Designing Developable Surfaces Zhao Hongyan [email protected] [email protected]...

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


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.


Generalized Cylinder

( , ) ( )u v u v X b p

where is a fixed point.p

Fig.2. A generalized cylinder over a cardioid.


Tangent Surface

( , ) ( ) ( )u v u v u X b b


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


Application Design of ship hulls Sections of automotive and aircraft bodies Pipework and ducting Shoes and clothing


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


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.


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.


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


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/


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


Geometric interpretation of the developability condition

1. de Casteljau algorithm



2. Developability condition:

Tangent lines and the corresponding ruling remain coplanar

0 IJ KL IK



0 IJ KL IK


Geometric interpretation of the developability condition Quadratic developable Bézier patch

Suppose

then



0 IJ KL IK



0 IJ KL IK

Solve the non-linear system of equations Return


Geometric interpretation of the developability condition Cubic developable Bézier patch


Geometric interpretation of the developability condition Cubic developable Bézier patch

0 IJ KL IK

Return


Counting DOF (degrees of freedom)

0 IJ KL IK

The second boundary curve


Counting DOF (degrees of freedom)

0 IJ KL IK

The second boundary curve B-curve


Counting DOF (degrees of freedom) Inherent scaling parameter

A scaling factor


Designing quadratic and cubic Bézier patches utilizing DOF Method Ⅰ


Designing quadratic and cubic Bézier patches utilizing DOF Method Ⅱ


Designing quadratic and cubic Bézier patches utilizing DOF Quadratic case

Method ⅠSubstitute

into the developability conditions, and there is


Designing quadratic and cubic Bézier patches utilizing DOF Quadratic case

Method Ⅱ

a)

b)

c)


Designing quadratic and cubic Bézier patches utilizing DOF Cubic case

Method ⅠSubstitute into the developability conditions.

Assume , and there are


Designing quadratic and cubic Bézier patches utilizing DOF Cubic case

Method ⅡSubstitute into the developability conditions.

Assume


Special cases of developable Bézier patches Generalized conical model

4 DOF


Special cases of developable Bézier patches Generalized cylindrical model

More than 5 DOF


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


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.



condition

2. Generating Bézier surface

3. Discussion

4. Application Interpolation



condition


3. Discussion



Generating Bézier surface



De Casteljau algorithm



condition


3. Discussion



Discussion(1)

？


Discussion(2)

？


Discussion(1)


Discussion(2)



condition


3. Discussion



Application: interpolation Problem description

:

:


Application: interpolation

The case left

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


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


Three cases

( ) 0u

( ) 0u 1( , ) ( ) ( ) ,u v u v b r 1( ) ,u b

a b

a b

Generalized cylinders

Generalized cones

Tangent surfaces

Parametric representation of a surface pencil with a common spatial geodesic

Wang GJ, Tang K, Tai CL

Computer-Aided Design 2004;36:447-459


Application Background


Problem Description


Tool– Frenet trihedron Frame


Method

are called


Theory Based on


Necessary and Sufficient Conditions

where


Designing the marching-scale functions


Constructing Developable surface


Application in Garment Industry

Designing Developable Surfaces Zhao Hongyan [email protected] [email protected]...

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