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Designing Developable Surfaces
Zhao [email protected]. 29, 2006
Developable Surface(1)
Ruled Surface A ruled surface is a surface that can be swept out by
moving a line in space.
where is called the ruled surface directrix (also
called the base curve) , is the director curve.
The straight lines themselves are called rulings.
( , ) ( ) ( )u v u v u X b δ( )ub
( )uδ
Developable Surface(2)
Generalized Cone
( , ) ( )u v v u X p δ
where is a fixed point which can be regarded as the vertex of the cone.
p
Fig.1. A generalized cone over a cardioid.
Developable Surface(3)
Generalized Cylinder
( , ) ( )u v u v X b p
where is a fixed point.p
Fig.2. A generalized cylinder over a cardioid.
Developable Surface(4)
Tangent Surface
( , ) ( ) ( )u v u v u X b b
Developable Surface(5)
Geometric Property A developable surface is a ruled surface having
Gaussian curvature . Developable surfaces include generalized cones, generalized cylinders and tangent surfaces.
It could be made out of a sheet material without stretching or tearing.
0K
Developable Surface(6)
Application Design of ship hulls Sections of automotive and aircraft bodies Pipework and ducting Shoes and clothing
Developable Surface(7)
Developable Surface(8)
Developable Surface(9)
Developability conditions
represents a developable
surface
( ) ( ) ( ) 0u u u b δ δ
( , ) ( ) ( )u v u v u X b δ
Design Developable Surface(1)
Problem Description Given a boundary curve
0
( ) ( ) ,n
ni i
i
w B w
A A
0
( ) ( ) ,n
ni i
i
w B w
B B
find the other boundary curve
so that the ruled Bézier surface
( , ) (1 ) ( ) ( ) (0 1, 0 1)t w t w t w t w X A B
is a developable surface.
Design Developable Surface(2)
Design Developable Surface
Designing Methods Solving equations
Solving nonlinear characterizing equations on the surfac
e control points to ensure developability. Using plane geometry
Using the concept of duality between points and planes in 3D projective space.
Based on de Casteljaus algorithm
Design Developable Surface
Designing Methods(1) : solving equations Theory based on
( ) ( ) [ ( ) ( )] 0w w w w A B A B
ReferencesAumann G. Interpolation with developable Bézier patches. Computer Aided Geometric Design 1991;8:409-20.Maekawa T, Chalfant JS. Design and tessellation of B-spline developable surfaces. ASME Transaction of Mecha-nical Design 1998;120:453-61.
Design Developable Surface
Designing Methods(2) : Using plane geometry Main basis
Dual Bézier or B-spline representations by Hoschek. References
Hoschek, J. Dual Bézier curves and surfaces, in BarnHill, R.E. and Boehm, W., eds., Surfaces in Computer Aided Geometric Design, North-Holland, Amsterdam, 1983, p.147-156.Bodduluri, RMC, Ravani, B. Design of developable surfaces using duality between plane and point geometries. Computer-Aided Design 1992;25:621-32.Pottmann, H, Farin G. Developable rational Bézier and B-spline surfaces. Computer Aided Geometric Design 1995;12:513-31.
Design Developable Surface
Designing Methods(3) : Based on de Casteljaus algorithm Main basis
de Casteljaus algorithm References
Chu CH, Séquin CH. Developable Bézier patches: properties and design. Computer-Aided Design 2002;34(7):511-27.Aumann G. A simple algorithm for designing developable Bézier surface. Computer Aided Geometric Design 2004;20:601-16.Chu CH, Chen JT. Characterizing degrees of freedom for geometric design of developable composite Bézier surfaces. Robitics and Computer-Integrated Manufacturing 2007;23(1):116-125.
Developable Bézier patches: properties and design
Chih-Hsing Chu, Carlo H. SéquinDepartment of Mechanical Engineering, University of California at Berkeley, Berkeley
Computer-Aided Design 34(2002), 511-527
Developable Bézier patches: properties and design
Author Infromation姓名: Chih Hsing Chu( 瞿志行 ) 職稱:國立清華大學 IEEM 副教授 學歷:美國加州大學柏克萊分校機械工程博士
E-mail :[email protected] 研究領域:協同設計、幾何模擬、產業電子化
Carlo H. Séquin
Professor, CS Division, EECS Dept., U.C. Berkeley, (Graphics Group)Associate Dean, Capital Projects, College of Engineering
Homepage: http://www.cs.berkeley.edu/~sequin/
Developable Bézier patches: properties and design
Outline1. Geometric interpretation of the developability
conditiona) Quadratic developable Bézier patch
b) Cubic developable Bézier patch
2. Counting DOF (degrees of freedom)
3. Designing quadratic and cubic Bézier patches utilizing DOFa) Method Ⅰb) Method Ⅱ
Developable Bézier patches: properties and design
Outline1. Geometric interpretation of the developability
conditiona) Quadratic developable Bézier patch
b) Cubic developable Bézier patch
2. Counting DOF (degrees of freedom)
3. Designing quadratic and cubic Bézier patches utilizing DOFa) Method Ⅰb) Method Ⅱ
Developable Bézier patches: properties and design
Geometric interpretation of the developability condition
1. de Casteljau algorithm
Developable Bézier patches: properties and design
Geometric interpretation of the developability condition
2. Developability condition:
Tangent lines and the corresponding ruling remain coplanar
0 IJ KL IK
Developable Bézier patches: properties and design
Geometric interpretation of the developability condition
0 IJ KL IK
Developable Bézier patches: properties and design
Geometric interpretation of the developability condition Quadratic developable Bézier patch
Suppose
then
Developable Bézier patches: properties and design
Geometric interpretation of the developability condition Quadratic developable Bézier patch
0 IJ KL IK
Developable Bézier patches: properties and design
Geometric interpretation of the developability condition Quadratic developable Bézier patch
0 IJ KL IK
Solve the non-linear system of equations Return
Developable Bézier patches: properties and design
Geometric interpretation of the developability condition Cubic developable Bézier patch
Developable Bézier patches: properties and design
Geometric interpretation of the developability condition Cubic developable Bézier patch
0 IJ KL IK
Return
Developable Bézier patches: properties and design
Outline1. Geometric interpretation of the developability
conditiona) Quadratic developable Bézier patch
b) Cubic developable Bézier patch
2. Counting DOF (degrees of freedom)
3. Designing quadratic and cubic Bézier patches utilizing DOFa) Method Ⅰb) Method Ⅱ
Developable Bézier patches: properties and design
Counting DOF (degrees of freedom)
0 IJ KL IK
The second boundary curve
Developable Bézier patches: properties and design
Counting DOF (degrees of freedom)
0 IJ KL IK
The second boundary curve B-curve
Developable Bézier patches: properties and design
Counting DOF (degrees of freedom) Inherent scaling parameter
A scaling factor
Developable Bézier patches: properties and design
Outline1. Geometric interpretation of the developability
conditiona) Quadratic developable Bézier patch
b) Cubic developable Bézier patch
2. Counting DOF (degrees of freedom)
3. Designing quadratic and cubic Bézier patches utilizing DOFa) Method Ⅰb) Method Ⅱ
Developable Bézier patches: properties and design
Designing quadratic and cubic Bézier patches utilizing DOF Method Ⅰ
Developable Bézier patches: properties and design
Designing quadratic and cubic Bézier patches utilizing DOF Method Ⅱ
Developable Bézier patches: properties and design
Designing quadratic and cubic Bézier patches utilizing DOF Quadratic case
Method ⅠSubstitute
into the developability conditions, and there is
Developable Bézier patches: properties and design
Designing quadratic and cubic Bézier patches utilizing DOF Quadratic case
Method Ⅱ
a)
b)
c)
Developable Bézier patches: properties and design
Designing quadratic and cubic Bézier patches utilizing DOF Cubic case
Method ⅠSubstitute into the developability conditions.
Assume , and there are
Developable Bézier patches: properties and design
Developable Bézier patches: properties and design
Designing quadratic and cubic Bézier patches utilizing DOF Cubic case
Method ⅡSubstitute into the developability conditions.
Assume
Developable Bézier patches: properties and design
Developable Bézier patches: properties and design
Special cases of developable Bézier patches Generalized conical model
4 DOF
Developable Bézier patches: properties and design
Special cases of developable Bézier patches Generalized cylindrical model
More than 5 DOF
Developable Bézier patches: properties and design
Conclusion
A simple algorithm for designing developable Bézier surfaces
Computer Aided Geometric Design
2003;20:601-619
Günter Aumann Mathematishes Institut Ⅱ, Universität Karlsruhe, Germany
A simple algorithm for designing developable Bézier surfaces
Restrictions of previous algorithms The characterizing equations can only be solved f
or boundary curves of low degrees. Only planar boundary curves are premitted. It is difficult to control singular points.
Developable Bézier patches: properties and design
Outline1. Geometric interpretation of the developability
condition
2. Generating Bézier surface
3. Discussion
4. Application Interpolation
Developable Bézier patches: properties and design
Outline1. Geometric interpretation of the developability
condition
2. Generating Bézier surface
3. Discussion
4. Application Interpolation
A simple algorithm for designing developable Bézier surfaces
Geometric interpretation of the developability condition
A simple algorithm for designing developable Bézier surfaces
Geometric interpretation of the developability condition
Developable Bézier patches: properties and design
Outline1. Geometric interpretation of the developability
condition
2. Generating Bézier surface
3. Discussion
4. Application Interpolation
A simple algorithm for designing developable Bézier surfaces
Generating Bézier surface
A simple algorithm for designing developable Bézier surfaces
Generating Bézier surface
De Casteljau algorithm
A simple algorithm for designing developable Bézier surfaces
Generating Bézier surface
Developable Bézier patches: properties and design
Outline1. Geometric interpretation of the developability
condition
2. Generating Bézier surface
3. Discussion
4. Application Interpolation
A simple algorithm for designing developable Bézier surfaces
Discussion(1)
?
A simple algorithm for designing developable Bézier surfaces
Discussion(2)
?
A simple algorithm for designing developable Bézier surfaces
Discussion(1)
A simple algorithm for designing developable Bézier surfaces
A simple algorithm for designing developable Bézier surfaces
Discussion(2)
A simple algorithm for designing developable Bézier surfaces
A simple algorithm for designing developable Bézier surfaces
Discussion(2)
A simple algorithm for designing developable Bézier surfaces
Developable Bézier patches: properties and design
Outline1. Geometric interpretation of the developability
condition
2. Generating Bézier surface
3. Discussion
4. Application Interpolation
Developable Bézier patches: properties and design
Application: interpolation Problem description
:
:
Developable Bézier patches: properties and design
Application: interpolation
The case left
Developable Bézier patches: properties and design
A simple algorithm for designing developable Bézier surfaces
Related Work
张兴旺 , 王国瑾 . 可展 Bézier 曲面的设计 . Wang GJ, Tang K, Tai CL. Parametric representation
of a surface pencil with a common spatial geodesic. Computer-Aided Design 2004;36:447-459.
可展 Bézier曲面的设计
张兴旺 , 王国瑾
可展 Bézier曲面的设计
Bézier surface
( , ) (1 ) ( ) ( ) ( ) ( ), ( , ) [0,1] [0,1],u v v u v u u v u u v r q
0
( ) ( )( ), [0,1],n
ni i i
i
u B u u
q p
1n
where
is a developable surface.
. 0
可展 Bézier曲面的设计
Three cases
( ) 0u
( ) 0u 1( , ) ( ) ( ) ,u v u v b r 1( ) ,u b
a b
a b
Generalized cylinders
Generalized cones
Tangent surfaces
可展 Bézier曲面的设计
可展 Bézier曲面的设计
Parametric representation of a surface pencil with a common spatial geodesic
Wang GJ, Tang K, Tai CL
Computer-Aided Design 2004;36:447-459
Parametric representation of a surface pencil with a common spatial geodesic
Application Background
Parametric representation of a surface pencil with a common spatial geodesic
Application Background
Parametric representation of a surface pencil with a common spatial geodesic
Problem Description
Parametric representation of a surface pencil with a common spatial geodesic
Tool– Frenet trihedron Frame
Parametric representation of a surface pencil with a common spatial geodesic
Method
are called
Parametric representation of a surface pencil with a common spatial geodesic
Theory Based on
Parametric representation of a surface pencil with a common spatial geodesic
Necessary and Sufficient Conditions
where
Parametric representation of a surface pencil with a common spatial geodesic
Designing the marching-scale functions
Parametric representation of a surface pencil with a common spatial geodesic
Constructing Developable surface
Parametric representation of a surface pencil with a common spatial geodesic
Application in Garment Industry