, P1, of the polygonand going along p1. Due to acuteness (Def. 4.4), the coordinates of the control verticesare: P0 = (�a;�b; 0)T ; P1 = (0; 0; 0)T ; P2 = (1; 0; 0)T ; and P3 = (c;�d; e)T ; wherea; b; c; d; e 2 IR+; a 6= 0; c > 1: The segment Q(t); t 2 [0; 1]; can be written as:P0M0(t) + P2M2(t) + P3M3(t) = 264 �a 1 c�b 0 �d0 0 e 375 [M0(t)M2(t)M3(t)]T (4.6)where Mk(t); k = 0(1)3 are the transformed basis functions, corresponding to N3j (u);j = i� 3(1)i:The cross product of the �rst and second derivative of the curve can be writtenL(t) = 264 Lx(t)Ly(t)Lz(t) 375 = 18(1� t0)(1� t1)(t4 � t1)t4t5 264 lx(t)ly(t)lz(t) 375 ; (4.7)wherelx(t) = be(t4 � t1)(1� t)tly(t) = �e[a(t4 � t1)(1� t) + (1� t0)(t� t1)]t (4.8)lz(t) = �d(1� t0)(t� t1)t� [ad+ b(c� 1)](t4 � t1)(1� t)t� bt5(t4 � t)(1� t)Furthermore 4.8imply that, for t 2 (0; 1), all three components of L(t) are of constantsign, namely Lx(t) is positive, and Ly(t); Lz(t) are negative. For t = 0; 1 the vector L(t)is vertical to the planes of (p0;p1) and (p1;p2); respectively. That implies that the anglebetween the binormal vectors of any two positions within the segment is less than or equalto �=2.In order to secure that this angle is always strictly less than �=2, the coordinate systemis rotated around the x-axis. In the new system, P1xy0z0, the xy0-plane is identical to

48 Chapter 4. Fairing of 3D curves under shape constraintsthe plane of (p1;p2) (Fig. 4.3). If we repeat the calculation of L(t), we can notice thatLx(t) and Ly0(t) are positive, while Lz0(t) is negative. From this second result and the�rst one, we conclude that L(t) lies within the subspace that is de�ned by the positivex-axis, the negative z-axis and the negative z0-axis. Then, all possible directions of L(t)form among them angles smaller than �=2, since none of them can be parallel to x-axis,as the algebraic expressions imply. So, the possible directions of binormal vector in thesegment lie within the convex spherical sector formed be the unit sphere at P1 and thepositive x�axis, the negative z�axis and the negative z0�axis.A similar result holds true for the normal vector:Proposition 4.2 If �di�3�di�2�di�1 is a 3D acute polygon, corresponding to a segmentof Q(u); u 2 [ui; ui+1], then the change of direction of the normal vector is always lessthan � therein.Proof: If one employs the same technique of rotating the coordinate system P1xyz, onecan express the �rst derivative and �nd out that the x� and z�components of _Q(t) arepositive in both coordinate systems. For the initial coordinate system, the �rst derivativecan be written as:_Q(t) = 264 _Qx(t)_Qy(t)_Qz(t) 375 = 3(1� t0)(1� t1)(t4 � t1)t4t5 264 qx(t)qy(t)qz(t) 375 ; (4.9)where qx(t) = a(t4 � t1)t4t5(1� t)2 + (1� t0)(1� t1)t5(t4 � t)t +(1� t0)t4t5(t� t1)(1� t) + (c� 1)(1� t0)(1� t1)(t4 � t1)t2 (4.10)qy(t) = b(t4 � t1)t4t5(1� t)2 � d(1� t0)(1� t1)(t4 � t1)t2 (4.11)qz(t) = e(1� t0)(1� t1)(t4 � t1)t2:The formulae for the second coordinate system, P1xy0z0, are similar.

4.4. Acuteness and local convexity 49z

z’

y

y’

q

q

q z’q

y’

qy

y’

qx

qy

z

��

��

��

��

��

��

xx

z yz’

y’

xy-plane

xz’-plane

xz-plane

xy’-plane

zz’

y

y’

y

l l

l

ll

z’ z

y’

Figure 4.4: Subspaces that the two coordinate systems can create, and possible positionsof the components of _Q(u) and L(u).The cross-product L(t)� _Q(t) can be analyzed in the cross-products of the componentsof L(t) and _Q(t), which lie on the axes of xyz and xy0z0 (Fig. 4.4). Calculating thenine cross-products, we see that the components of L(t) � _Q(t) lie within a subspaceformed by x-axis, the negative z0-axis and the positive z-axis. Then the only case thatan in ection could happen would be if the vector lied in x-axis, which is incongruous,since the y-component of that vector, Lz(t) _Qx(t)� Lx(t) _Qy(t); is always non-zero. Thatis true, because Lz(t) _Qx(t) is always negative, while Lx(t) _Qy(t) is non-positive. So, thepossible directions of normal vector in the segment lie within the convex spherical sectorformed be the unit sphere at P1, the positive x�axis, the positive z�axis and the negativez0�axis.Based on the results above, we can state that, if the control polygon of a B�spline curveis locally acute, within the segment it in uences neither an in ection nor even a latentin ection occur. Furthermore, the Ff changes predictably therein.

50 Chapter 4. Fairing of 3D curves under shape constraints4.5 Formulation of the problemIn former sections the su�cient conditions for the control polygon are set, so that themanipulation of the shape of the curve is guaranteed in any segment. It is also wellknown how one can create planar segments (zero torsion) or straight-line segments (zerocurvature) by enforcing a special con�guration on the control polygon (see, e.g. [Farin '93],[Hoschek & Lasser '93]). Having all these in mind, we set the following problem:Problem (Pshape): Let Q(0)(u); u 2 [u3; uN�3]; be a C2 cubic B�spline with a knotvector U = fui; i = 0(1)Ng and a control polygon D(0) = fd(0)i ; i = 0(1)Mg; N = M +4.Construct a cubic B�spline Q(u) with the same knot vector U and a new control polygonD, which satis�es the �delity criterion:MXi=0 kdi � d(0)i k2 = mind̂i2IR3 MXi=0 kd̂i � d(0)i k2; (4.12)and ful�ls the shape requirements (i),(ii) and the tolerance constraints (iii):(i) Torsion constraints: Let It = fr ; 3 � r � N � 4g be a set of indices, specifyingthe intervals [ur; ur+1], where the sign of torsion can be predetermined. For the cor-responding segments the torsion can be positive, negative, or even zero (planarcase), that is: j�dr�3�dr�2�dr�1j ><= 0: (4.13)(ii) Ff constraints: Let If = fl ; 3 � l � N � 4g be a set of indices, specifyingthe intervals [ul; ul+1], where the user desires a controlled change of the Ff. Forthe corresponding segments, the related control vertices can either form an acutepolygon:�dl�3 ��dl�2 > 0; �dl�2 ��dl�1 > 0; and (�dl�3 ��dl�2) � (�dl�2 ��dl�1) > 0;(4.14)

4.6. Examples 51or be co-linear: �dl�3 ��dl�2 = 0; and �dl�2 ��dl�1 = 0: (4.15)(iii) Tolerance constraints: each control point, di, should lie within a sphere withcentre at d(0)i and user-speci�ed radius, ri:kdi � d(0)i k2 � ri: (4.16)(Pshape) is a Non-Linear Programming (NLP) problem, consisting of a quadratic objectivefunction, bi- tri- and quarti-linear shape constraints and quadratic tolerance constraints.Because of the tolerance constrains, (Pshape) is not always solvable. The acuteness ofthe control polygon is materialized by replacing zero in the strict shape constraints withsu�ciently small numbers �: All these � and ri are parameters for the problem. Anyfeasible data for the optimization problem guarantees the non-existence of latent points.If no feasible point is found, one should permit larger deviations from the initial controlpoints and/or decrease of the �-values.(Pshape) is solved with the Sequential Quadratic Programming (SQL) technique , wherethe whole problem is analysed in a number of Quadrartic Programming (QP) problems(see, e.g., [Gill et. al '81]). The software implementation that has been used for solving(Pshape) is provided by [Spellucci '95].4.6 ExamplesIn order that the performance of the algorithm can be tested, the employed data sets arethe same to those of Chapter 3. Again, the initial curve is depicted with dashed line, whilethe �nal one with solid line. The same convention stands for curvature and torsion plots.In both examples, the �rst and the last control points remain unchanged. The �guresare in exactly the same order as in the previous chapters, namely �rst the orthogonal

52 Chapter 4. Fairing of 3D curves under shape constraintsprojections, second the curvature and torsion plots and, �nally, the distributions of thebinormal vector and the curvature vector (scaled).Example 1: For the chine data, the permissible deviation for the control points is setto ri = 0:050m; i = 2(1)14 (reference length: 10:000m), and the sign of torsion has tobe negative. The maximum deviation between the initial and the �nal curve is foundto be 0:029m, the maximum nodal deviation 0:020m and the sum of nodal deviations is� = 0:123m. The resulting curve is almost planar, with no in ections or latent in ections(Figs. 4.7, 4.8), though the changes in curvature plot are not impressive (Fig. 4.6). InFig. 4.5, one can hardly distinguish the initial curve from the �nal one.Example 2: The permissible deviation for the control points of the GM data is kept tothe industry standards, namely ri = 1:000mm; i = 2(1)21 (reference length: 700mm), andthe sign of torsion is set negative. The resulting curve exhibits maximum nodal deviation0:660mm and sum of nodal deviations � = 4:729mm. The quality of the resulting curvecan be checked through the torsion plot in Fig. 4.10, as well as with the distributions inFigs. 4.11, 4.12.4.7 Conclusions - Future workObserving the results of existing fairing methods, the one of Chapter 3 included, one canrealise that ill-conditioned areas on a curve, like latent in ections, are very persistent andchange with great di�culty. The proposed method is { in some sense { constructive, sinceit forms a locally acute control polygon, and guarantees the elimination of the problem.Though the non-linearity of the inequalities sometimes make the solving process uncertainand time-consuming, the solution is very satisfactory and appropriate for re�ned dataclose to planar, which, in most cases, exhibit acceptable curvature and local aberrationsof torsion.

4.7. Conclusions - Future work 53Linearization of the inequalities would be an obvious improvment for the method, sincethe problem would become a QP one, which guarantees the existence of solution. Also,this will enable the acceleration of the process and the determination of a good startingpoint for the existing algorithm.

54Chapter4.Fairingof3Dcurvesundershapeconstraints

0

1.6

0 1.60

1.6

0 10

0

1.6

0 10

Figure 4.5: Orthogonal projections of the initial (dashed) and �nal (solid) chine curve.

Figures of examples 55

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15

-120

-100

-80

-60

-40

-20

0

20

u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15

-15

-10

-5

0

5

u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15Figure 4.6: Curvature (upper) and torsion (lower) plots of the initial (dashed) and �nal(solid) chine curve.

56 Chapter 4. Fairing of 3D curves under shape constraintsS

010

0

6

-2

0

2

S

010

0

6

-2

0

2

Figure 4.7: Binormal vector distribution of the initial (upper) and �nal (lower) chinecurve.


10

0

6

-2

0

2

S

10

0

6

-2

0

2

Figure 4.8: Curvature vector distribution of the initial (upper) and �nal (lower) chinecurve.

58Chapter4.Fairingof3Dcurvesundershapeconstraints4650

5050

950 10504650

5050

0 700

950

1050

0 700

Figure 4.9: Orthogonal projections of the initial (dashed) and �nal (solid) GM curve.

Figures of examples 59

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


-1

-0.5

0

0.5

1

1.5

2


0

0.002

0.004

0.006

0.008

0.01

0.012


Figure 4.10: Curvature (upper) and torsion (lower) plots of the initial (dashed) and �nal(solid) GM curve.

60 Chapter 4. Fairing of 3D curves under shape constraintsS

4600

5100

0

700

650

1050

S

4600

5100

0

700

650

1050

Figure 4.11: Binormal vector distribution of the initial (upper) and �nal (lower) GMcurve.


4600

5100

0

700

650

1050

S

4600

5100

700

650

1050

Figure 4.12: Curvature vector distribution of the initial (upper) and �nal (lower) GMcurve.

Chapter 5Fairing of 2D curvesunder design constraints5.1 IntroductionThe subject of this chapter is how one can produce fair planar curves, which possess anumber of properties of practical interest. These properties may be of geometric nature {i.e., the convexity of the curve locally, the deviation of the curve from a given point set,the end conditions of the curve, but also general design constraints { i.e., the area undera curve and the area centroid as well as higher moments of it. This problem has beenpartially faced by [Nowacki et al. '90], [Nowacki & L�u '94], whereas here is set and solvedin a more general frame. The resulting algorithm can be e�ciently used in the context ofCAShipD: it o�ers the ability of determining ship lines when only a rough description ofgeometry of the hull is given and the integrated characteristics (hydrostatics) of the shipare available. The proposed method is presented through a detailed practical example.5.2 PrerequisitesSome aspects connected with properties of B�splines are examined, so that the con-straints of the fairing problem can be set in terms of the control polygon of the curve, not62

5.2. Prerequisites 63a�ecting the initial parametrization.5.2.1 End conditionsBased on eqs. (2.21), (2.22) and (2.23), it can be shown (see [Kapniaris '95]) that the �rst(second) derivative at both ends is controlled by only two (three) control points, providedthat the curve is open and the end knots are of multiplicity n + 1. More speci�cally, the�rst-order boundary derivatives are given by:_Q(un) = nun+1 � u1�d0; (5.1)and _Q(uM+1) = nuM+n � uM�dM�1: (5.2)Now, for the second-order bounaryderivatives, we have:�Q(un) = n(n� 1)un+1 � u2" 1un+2 � u2�d1 � 1un+1 � u1�d0#; (5.3)and�Q(uM+1) = n(n� 1)uM+n�2 � uM�1" 1uM+n�1 � uM�1�dM�1 � 1uM+n�2 � uM�2�dM�2#: (5.4)Finally, for closed curves, periodic conditions (d`Q=du`ju=u0= d`Q=du`ju=uM+n+1; ` =0(1)n� 1) can be achieved when:d = dM�n+i+1; �u = �uM+n�i�1; i = 0(1)n� 1; (5.5)with �ui = ui+1 � ui:5.2.2 Calculation of integrated measursIn case that the curve, Q(u) = (Qx(u); Qy(u))T , is a Jordan curve that bounds a pla-nar compact domain D with boundary @D, the area of that domain and its �rst- and

64 Chapter 5. Fairing of 2D curves under design constraintssecond-order moments, with respect to the axes of a Cartesian co-ordinate system, canbe eventually expressed through the control polygon of the curve with the aid of thetwo-dimensional Stokes formula:ZZD�@Q(x; y)@x � @P (x; y)@y � dx dy = � I@D�P (x; y) dx�Q(x; y) dy�; (5.6)where P (x; y)i+Q(x; y)j is a di�erentiable vector �eld and `H ' denotes clockwise contourintegration.The area A of D can be obtained from the above formula by setting Q(x; y) = 0 and@P=@y = �1, which gives:A = ZZD dx dy = I@D y dx = ZU Qy(u) _Qx(u) du = M+nXi=0 Z ui+1ui Qy(u) _Qx(u) du= M+nXi=0 iXj;k=i�ndyj dxk Z ui+1ui Nnj (u) _Nnk (u) du: (5.7)The �rst-order moments of D with respect to the Ox�/Oy�axis are obtained by settingQ(x; y) = 0 and @P=@y = �y /�x, respectively:Mx = ZZD ydxdy = 12 I@D y2 dx = 12 ZU(Qy(u))2 _Qx(u) du (5.8)= �12 M+nXi=0 Z ui+1ui (Qy(u))2 _Qx(u) du= 12 M+nXi=0 iXj;k;l=i�ndyj dyk dxl Z ui+1ui Nnj (u)Nnk (u) _Nnl (u) du; (5.9)and My = ZZD xdxdy = I@D x y dx = ZU Qx(u)Qy(u) _Qx(u) du= M+nXi=0 Z ui+1ui Qx(u)Qy(u) _Qx(u) du= M+nXi=0 iXj;k;l=i�ndxj dyk dxl Z ui+1ui Nnj (u)Nnk (u) _Nnl (u) du: (5.10)

5.2. Prerequisites 65The second-order moments, also called inertia, of D with respect to the Ox� /Oy�axisare (Q(x; y) = 0 and @P=@y = �y2 /�x2; respectively):Ix = ZZD y2 dx dy = 13 I@D y3 dx = 13 ZU(Qy(u))3 _Qx(u) du= 13 M+nXi=0 Z ui+1ui (Qy(u))3 _Qx(u) du= 13 M+nXi=0 iXj;k;l;r=i�ndyj dyk dyl dxr Z ui+1ui Nnj (u)Nnk (u)Nnl (u) _Nnr (u) du; (5.11)and Iy = ZZD x2 dx dy = I@D x2 y dx = ZU(Qx(u))2 Qy(u) _Qx(u) du= M+nXi=0 Z ui+1ui (Qx(u))2 Qy(u) _Qx(u) du= M+nXi=0 iXj;k;l;r=i�ndxj dxk dyl dxr Z ui+1ui Nnj (u)Nnk (u)Nnl (u) _Nnr (u) du: (5.12)Now, if the curve, Q(u), is open and represents a function with respect to the Ox�axis,then formulae (5.7)-(5.12) have to be modi�ed by simply restricting the range of indexi from i = 0(1)(M + n) to i = 1(1)M: Furthermore, if the limits of integration do notcoincide with knots of the given parametrization, U , then knot insertion can be employed(see, e.g., [Hoschek & Lasser '93], pp. 190-195), so that the curve remains unchanged andthe integration limits become knots. The integrals of the basis functions can be calculatedexactly with Gauss quadrature (see, e.g., [Hildebrand '56], ch. 8), which exhibits no lossof signi�cance for any order or knot sequence, and the evaluation cost per segment is quitesatisfactory [Vermeulen et al. '92]. More about calculations of area and moments can befound in [Kapniaris '95].

66 Chapter 5. Fairing of 2D curves under design constraints5.2.3 Local convexity of planar cubic B�splinesReferring to local convexity we mean the convexity of a segment of Q(u), namely theconstancy of sign for the signed curvature in that segment. From eq. (2.10) one cansee that the sign depends on the numerator of curvature, namely on L(u), which, for aB�spline of arbitrary degree, n, can be expressed as:L(u) = n2(n� 1) i�1Xj=i+1�n iXk=j+1�dj�1 ��dk�1Anj;k(u); (5.13)where Anj;k(u) = 1(uj+n � uj)(uk+n � uk) �h (uk � uj)Nn�2k (u)Nn�2j (u)(uk+n�1 � uk)(uj+n�1 � uj) + (uk+n � uj+n)Nn�2k+1 (u)Nn�2j+1 (u)(uk+n � uk+1)(uj+n � uj+1) +(uj+n � uk)Nn�2k (u)Nn�2j+1 (u)(uk+n�1 � uk)(uj+n � uj+1) � (uk+n � uj)Nn�2k+1 (u)Nn�2j (u)(uk+n � uk+1)(uj+n�1 � uj) i; (5.14)and j = i + 1� n(1)i� 1; k = j + 1(1)i:The function Anj;k(u) is positive if the parametrization is uniform, because in that casethe multiplier of the negative term in the right-hand side of (5.14)equals to the sum ofthe multipliers of the positive terms, while the products of the basis functions for thepositives terms are greater or equal to the product of the negative term.Proposition 5.1 The signed curvature of a planar B�spline curve, Q(u), of degree nand uniform parametrization, is of constant sign in [ui; ui+1]; if all cross products �dj�1��dk�1; j = i+ 1� n(1)i� 1; k = j + 1(1)i; are of the same direction.In the particular case of n = 3, L(u) is analysed according to (4.2) and (4.3). Baring inmind that �j(u) � 0; j = 1; 2; 3; for an arbitrary parametrization, it can be stated thatProposition 5.2 The signed curvature of a cubic B�spline curve, Q(u), is of constant

5.3. Fairing of B�spline curves under constraints 67sign in [ui; ui+1]; if �di�3 ��di�2; �di�2 ��di�1; and �di�3 ��di�1 are of the samedirection.So, if the sign is predetermined, then the unit vector �i = (�1; 0; 0)T can be introduced forde�ning the sense that the cross products should share in order that the signed curvatureis positive/negative in [ui; ui+1]:�i(�di�3 ��di�2) � 0; and �i(�di�2 ��di�1) � 0; and �i(�di�3 ��di�1) � 0: (5.15)Also, it can be shown that (see Theorem 2 in [Pigounakis & Kaklis '94]):Proposition 5.3 If jj�di�3jj = jj�di�2jj = jj�di�1jj, the knots uj; j = i � 2(1)i + 3;are equally spaced and the cross products �di�3��di�2; �di�2 ��di�1 are of the samedirection, then the signed curvature of the cubic B�spline curve, Q(u); is of constant signin [ui; ui+1]:5.3 Fairing of B�spline curves under constraints5.3.1 The problem (Pdesign)The problem of fairing with constraints is approached as an optimization problem andset as follows:Problem (Pdesign) : Let Q0(u) be a C2 cubic B�spline curve, and D0, U its controlpolygon and knot vector, respectively. Retaining U , �nd Q�(u), which minimizes thefunctional:J(Q) = w0J0(Q) + wr1Jr1(Q) + wr2Jr2(Q); w0 +wr1 +wr2 = 1; w0;wr1;wr2 � 0; (5.16)with J0(Q) = MXi=0 kdi � d0i k2; (5.17)

68 Chapter 5. Fairing of 2D curves under design constraintsand Jr(Q) = Z uM+1un kQ(r)(u)k2du = Z uM+1un Q(r)x 2(u) +Q(r)y 2(u)du; (5.18)under the following constraints:(i) Boundary constraints: For closed curves, conditions (5.5) should hold. For opencurves, the �rst- and/or second-order derivative at each end, as presented in (5.1)-(5.4), should be equal to speci�ed vectors.(ii) Area constraints: For closed curves, the area of the included domain is given by(5.7), whereas for open curves the range of index i in (5.7) should be restricted fromi = 0(1)(M + n) to i = 1(1)M: The area should not deviate from a speci�ed valuemore than a certain tolerance.(iii) Moments-of-area constraints: For closed curves, �rst- and second-order mo-ments are calculated using the formulae (5.9)-(5.12), whereas for open curves therange of i in (5.9)-(5.12) should be restricted from i = 0(1)(M + n) to i = 1(1)M:The moments should not deviate from speci�ed values more than a certain tolerance.(iv) Local-convexity constraints: For cubic splines, inequalities (5.15) can be set forevery segment that its curvature sign �i is predetermined. Also, for cubic splineswith almost equally spaced vertices and uniform parametrization, it is safe to takeadvantage of Corollary 5.3. For arbitrary degree, the parametrization should beuniform in order to bene�t of Proposition 5.1.(v) Tolerance constraints: The deviation of nodal points, Pij=i�n djNnj (ui), from thenodal points of the initial curve should be bounded:j iXj=i�n(dxj � dxj 0)Nnj (u)j � �xi; j iXj=i�n(dyj � dyj 0)Nnj (u)j � �yi; (5.19)where i = n(1)M + 1, and �xi; �yi are user-speci�ed tolerances.

5.3. Fairing of B�spline curves under constraints 69The functional J0(Q) in (5.16) measures the �delity of the curve Q(u) to the initialone, Q0(u), while the other two functionals, Jr(Q); r � 3, measure the smoothness ofQ(u): J2 and J3 express the (simpli�ed) bending energy and the (simpli�ed)jerk of thecurve, respectively, and are commonly for global fairing (see, e.g., [Eck & Hadenfeld '95a],[Nowacki & L�u '94], [Reinsch '67], [Reinsch '71]). Note that, in terms of the control poly-gon, D, the smoothness functionals are expressed as below:Jr(Q) = MXi=n Z ui+1ui iXj=i�n dxjNnj (r)(u) iXk=i�n dxkNnk (r)(u)+ iXj=i�ndyjNnj (r)(u) iXk=i�ndykNnk (r)(u)du;(5.20)and �nally:Jr(Q) = MXi=n iXj;k=i�ndTj dk Z ui+1ui Nnj (r)(u)Nnk (r)(u)du; dTj dk = dxjdxk + dyjdyk: (5.21)5.3.2 Solution of the problem (Pdesign) - An example(Pdesign) is an NLP problem with respect to the vertices of the control polygon, D, of thesought-for curve Q(u). More speci�cally, the objective function, J(Q), is quadratic withrespect to the control vertices, di; i = 0(1)M , while the family of constraints includeslinear (boundary, tolerance), bilinear (local convexity), quadratic (area), cubic (�rst-ordermoments of area) and quartic (second-order moments of area) inequalities.If the feasible space, de�ned by the constraints of the problem, is non-void, (Pdesign)is numerically solved by employing the SQP technique (see, e.g., [Gill et. al '81]) in its[NAg '90] implementation. In this implementation, the user should provide an initialpoint that is feasible with respect to the linear constraints of the problem. In our case, thestarting point is chosen to be the C2 cubic B�spline, that interpolates the initial nodalpoints under the prescribed boundary conditions, thus satisfying the linear (boundary,tolerance) constraints of (Pdesign).

70 Chapter 5. Fairing of 2D curves under design constraintsThe various integrals, appearing in the analytic expression of the objective function (seeeqs. (5.18), (5.21)), and the area and moments-of-area constraints (see eqs. (5.7)-(5.12)),retain constant values throughout the optimization procedure since the parametrizationU remains �xed and they are calculated accurately by means of the Gaussian quadrature.Finally, it is worth noticing that, when U is close to be uniform and the lengths of thelegs of the initial control polygon, D0, are almost equal, then it is rather safe to appeal toProposition 5.3 for simplifying (Pdesign) with respect to the local-convexity constraints.A number of examples are presented in [Kapniaris '95], but no inertia constraints are em-��

��

xO��

��

��

��Figure 5.1: Planar symmetrical section of a symmetric body. Axis of symmetry: Ox.ployed there. Here, the algorithm is tested for a curve which represents half the boundaryof a section of a symmetric body; see Fig. 5.1. The curve is a C2 cubic B�spline inter-polating eighteen (18) data points points obtained with digitizing, so the control polygonconsists of twenty (20) vertices. The parametrization is with respect to chord length ofthe data points, and the end conditions are _Q(u3) = (1; 0)T and _Q(u20) = (0;�1)T . Thetask for this curve is to retain its shape and the sense of the end conditions and to extendit to the x-direction from 82m to 92m, while the y-direction should not extend more than6m (half breadth). Furthermore, the area centroid should be located at about 52% of thex-length of the curve, namely at 47:84m from the starting point, and the area concludedby the new section should be A = 840m2. So, the moment of sectional area should equalto Myy = 40200m � m2. The inertia of the �nal section cannot be estimated so far. InFigs. 5.2, 5.3 at the top and with long dashes the control polygon and the curve is pre-

5.3. Fairing of B�spline curves under constraints 71sented, respectively. The curvature plot can be found in Fig. 5.4 also with long dashedline in two di�erent scalings. The characteristics of that initial curve is shown in Table5.1.Since the curve is aB�spline, an obvious solution for extending the x-direction would be totransform the control polygon of the curve in the x-direction. Due to B�spline properties,such a transformation results in a new curve, which retains the linear part around themiddle and the sense of the end conditions. The characteristics of the transformed curveare given in Table 5.1 and the control polygon, the curve and its curvature plot are givenin Figs. 5.2, 5.3 and 5.4, respectively, denoted with short dashed line.The transformed curve is the initial `vector' for the optimization algorithm. First, thedeviation from the transformed curve is not bounded, as we are interested only in thearea and the centre of area of the curve. The constraints for those are considered satis�edif the values exhibit a deviation of 1%. Then the algorithm gives Solution 1 (see Table5.1) shown in Figs. 5.2 (control polygon) and 5.3 (curve), third from top with thin solidline. The curvature plot is given also with thin solid line in Fig. 5.4. As one can observe,Solution 1 is fair enough and ful�ls the area and centroid constraints but its shape doesnot resemble to that of the transformed curve. The reason is that neither toleranceconstraints nor inertia constraints are active. In order that the shape of the transformedcurve is kept, tolerance constraints are set for the parallel part as well as for the areanear the second end. The tolerances of the nodal points for the parallel part are 0:050m(y-direction) and 5:000m (x-direction), and for the parts of the curve near the ends are0:100m (y-direction) and 2:500m (x-direction). Furthermore, the inertia constraints areactivated and set to Iyy = 1:7� 104 m�m�m2 and Ixx = 7:5� 105 m�m�m2 for thewhole sectional area, and they are considered satis�ed when the values deviate up to 2%.The values for the inertia constraints are set so, because the distribution of area shouldbe changed, i.e., area should be moved away of the axis of symmetry and gathered nearthe middle of the section along its length. The result of this run is Solution 2 (see Table

72 Chapter 5. Fairing of 2D curves under design constraints5.1 and Figs. 5.2, 5.3, 5.4).Both solutions are achieved within a limited number of iterations (between 10 and 20)and the real time neaded for optimization is about 30secs. The weight of the three termsin (5.16) are equal. Though (Pdesign) is not always solvable, an experienced user can setthe constraints in such a way that the desirable solution is found in relatively short time.Curve Initial Transformed Solution 1 Solution 2 DeviationArea A 711.9 800.8 844.2 838.2 0.2%Moment Myy 28567. 36175. 20009. 40128.2 >0.1%Centre Myy=A 40.13 45.18 47.40 47.88 0.2%Inertia-x Ixx 13830. 15558. 16801. 17088. 0.5%Inertia-y Iyy 479630. 682672. 774024. 743586. 0.9%P jjQ� �Q0jju=ui | | 6.499 15.382R jj�Qjj2du 2.068 2.356 1.873 2.169R jj ...Qjj2du 3.118 3.567 1.854 3.436Table 5.1: Characteristics of the boundary curves of the example section.5.4 Conclusions - Future workIn this chapter we presented a method of fairing planar curves under constraints which aresound for design and cover end, tolerance, area, moments of area and shape requirements.The innovation of the method lies mainly on the number of kinds of constraints andtheir nature (linear / non-linear). The performance of the developed algortihm is quitesatisfactory, so it can be used in a CAD system, or even become a kernel of a morecomplex method for handling curve meshes (e.g. ship lines). This makes the particulartool very useful, as one can also see in the next chapter.It would be interesting if the geometric conditions of the method would enriched withconditions of uid mechanics, so that the solution would be optimalfor the ow round the

5.4. Conclusions - Future work 73sought-for section. Also, the linearization of the problem would result to faster perfor-mance.

74Chapter5.Fairingof2Dcurvesunderdesignconstraints

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-5 95Figure 5.2: Control polygons: Interpolant of the initial data (long dashed), A�nely transformed curve (short dashed),Solution 1 (thin solid), Solution 2 (thick solid).

Figuresofexample75

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-5 95Figure 5.3: Curves: Interpolant of the initial data (long dashed), A�nely transformed curve (short dashed), Solution 1(thin solid), Solution 2 (thick solid).

76 Chapter 5. Fairing of 2D curves under design constraints

-5

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u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15 u16 u17 u18u19u20

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u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15 u16 u17 u18u19u20Figure 5.4: Curvature Plot: Interpolant of the initial data (long dashed), A�nely trans-formed curve (short dashed), Solution 1 (thin solid), Solution 2 (thick solid).

Chapter 6Applications inComputer Aided Ship Design6.1 IntroductionThe majority of designers still prefer the traditional way of hull description, i.e. through amesh of planar curves, to surface de�nitions, which are currently available in commercialsystems [Wake '93]. There are at least two reasons for that: (i) the inherent di�cultyin handling a large and complex surface, like the one representing the ship hull, and(ii) the methodologies of naval architecture, which make extensive use of hull sections[Comstock-PNA '67], [Rawson & Tupper '68]. Nevertheless, Computational Geometry[Nowacki et al. '95], [Su & Liu '89], has started showing o� methods that naval architectscan bene�t of in Ship Design.Taking advantage of the previously presented algorithms, this chapter proposes a process,which is a compromise of the old and the new trend in hull design. In order to describe thehull of a ship, sections of the hull along di�erent directions are developed, forming a densewireframe of fair planar lines { stations, waterlines, buttocks, which satisfy requirementsfor shape and hydrostatics. 77

78 Chapter 6. Applications in Computer Aided Ship Design6.2 Ship design and hull geometryShips are quite expensive and complex constructions, with special needs and tasks, whichservice mainly in an inhospitable environment. Thus, a new building is decided only if anactual or forecast situation shows the ship's necessity and makes the investment sound.For merchant ships, the basic need is for transporting a payload { whether it is passengersor cargo { under certain speci�ed conditions of speed, distance and environment. Thisinformation is coded in the term owner's requirements. The variety of payloads and serviceconditions leads to di�erent types of ships, e.g. passenger ships, ferry boats, oil tankers,bulk carriers, etc., resulting to di�erences in hull geometry.The naval architect applies special knowledge to meet the owner's requirements and ensurethat the ship is optimum, from the engineering point of view. Hydrodynamics, structuralanalysis, as well as economics form the framework of the design, generating several inter-depended studies and projects to be carried out by the design team. The endeavour isconveyed by the design spiral, each coil of which is of smaller uncertainty than the pre-vious one, and �nal design lies in the centre [Papanikolaou '89], [Rawson & Tupper '68].Starting from a feasibility study, called concept design, the spiral goes on to preliminarydesign, where all main characteristics are determined, employing standard methodologiesand techniques of naval architecture. The next stage is known as contract design, when thedesign team of the shipyard are bound for the successful materialization of their choices,and the spiral ends with the detailed design.As one can realize, preliminary design is vital for the success of the whole undertaking.According to [Rawson & Tupper '68], an accuracy of about 5 per cent is desirable forthis stage, where the naval architect concentrates in space allocation, main dimensions,dislacement and deadweight, form parameters, weather deck layout, and machinery typeand layout. After having estimated the principle dimensions and the displacement, theclassical approach demands the determination of the curve of areas for the below water

6.3. Development of ship lines 79portion of the hull. This may be based on parent ships, or on systematic series data,which also provide the required powering characteristics. The curve of areas, or sectionalarea curve (SAC), must then translated into a hull geometry, that is, into a `body plan'(sections of the hull along the ship's length), which is also based on parent ships orsystematic series (e.g. [Guldhammer-FORMDATA]). Finally, other sets of sections alongthe rest principle dimensions are calculated and drawn. That phase is known as linesdevelopment.6.3 Development of ship linesAs stated above, the hull of a ship is traditionally described through a set of curves,called ship lines. These curves are mainly planar and represent sections of the ship hullalong her length, her draught and her breadth and called stations, waterlines and buttocks,respectively. Sectional curves in the same direction are drawn together, namely stationscompose the body plan of the hull, waterlines compose the half-breadth plan, and buttockscompose the sheer plan. Along with these lines, curves that represent intersections of thehull with inclined planes are also typical for the lines plan. Such curves are the diagonallines, or the boundary curves of the at of side (FOS) and the at of bottom (FOB).Furthermore, a few spatial curves are also present, like deck or `knuckle' lines, whichdelimit regions of the ship. Details for the lines plan can be found in [Comstock-PNA '67].Baring in mind the capabilities of Computational Geometry, the problem of designing ahull is set as follows [Nowacki et al. '95]:Problem (Pmesh) : Develop a set of curves, most of them planar curves, such that themesh of ship lines is formed by which a ship surface geometry of some desirable shape isde�ned.The characteristics of the 'desired shape' are related not only to the geometric shape,

80 Chapter 6. Applications in Computer Aided Ship Designbut mainly to form-parameters of the individual curves and some global properties of thehull. These are the hydrostatic characteristics of the hull, also called hydrostatics, whichare to be determined before the development of lines, since they are closely related tothe performance of the ship and derive almost explicitly from the owner's requirements.Among hydrostatics, the Bonjean curves are of vast importance, because they expressthe volume distribution of the hull. In the �rst step of lines development, the resultingstations should satisfy at least the SAC, which expresses the intersection of the Bonjeancurves with the still water plane. Starting from the body plan, one can obtain informationand form the half-breadth plan and the sheer plan.In order to obtain the desirable hull description, the designer can either start an ab initiodesign, or distort an existing successful one. The �rst choice used to be not so favourite,since testing a totally new hull is a rather di�cult task. Nonetheless, modern CAShipDsystems o�er such an opportunity for quick and relatively accurate calculations of theperformance of a new design, though hydrostatics are checked a posteriori and cannot beevaluated before or during the design process. The second choice seems more secure andis still followed by many designers, though experience has shown that small and mediumsized CAShipD systems handle with di�culty distortion problems, as they are orientedto the previous approach, and rely on the capabilities of the user. The use of systematicseries can be also classi�ed in the latter case, since non-dimensional descriptions of amodel are adapted to the needs of the current designer. Systematic series provide goodestimates for the shape, the hydrostatics and the resistance of the ship hull.In the proposed process, the designer can start either ab initio, or based on existingdata. First, the main dimensions and hydrostatics are decided, and then any of the threeset of lines of the hull is faired, subjected to the afore mentioned characteristics, whichare de�ned in the following subsections. A more complete approach would employ acontinuous description of the Bonjean curves, namely a Bonjean surface. Such a surfaceincludes all information needed for the development of a hull, so, along with shape, it

6.3. Development of ship lines 81would consist the constraint for the hull surface, and would o�er the ability of solving theproblem (Pmesh) in a one-step process.6.3.1 Main characteristicsThe basic geometric characterists of a ship are her main dimensions, namely� length LOA=LW : overall or measured on the design waterline (DW),� draught T : vertical distance between the DW and the basic reference plane that theship is laid on, measured at the middle of her length (midship section),� breadth B=BW : maximum breadth of the midship section / maximum breadth ofthe DW, respectively,� depth D: vertical distance between the basic reference plane and the main deck,measured at the midship section.From now on, reference to the main dimensions implies dimensions measured on or fromthe design waterline.The hydrostatics that are taken into consideration for the proposed process are:� Displacement, r, block coe�cient, CB : The displacement is the volume of the underwater portion of the hull and CB = r=(LW �BW � T ).� Maximum sectional area, AX , maximum section coe�cient, CX : The maximumarea of station up to DW, and CX = AX=(BW � T ).� Sectional area curve (SAC) : The area of station up to DW vs. the length of theship.

82 Chapter 6. Applications in Computer Aided Ship Design� Sectional leverarm curve (SLC) : The vertical moment leverarm of transverse sectionup to DW vs. the length.� Longitudinal centre of buoyancy (LCB) : The longitudinal position of the centroidof the displacement.� Area of the design waterplane, AW , waterplane coe�cient, CW = AW=(B � LW ).� Longitudinal centre of otation (LCF) : The longitudinal position of the centroid ofAW .� Inertia moments of AW , Ixx; Iyy.6.3.2 Process for solving the problem (Pmesh)The procedure proposed here for solving (Pmesh) is mainly based on the algorithm pre-sented in Chapter 5, henceforth called the core algorithm, and follows a number of steps:1. Determine or estimate the following characteristics: LW , BW , T , D, r, SAC, LCB,SLC, AX , AW , LCF.2. Determine the necessary control lines, i.e. FOS, FOB, knuckle lines, deck line, pro�leline etc., and fair them.3. Choose positions for developing a set of sections. Estimate end conditions and shapeand provide/obtain an initial point set for each section with the aid of the controlcurves of Step 2. Interpolate all data sets and fair the resulting curves with the corealgorithm, so that all constraints, derived by the hydrostatics or the shape and theends of each curve, are satis�ed within a tolerance.4. Choose positions for developing a set of sections, in a vertical direction to the one ofthe already developed one(s). For each position, obtain an initial point set from the

6.4. A CAShipD example 83already developed set(s) and interpolate appropriately. Using the core algorithm,develop a fair set of sections while retaining within a tolerance all the requestedcharacteristics.5. Check the nodal points of the so far derived mesh. If the deviation between the setsof curves is less than a tolerance, stop the process, else return to Step 4.As it is mentioned, from the very start of a design the characteristics of Step 1 canbe evaluated and remain more or less unchanged throughout the whole process. Theestimates and choices of Step 2 remain unchanged, too. In Step 3 an initial geometry,compatible to the curves of Step 2, should be provided by the user or by a data base andimported into the system. Along with geometry, the available hydrostatics should be setas constraints. If no hydrostatics or distribution curves are available, the ones of the inputgeometry can be used. Step 4 is repeated for any set of sections until all sets coincide atthe nodal points of the mesh (within a tolerance). This checking is done in Step 5.Though there is no evidence for convergence, this process has been used for the develop-ment of lines of a few hulls, which have presented good quality of the resulting meshesand satisfactory hydrostatics (deviation from the required ones less that 2 per cent).6.4 A CAShipD exampleThe development of a lines plan with the aid of a procedure, similar to the one proposedhere, has been successfully applied in [Kapniaris '95]. The derived lines plan is of highquality and has been developed in two loops of the process. In forth, another exampleis presented, where the hull of an existing ship is available in electronic form (digitized),as well as her main characteristics; see Table 6.1. The task is to develop a new hull,with greater displacement of about 20 � 25%, say 530m3 and retain the breadth of theship. For this purpose, Chapter 66{'Steps in the Preliminary Design' of [Saunders '57] is

84 Chapter 6. Applications in Computer Aided Ship Designfollowed. From Fig. 66E, p.470 (ibid.), the length of the hull is estimated to be about 92m,which means an enlongation of 10m. According to Fig. 66A (ibid), the displacement-length quotient implies that the Froude number should be about Fn = 0:301 and theprismatic coe�cient should be near to CPX = 0:60. The midship section coe�cientis determined from the existing initial hull and set to CX = 0:894. Then, the blockcoe�cient is about CB = 0:54 and the draught (DW) is calculated at T = 4m3. Also,according to [Saunders '57], LCB should be located at 48:5�50:5% of LW , measured fromstern end (Fig. 66.N), while the midship section should be located at 46 � 50% of LW(Fig. 66L). The centre of otation is normally between the midship section position andthe position of LCB. The higher the Froude number, the more abaft all three positionsshould lie. Additionally, an estimation can be made for the waterline coe�cient and theinertia moments, but these magnitudes can vary more than others. The depth of the newhull remains unchanged, since the change of draught is rather small, and the remainingfreeboard is judged enough, according to holding regulations.The desirable characteristics of the �nal hull are also given in Table 6.1, and mean valuesare chosen in the cases of magnitudes within a range.Analysis of the design processIn this example only the body plan and the half-breadth plan are involved in the designprocess. This choice is based on the observation that, usually, no special requirementsfor areas or moment of areas are set for buttocks. Additionally, the shape cannot bepredetermined, at least not so much as the shape of a waterline or a station. Therefore,buttocks are not adequate for evaluating the process. On the other hand, an examplethat makes use of the sheer plan can be found in [Kapniaris '95].First, the body plan of the initial ship is digitized and the derived points are interpolated1Due to resistance reasons, the area near this speci�c Froude number is generally avoided in designs.Nonetheless, the derived hull is geometrically acceptable, and the ship can �nally service with a speed of20 knots, i.e., Fn = 0:34, out of the undesirable area.

6.4. A CAShipD example 85Characteristics Initial ship hull New ship hullLW 82:00m 92:00mB 14:65m 14:65mT 3:85m 4:00mD 5:60m 5:60mr 2368m3 2900m3CB 0:516 0:54LCB 42:43m 45:0mAX 50:40m2 52:75m2CX 0:894 0:894CPX 0:578 0:60AW 881:2m2 1025m2CW 0:734 0:76LCF 40:14m 43:5mIxx 11485m4 14000m4Iyy 328900m4 500000m4Table 6.1: Initial and required characteristics of the hull.Characteristics Hull0.1 Hull0.2 Hull1.1 Hull1.2 Hull1.3 DeviationLW [m] 81:71 81:71 92:07 92:07 92:07 > 0.1%B [m] 14:57 14:54 14:62 14:60 14:64 > 0.1%T [m] 3:85 3:85 4:03 4:03 4:03 0.8%D [m] 5:60 5:59 5:58 5:58 5:58 0.4%r [m3] 2368 2378 2904 2910 2909 0.3%CB 0:517 0:520 0:535 0:537 0:536 0.7%LCB [m] 42:14 41:04 45:01 45:11 45:08 0.2%AX [m2] 50:60 50:04 52:70 52:71 52:78 > 0.1%CX 0:902 0:894 0:894 0:896 0:895 0.1%CPX 0:573 0:582 0:598 0:598 0:599 0.2%AW [m2] 863:2 864:1 1030:8 1029:7 1030:4 0.5%CW 0:725 0:727 0:766 0:766 0:764 0.5%LCF [m] 40:24 40:01 43:53 43:49 43:58 0.2%Ixx [m4] 11139 11129 13932 13750 13892 0.8%Iyy [m4] 318073 319333 515177 520491 516572 3.3%Table 6.2: The characteristics of hulls calculated through the development process.(St0.0) and faired (St0.1) with the core algorithm. The deck line has been also fairedby employing the algorithm of Chapter 2 (Fig. 6.1). For St0.0, the curves SAC (SAC-

86 Chapter 6. Applications in Computer Aided Ship DesignI) and SLC (SLC-I) are calculated and retained throughout the fairing process, whichresults St0.1. From St0.1, the derived waterline points are interpolated with appropriateend conditions (Wl0.1). St0.1 and Wl0.1 consist the mesh Hull0.1, the characteristics ofwhich are given in Table 6.2.Wl0.1 is faired and results Wl0.2. In Figures 6.3, 6.4 the comparison of Wl0.1 and Wl0.2is given, where one can notice how the core algorithm changed the undesirable areasaftwords and forewards. Next, Wl0.2 is used for deriving sections again (St0.3), whichpossess slightly di�erent distributions, SAC-II, SLC-II (see Fig. 6.5). From St0.3, SAC-IIand SLC-II, a denser set of faired stations is derived and, since the tolerance constraintsare rather strict (2:5cm globally), one can consider that the waterline characteristics areretained, and the sets St0.4 (Fig. 6.2), and Wl0.2 form Hull0.2, the characteristics ofwhich are also given in Table 6.2.Hull0.2 is considered the parent hull mesh, the stations of which are used for the generationof the �rst body plan, St1.1, of the new hull (Fig. 6.7). St1.1 enables the calculation ofa set of waterlines, Wl1.1 (Fig. 6.10), and the two sets form Hull1.1, which exhibits thedistribution curves, SAC-1, SLC-1. The characteristics of Hull1.1, are shown in Table 6.2.Fairing Wl1.1, we get Wl1.2 (see Fig. 6.10), and from that new set the distribution of thearea of waterlines, as well as the longitudinal centroid and the inertia moments around thetwo basic axes, Ox and Oy, are calculated and faired. The resulting distributions, G, aregiven in Figure 6.9. These magnitudes are going to be active constraints for waterlines.FromWl1.2 a new set of stations is derived, St1.2, which, �nally results to St1.3, satisfyingSAC-1, SLC-1 (Hull1.2- Table 6.2).Next, Wl1.3 results from St1.3, and it is faired to Wl1.4, satisfying G and tolerance of1cm between the corresponding nodal points. Wl1.4 is used for St1.4, which, after fairingwith tolerance 1cm, results St1.5. The deviation between St1.4 and St1.5 is less than 1cmfor all nodal points, so St1.5 and Wl1.4 form the �nal mesh Hull1.3, (see Figs. 6.8, 6.11),

6.5. Conclusions - Future work 87which satis�es G, and its contributions, SAC-F and SLC-F, are almost identical to SAC-1and SLC-1, respectively (see Fig. 6.6). The characteristics of the �nal hull are given alsoin Table 6.2.6.5 Conclusions - Future workThe proposed process enables the generation of curves from any existing data, thoughin few cases the core algorithm cannot reach an optimum, due to stringent conditions.Another drawback is the interpolation phase, which depends on the end conditions, nowgiven as an input. For these two reasons, the user must be fully aware of the CAGDprinciples and techniques. Nevertheless, comparing the results of this process to thoseof other proposed methods { e.g., [Standerski '88], one ascertains that the quality of thedeveloped lines is higher and the requirements for integrated quantities are met verysatisfactorily.The process can be automated and employed either for modi�cation of an existing hull (theexample above), or for ab initio design, based, e.g., on some systematic series (examplein [Kapniaris '95]). An open data base, where meshes of existing ships and/or systematicseries forms would be stored, could enable the fast and accurate lines development, atleast for the needs of the preliminary design.

88 Chapter 6. Applications in Computer Aided Ship Design7

1

2

5 62 345 23

6

5

4

3

1 718 7

1

2

84

6

7

5

4

3

38

812

80

80

BLBL

CL

-276

70

5258

64

44

6

78

2

3238

2824

2016

Figure 6.1: Initial Hull: Body plan St0.1 of Hull0.1.

40

81

82

-2

52

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6764

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L

76

7880

0

58

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79

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1

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83 42 7651L

6

5

4

3

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1

2

7

34 2 18 7 6 5Figure 6.2: Initial Hull: Body plan St0.4 of Hull0.2.

Figuresofexample89

64

70

73

58

38

44

52

78

C

8

8

L

80

76

-2 520 8 32

28

24

20

16

12

Figure 6.3: Sets of waterlines of the initial hull: Wl0.1 (upper) and Wl0.2 (lower).

C

7064

5.5675.0003.8503.000

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5.000

8

880787673

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0.500

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4.250

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3.000

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0.500

0.500

1.500

1.000

3.850

0.750

1.500

1.000

3.000

2.5002.000

Figure 6.4: Aft (left) and fore (right) of Wl0.1 (upper) and Wl0.2 (lower).

90Chapter6.ApplicationsinComputerAidedShipDesign

Leverarm

Sectional Area

00-10

60

50

10

20

30

40

10 9080706020 30 40 50Figure 6.5: Initial (dashed-I) and modi�ed (solid-II) distributions, SAC and SLC, of the initial hull.

60 65 757045 555040

Leverarm

Sectional Area

95908580

40

-5

60

50

30

20

10

00 3530252015105Figure 6.6: Comparison between SAC-1/SLC-1 (dashed) and SAC-F/SLC-F (solid).

Figures of example 91

543

90

90

88

85

38 6 5 4 2112

0

3228

5

7 876

2

82

49.558.5

62.25

45

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1 1

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5

6

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2

3

4

5

6

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B B

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LCLFigure 6.7: Initial Hull: Body plan St1.1 of Hull1.1.

5860

62

56 53 50

68

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72

70

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1822

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2

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7

6

5

4

Figure 6.8: Final Hull: Body plan St1.5 of Hull1.3.

92 Chapter 6. Applications in Computer Aided Ship Design

500200 250150 350 400 450 650

4241.54140.540

600550

I

/2

1

2

LB

3

4

LCF

300

T [m]

yy

xx

wlA

I

4816 24 32 40 56 64 728I [x10 m ]

I [x10 m ]

yy

xx

0

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A /2 [m]wl

80

2.50

4544.542.5 43 43.5 44

5 2522.52017.57.5 10 12.5 152 4

4 4

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5

Figure 6.9: Distributions of Aw, LCF , Ixx and Iyy with respect to the draught of theship (G-distributions).

Figuresofexample93

1.5

4.55.55.0

3.0 1.0 0.51.0

2.00.3

8

95-5

8

4.033.53.02.5

Deck line

CL

5.05.54.5

3.54.03

0.51.52.02.5

Figure 6.10: Sets of waterlines: Wl1.1 (upper) and Wl1.2 (lower).

8

8

-5 95

Deck line

4.53.53.0 1.0

Flat bottom line0.5

5.5

0.51.52.02.54.0

35.0

1.02.5

CL

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Figure 6.11: Waterlines: Wl1.4 (upper and lower).

Bibliography[Anderl & Mendgen '96] Anderl, R. and Mendgen, R., \Modelling with constraints: the-oritical foundation and application", Computer -Aided Design, Vol. 28, No. 3, pp.155-168, 1996.[AUTOSHIP '96] AUTOSHIP, User's Guide, Release 6, 1996.[Birkho� '33] Birkho�, G., Aesthetic Measure, Harvard University Press, 1933.[Birmingham & Smith '95] Birmingham, R. and Smith, T., \A practical approach to formtransformation for software generated designs", in the Proceedings of InternationalConference CADAP '95: Computer Aided Design & Production for Small Crafts,Pa. 13, The Royal Institution of Naval Architects, Southampton, 26-27 September,1995.[Bedi & Vickers '89] Bedi, S. and Vickers, G.W., \Surface lofting and smoothing withskeletal lines", Computer Aided Geometric Design, Vol. 6, No. 2, pp. 87-96, 1989.[Bercovirer & Jacobi '94] Bercovirer, M. and Jacobi, A. \Minimization, constraints andcomposite B�ezier curves", Computer Aided Geometric Design, Vol. 11, pp. 533-563,1994.[B�ezier '90] B�ezier, P., \Style, mathematics and NC", Computer -Aided Design, Vol. 22,No. 9, pp. 524-537, 1990.[Boehm '80] Boehm, W., \Inserting new knots into B-spline curves", Computer -AidedDesign, Vol. 12, pp. 199-201, 1980.[Boehm '88] Boehm, W., \On the de�nition of geometric continuity", Computer -AidedDesign, Vol. 20, No. 4, pp. 370-372, 1988.[Brand '84] Brand, L., Advanced Calculus (greek edition), Hellenic Society of Mathemat-ics (E.M.E), 1984.[Burchard et al. '94] Burchard, H.G., Ayers, J.A., Frey, W.H., Sapidis, N., \Approxi-mation with aesthetic constraints", GM Research Publication GMR-7814, also in:N. Sapidis (ed.) Designing Fair Curves and Surfaces: Shape Quality in GeometricModelling and CAD. pp. 3-28, SIAM: Geometric Design Publications, Philadelphia,1994. 94

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Documents

Fairing Methods of Planar and Space Curves Under Design Constraints - Applications in Computer-Aided Ship Design