Surface Fitting Three-Dimensional Bodies

8/13/2019 Surface Fitting Three-Dimensional Bodies

1/28

SURFACEFITTINGTHREE-DIMENSIONAL BODIESFredR.DeJarnetteand C.Phillip FordIII

North Carolina State University

SUMMARY

Thegeometryofgeneral three-dimensional bodiesisgenerated from coordi-natesofpointsinseveral cross sections. Since these pointsmay not besmooth, theyare dividedinto segmentsandgeneralconicsectionsarecurvefitinaleast-squares sensetoeach segmentof across section. Theconic section*Iarethen blendedin thelongitudinal direction through longitudinal curvesvhichmay beusedtodefinetheconicsectionsin thecross-sectionalplanes.Boththecross-sectionalandlongitudinal curvesmay bemodifiedbyspecifyingparticular segmentsasstraight linesorspecifying slopesatselected points.Slopesmay becontinuousordiscontinuousandfiniteorinfinite.

This methodwasusedtosurfacefit a 70slab delta wingand the HL-10Lifting Body. Theresultsfor thedelta wing were very closeto theexactge-ometry. Although thereis noexact solutionfor thelifting body,thesurfacefitgeneratedasmooth surface with cross-sectional planes very closetopre-scribedcoordinate points.

INTRODUCTION

Many disciplines requireamathematical descriptionofthree-dimensionalsurfaces which cannotberepresentedbysimple mathematical expressions. Thelocationandslopeofpointson abodyareneededin theanalysisofstructuresandinviscid flow fields. Viscous flow-fieldanalysesrequirethebodycurva-tureinadditiontolocation andslopes. Thegeometrical propertiesofsimpleshapes like spheres, cones, ellipsoids,andparaboloidscan bedescribedby re-latively simple mathematical equations. However, many configurationsofinter-esttoday, suchas thespace shuttle,arecomplex three-dimensional shapeswhose geometry cannotbedescribed veryeasily. Inmany casesall theinforma-tion thatisknownis adrawing withaplan view,aside view,andseveralcrosssectionsof thebody. Sometimes modelsof avehicleareavailableandcoordinate positionscan beaccurately measuredonthem. On theotherhand,


2/28

Eachofthese methodshasundesirable featuresin theforms used previously.Flatsurfaces are completely unacceptable if radii of curvature or continuousslopes are needed. Cubic or higher order polynomials often lead to unwantedwigglesandbulges since theyallowpointsofinflection. Inref.1,DeJarnette applied the method of double splines, which used bi-cubic interpola-tion, to surface fit the coordinates of points in several cross-sectionalplanesof three-dimensional bodies. However, it was found that bulges and/ordimples occurred in these surfaces, particularly when the thickness was muchsmaller thanthespan.

Reference2approximatedtheshapeof aspace shuttle with ellipticalcrosssections with different ellipticityon thewindwardandleeward sides.Cubicpolynomialswere usedtodefine segmentsof theplanandthickness dis-tributions. Thecoefficientsofthese polynomials were chosentomaketheslopes continuous across boundaries of the longitudinal segments. However,pointsof inflections were found to occur inside the segments and thus gave un-desirable bumps and wiggles, which in turn significantly affected the surfacepressureandheating distributions calculated with this geometry.

Forsometime conic sections have been usedtodescribe segmentsofair-craft contours (ref.3). Both longitudinalandtransverse contours were repre-sentedbyconic sections,but theslopesat theendsofeach segmenthad to bemeasured. Asmentioned earlier, slopesaredifficulttomeasure accurately.Onthe otherhand,conic sections cannot have inflection points, andthisfea-tureisusefulineliminating undesirable wiggles.

Coons (ref.4)developedatechniquetodescribe three-dimensional sur-faces by using blending functions to blend the surface between the boundarycurves of each patch. A major difficulty of applying Coons' method is that theuser must supply the coordinates, slopes, and twists (cross derivatives) at allfour corners of eachpatch. This information is generally difficult to deter-mine,particularlythecross derivatives (See ref.5). Craidon (ref.6)usedthis method with the twists arbitrarily chosen to be zero to surface fit three-dimensional geometries.

Themethod presentedhereuses data pointsincross-sectional planestocurve fit segments of generalconicsections in a least-squares sense. Theconicsectionsarethen blendedin thelongitudinaldirectionby use ofcoordi-nate points which definetheconicsectionsin thecross-sectional planes.This technique has the advantage of allowing the user to continually modify thecross-sectional curves and the longitudinal curves until the body shape has thedesiredfeatures. Also,discontinuous slopes in both the circumferential and


3/28

A, B , C coefficientsofconic section givenby Eq. (l) inlocalcoordinatesJ J JV V Jd constant vectorin Eq. (19)

G matrix usedin Eq. (19)

K data point numberoffirst control pointinsegmentJ

m, n slopes definedbyEqs.(5) and (6)J JN number of segments in a cross sectionP , Q terms definedbyEqs. (A2)and(A3)J J

r , residualinEqs. (Bl)and (B2)

R term defined by Eq. (A5)*J

x,y, z Cartesian coordinates,seefigure2y, z local coordinates,seefigure3a, 3., Y, coefficients definedbyEqs. (15), (l6),and (17)J J J6 slope of segment j, see figure 5JA9, 6. - 6,J J J-lSubscripts:h intermediate pointin across-sectionalsegmentj segment numberin across sectionk data point numberin across section


4/28

surfacemust thenbe fitthroughall thecross sections. In themethod pre-sented here,thedata pointsineach cross sectionaredividedinto segmentsand general conic sections are fit to the data points in each segment as showninfigure 1. A three-dimensional surface is then generated by blending thecross-sectional curves in the longitudinal direction. Consider first the curve-fitting of data points in a cross-sectional plane.

Curve-Fit in a Cross-Sectional PlaneThe data points in a cross-sectional plane are generally not completely

smooth,and in these cases a smooth curve cannot be made to pass through everydata point. Therefore, the data points are divided into segments, and a gener-al conic sectionis fit in aleast-squares sense throughthedata pointsinthat segment. The conicsectionisconstrainedto gothroughthecontrolpoints (end points of a segment) and also have a continuous slope at each con-trol point unless otherwise specified (See figure l).

A three-dimensional coordinate system x, y, z is used with x in the longi-tudinaldirectionand x =constantis across-sectional plane (See figure2).Let j denote the segment number in a cross-sectional plane asshownin figure 1,with the first segment starting on the positive y - axis and J increasing clock-wise. It is convenient to use alocalcoordinate system y,z for each segmentwith the origin on the first control point and the positive y - axis passingthrough the control point at the other end of the segment (See figure 3). Inthislocalcoordinate systemit iseasytoinvestigate possibilitiesofcomplexroots and interpret the coefficients of a generalconicsection geometrically,whereas it isdifficult tointerpret them geometricallyin theglobal coordi-nates y,z. It is assumed that the local coordinate z is single-valued in eachsegment.

The equation for a general conicsectionfor the j segment is given byref.7 as

A.y2 + B yz + C z2+ D y + E z + F = 0 . (l)J J J J J JOnlyfive of the six coefficients are independent since the equation may bedividedby any non-zero coefficient. The constraints that the conic sectionspass through the two control points

y = 0, z = 0 and y = y , z = 0


5/28

Ay - 2A y - B zdy' By +2C.z+ EJ J J

The slopes at the control points are generally notknown,so define

mj= ay] at y =' = (5)and

Byusing thesetwoequationsin Eq. (k) itfollows that

and

(8)

These equations show that the end slopes do not affect the coefficient Cj. Onthe other hand, the productAjC 0the conicishyperbola. Ofparticular interestis thepossibility ofcomplexroots when solvingfor z as afunctionof y in theregionofinterest. It isshowninAppendixAthatzwillnot becomplexin theregion0 < y < y ifJ

For prescribed slopes m,, n, figure U shows how the productA^C^affects aconicsection.Unless a slope is specified at a control point, the method used here con-

strainstheslopeinglobal coordinates (y,z)to becontinuousat acontrolpoint. This makes theconicsection in one segment dependent on data pointsinother segments aswellas its own. From figure 5 it can be seen that con-tinuityofslopeatcontrol pointjrequires


6/28

Because this last equationisnon-linear in thecoefficientsA 1for J = 1 (15)

= 0 and the -sign mustbeusedif(A/m)j< 0.

OncethecoefficientsA< and C* aredetermined,all the conicsectionsforthat cross sectionarecompletely defined. Inorderto putthese results intoa form suitable for blending the cross sections in the longitudinal direction,theconicsectionsforeach segmentareredefinedintermsof hpoints: thetwo control points at the end of the segment, a slope point which determinestheslope at the end points, andfinallyan intermediate point on the curve be-tweenthe endpoints(Seefigure6). The 3pointson thecurveand the twoslopes at the end points of a segment are sufficient to determine new coeffi-cients AI, A2, Aj,Aij,and A^ for the general conic section*,

A^2 + A2yz +A3z2+ A^y + A z + 1 = 0 (l8)

inglobal coordinatesy,z. This process is applied to each segment in a crosssection, and therefore the 5 coefficients become functions of the longitudinalcoordinate x when the segments of a cross section are blended with correspond-ingsegments in the other cross sections.

Longitudinal Variation of Cross Sections


8/28

inthe x, y and x, zplanes; hence,twoplanar curvesareusedtorepresenteach three-dimensional curve. Theparameteric methodofcubic splines (ref.9)isusedtocurve-fiteach planar curve, viththechordal distance betweenthecoordinate pointsas theparameter. Theparametric splineallowsinfiniteslopeswhereastheregularspline does not.

This methodworkedverywellforsome test casesbut hadtrouble withothers becauseof thebumpsandwiggles inherenttocubic splines. These bumpsand wiggles can be corrected by specifying slopes in the longitudinal directionatcertain cross sections. Also parts of thelongitudinalcurves can be speci-fiedas straight lines.

Considernow thelongitudinal variationof aconic section. Ineachcross-sectionalplane,Eq.(18)willholdbut thecoefficientsA^, Ag, Ao, A^, and Acwillvary withx. Asmentioned previously, these coefficientsaredeterminedby k defining points (the two control points, an intermediate point, and theslope point). For each segment, the 5 equations used to determine the coeffi-cients Ag(q= 1,,5) areformedbyapplyingEq.(l8)to the 3pointsonthecross-sectional curve (thetwocontrol pointsyo, ZQand yls z^, theinter-mediate point y^, z^) and the slopes at the ends of the segment using the slopepointys, zg(See figure6). This procedure yieldsthefollowing5equations:

Atanylongitudinal position,Eq.(19)can besolvedby anystandard matrixin-version routinetodeterminethecoefficientsAQ. Thederivatives dAQ/dxandd^A /dx can beobtainedbydifferentiatingEq.(19)andsuccessively solvingtheresulting system of linear equations. The elements of Gpqand their deri-vativeswith respectto x areobtained fromthethree-dimensional curves whichwere spline-fit throughthe hpoints usedtodefinetheconic sectionforthatsegmentineach cross-sectional plane.

COMPUTATIONAL ALGORITHM

Given the set of data points (yvzjr) ncross-sectional planes at severallongitudinal stations,

(1) Foreach cross-sectional plane,dividethedata points into seg-


9/28

specifiedfor a segment by prescribing the slopes at the ends ofthe segment and using only one datum point between the end con-trol points.(3) Representtheconicsectionforeach segmentin a cross-sectionalplane in terms of the tvo control points at the ends of the seg-ment,anintermediate point,and theslope point (See figure6).(U) Foreach point foundinstep (3), spline-fita three-dimensionallongitudinal curve throughit and thecorresponding pointsinother cross-sectional planes (See figure7).(5) If thelongitudinal curvesare notsatisfactory, modify thembyoneormoreof thefollowing methods: (a)specifyslope(s)atlongitudinal station(s) (slopes may be finite or infinite andcontinuousordiscontinuous), (b)specify selected longitudinalsegmentsasstraight lines,(c)redefinetheboundaries(controlpoints) of the segments in the cross-sectional planes so thatthepoints used for the longitudinal spline-fit form a smoothcurve.(6) The geometrical properties of the surface may be computed inpolar or Cartesian coordinates at any position x, y by the fol-lowing steps:

a. Locatexbetweentwoconsecutive longitudinal stations,andthen locate the cross-sectional segment which con-tainsy.

b. Use the spline function to calculate the coordinates,slopes, and second derivatives of the h longitudinalcurvesforthis segmentat x(See figure 7).c. Calculate the coefficients A~(q =1, , 5) of theconicsectionat this location by use of Eq. (19).Determinethefirstandsecond derivativesof Aq withrespectto xfromthefirstandsecond derivativesof

Eq. (19).d. Calculatethebody positionzfromEq.(l8), and thederivativesof zwith respectto x and yfrom deriva-tivesof Eq. (18).


10/28

APPLICATIONTO 70DELTA WING

The surface fitting method was applied to the 70 slab delta wing shown infigure8. This example was chosen because it illustrates many of the optionsavailabletomodifythelongitudinalcurvesand because theresultscan becom-pared with an exact solution.

Cross-sectional data was used as shown in figure 8, and due to symmetryonlythefirst quandrant isused. Twosegments (three control points)areneeded to represent the cross section of x = 10. The first segment is astraight lineand thesecondisone-fourthof anellipse. Theleast-squarescurve-fit technique represents the ellipse exactly by specifying a zero slopeatcontrol point j = 2, an infinite slope at control point j = 3, and one datumpoint between thesetwocontrol points. Althoughtwosegments must alsobeusedfor theothertwocross sections, onlyonesegmentisnecessarytospecifythe circleat x =0.65798and theellipseat x =1.0. Therefore,thefirsttwodata points, which are also control points, are made coincident. Then the ex-actcurves are calculated from the least-squares curve-fit by specifying a zeroslope at control point j = 2, an infinite slope at control point j = 3, and onedatum point between these two control points. The exact location of this datumpointisirrelevant except thatitmustlie on thedesired curve.

Thethree-dimensional longitudinal curveisrepresentedby itsprojectionsinthe x,y and x,zplanes. Therefore,16longitudinal planar curvesareusedforthisexample(8 foreachcross-sectional segment). Modificationsweremadeto theinitial spline-fitsto lU ofthese curvessinceexact slopesarereadilyobtained from figure8.

The geometrical properties and their derivatives were calculated at fourcircumferential positions for x = 0.3, 1.0, 2.0, and 5. The results atx =1.0, 2.0,and 5.0 areexact (withintheaccuracyofsingle precisionon theIBM360/175computer), whereas some inaccuracies were noted at x = 0.3.

APPLICATION TO HL-10 LIFTING BODY

The surface fitting method was also applied to part of the HL-10 liftingbody. Figure 9 shows a view of this body and the six cross sections used togenerate the surface fit. Tabulated values of the coordinates of surfacepoints were availableformany cross sectionsinadditionto the sixactually


11/28

Aftersatisfactory curve-fits were obtainedfor all sixcross sections,usingthree segments in eachone,the cross sections were blended in the longi-tudinaldirectionby use of theparametric cubicsplines. Since each three-dimensionallongitudinal curve is represented by its projections in the x,y andx,zplanes,atotal of 2Uplanar curves were usedtorepresentthefour longi-tudinal curves required for each of the three segments in a cross section. Ofthese2k curves, seven of the initial spline-fits were found to have wiggles,andhencethey were modifiedbyspecifying theirslopesat thenose(x = 0).Someof therevised longitudinal curvesareshowninfigure9, andtheyareactually smooth althoughtheplotter makes themappeartohave wiggles.

Asa check on the accuracy of the finalsurfacefit,cross sections werecalculated at axial stations in between those used to generate the surfacefit.Figure12comparesthecalculated cross sectionat X *1*3.656withtheactualcoordinatesforthisbody. Other cross sections were found to compare'evenmoreclosely than this one with the actual coordinates.

CONCLUDING REMARKS

Analgorithmhasbeen developedforsurface fitting three-dimensionalbod-iesfrom data pointsinseveral cross-sectional planes. This methodwasfoundtogivesatisfactory surface fitsto the 70slab delta wingandHL-10iftingbody. The surface geometry can best be analyzed through the use of an inter-activecomputer graphics environment.Afterasatisfactory surfacefit to abodyhasbeen obtained,arelativelysimplegeometry subroutine package can be formed to use in other computerpro-gramsrequiring a mathematical model of the geometry. It will calculate thebodycoordinates, slopes,andsecond derivativesat anypositionon thebody.Thismethodcouldbeusedtogeneratetheinput dataforCoons' pathing methodifdesired. Computational time is short, and the amount of storage required isrelatively small. A copy of the computer program can be obtained from theauthors.

ACKNOWLEDGMENTS

ThisresearchwassupportedbyNASA Langley GrantNo. NCR3 -002-193. TheNASATechnical Officer for this grant was Mr. H. Harris Hamilton of the Ad-


12/28

APPENDIX A

Solution forConicSection EquationSubstituteEqs.(2), (3), (7),and (8)intoEq. (l) toobtaintheconicsection in thelocal coordinate system in the form

Cz 2+ P z + Q 0 (Al)J J Jwhere

and= - A / m ) j U / n ) y A / n O y (A2)

l j= Y y-yJ. (A3)The solution of Eq. (Al)is

1/21z = |- P, R, '|/2C. for C j 0J (Ah)forC = 0, P ? 0J J

wherethediscriminate(Rj)isgivenby

z = - Qj/Pj for Cj = 0, Pj

In ordertoobtain real rootsfor zfromEq.(AU),Eq.(A5)must giveR^ 0for 0 < y < yj. Note that Eq.(A5)gives Rj > 0 when AjCj > 0.

Also,

andR =(A/m)2y2> 0 at y = 0J J J


13/28

R

R > 0,real rootsJRegion ofComplex roots

The minimum value ofingcase shown below. whichstillgives real roots corresponds to the limit-

Theminimum valueof R


14/28

and thusz = 0requiresthe +signif(A/m)j> 0 and the -signif(A/m).t < 0.Atthesecond control point,y = y 0 and the +signif(A/n)j < 0.These conditionsare all compatible becauseaconicsection passing throughthetwocontrol pointsandsingle-valued in zwill haveHJ < 0 if mj > 0 and n > 0ifmj < 0. The sign given by these conditions can be used in Eq.(AU)for allvaluesof y in therange0 < y < yz = zk'^-nsegmentj; and r j iscalledtheresidual. Theleast-squaressolution of the overdetermined system of equations determines the coefficientsCj, C. , Aj (j = 2,,N)which minimizethe sum of theresiduals squaredSeeref. 8). DefineKj as thedata point number which correspondsto thefirstcontrol point in segment J. Square Eq.(Bl) and sum over all the data pointsincontinuous segmentsJ = 1,,N toobtain

N KJ+1 r -12 N K1+lV f /.-.r- \J-lWCj J k -31 J>k J>k k J=lk=Kj

The right side of Eq.(B2) is minimized by the system of equations obtained by


15/28

The result ofsetting partial derivatives with respectto C - j _ ,, Cjj equal tozerogivesthefollowingadditional system of equations:b nA , + b 0A + b _A ._ + b ,C =0m,l m-1 m,2 m m,3 m+1 m,U m

m = 1, , N . (Bit)Thecombined systemofEqs. B3)and (Bit) givesasystemof (2N - 1)linearequations for (2N - l) coefficients. The parameters used inEqs. B3)and (Bit)are defined asfollows:

Kman, 1= I m 1 V Y Tv (B5)m,.L i , v mJ.,K m J _ , KK=iv _m-1K K_m m+1

+ I a kB (B6)m-1 m

Km Km+l Km+2a = y Y + y e + y a (BT)m,3 , rm-l,k .Lv m,k . m+l,kk=K ^ k=K k=K nm-1 m m+1a i =a*i o ^B8) a ; = a0.0 i (B9)ffijU m+1,2 m,5 m+2,1

K Km p m+1am, = I z^m-,v (BIO) a = z^ 6m (Bll)(:) v- * m-l,K m,Tvz^r k m,KiC ~lv n K.lvm-1 mKm+2a p = ^ z, a , . (B12) b , = a , ft (B13)i n o - \ T J - - ni+j. K m j - L r o X 9 om+1

bm,2=am,T (B11 bm,3=am+l,6 (B15)


16/28

theresulting system,andrecall thatA^ = 1unlesstheconic section requiresAT = 0.

REFERENCES

1. DeJarnette, F. R.: Calculation of Inviscid Surface Streamlines and HeatTransfer onShuttle Type Configurations. NASACR-111921, Aug., 1971.

2. Rakich, J. V. , and Kutler, P.: Comparison of Characteristics and ShockCapturing Methods with Applicationto theSpace Shuttle Vehicle. AIAAPaperNo.72-191, Jan., 1972.

3. Bartlett,D. A.: Computer UtilizationforAircraft Contour Determination.Society of Automotive Engineers,700202,National Business Meeting,Wichita,Kansas, March18-20, 1970.

U. Coons, S. A.: Surfaces for Computer-Aided Design of Space Forms. AD66350UMAC-TR-lH,MIT,June,1967.

5. Bezier, P.: Numerical Control Mathematics and Applications. John WileyandSons,1972.

6. Craidon, C. B.: A Computer Program for Fitting Smooth Surfaces to anAir-craft Configuration and Other Three-Dimensional Geometries. NASATMX-3206,June,1975.

7. Mason,T. E., andHazard,C. T.: Brief Analytic Geometry. GinnandCo.,19U7.8. Scheid, F.: Theory and Problems of Numerical Analysis. Schaum's Outline

Series, McGraw-Hill BookCo., 1968.9. Ahlberg,J. H. ,Nilson,E. N., andWalsh,J. L.: TheTheoryofSplinesand

Their Applications. Academic Press,1967.


17/28

k = 1 0

k = 8

k = 7

O C O N T R O L P O I N TA D A T A P O I N T

k = 6


18/28


19/28

F i g u r e 3 . L o c a l c o o r d i n a t e s y s t e m , i l l u s t r a t e d f o r s e g m e n t j = 2 .


20/28

Equation for General Conic SectionA.y + B yz + C z + D y + E z + F - 0

LEGEND1 . A.C. < (A/m) .(A/n). h y p e r b o l a w h i c h g i v e s c o m p l e xJ J J J rootsfor z2 . AjCj = (A/m).(A/n)..3 . A C . > (A/m)(A/n).

J J J J


21/28

A 8 . , H e . -


22/28

f i r s t c o nt r o l p o i n t

i n t e r m e d i a t e p o i n t

s l o p e p o i n tVs

e n d c o n t r o l p o i n ty r I ,

F i g u r e 6 . F o u r p o i n t s u s e d t o d e f i n e a s e g m e n t o f a c o n i c s e c t i o n ,


23/28

O C O N T R O L P O I N T , (yQ,zo)and (^0 I N T E R M E D I A T E P O I N T , (yh,zh)

S L O P E P O I N T ,(ys.zs)


24/28

R = l

0.65798-

1 0

O C O N T R O L P O I N TA D A T A P O I N T

-* - zx=0.65798 ( c i r c l e ) x = l ( e l l i p s e )


25/28

0

4


26/28

oowooooCOCOCOo-p-pCOfo

0

CM

CM

SXU-A

4


27/28

sxz

4


28/28

nHPonroHJCH0O

OfaoOHHPhoW0

3oCOtoa

OP

*nn

OT

O0

OCO

too

H0

OoOJHCH

sxz

4

Documents

Surface Fitting Three-Dimensional Bodies