Surface modeling through geodesic

1. Surface modelingthrough geodesic Reporter: Hongyan Zhao Date: Apr. 18th Email: Hongyanzhao_cn@yahoo.com.cn

2. Surface modelingthrough geodesic • Guo-jin Wang, Kai Tang, Chiew-Lan Tai. Parametric representation of a surface pencil with a common spatial geodesic. Computer Aided Design 36 (2004) 447-459. • Marco Paluszny. Cubic Polynomial Patches though Geodesics. • ***, Wenping Wang, ***. Geodesic-Controlled Developable Surfaces for Modeling Paper Bending.

3. Background • Geodesic • A geodesic is a locally length-minimizing curve. • In the plane, the geodesics are straight lines. • On the sphere, the geodesics are great circles. • For a parametric representation surface, the geodesic can be found …… http://mathworld.wolfram.com/Geodesic.html

4. Background • Applications of geodesics(1) • Geodesic Dome • tent manufacturing

5. Background • Applications of geodesics(2) • Shoe-making industry • Garment industry

6. Surface modelingthrough geodesic • Guo-jin Wang, Kai Tang, Chiew-Lan Tai. Parametric representation of a surface pencil with a common spatial geodesic. Computer Aided Design 36 (2004) 447-459. • Marco Paluszny. Cubic Polynomial Patches though Geodesics. • ***, Wenping Wang, ***. Geodesic-Controlled Developable Surfaces for Modeling Paper Bending.

7. Parametric representation of a surface pencil with a common spatial geodesic Computer Aided Design 36 (2004) 447-459 Guo-jin Wang, Kai Tang, Chiew-Lan Tai

8. Author Introduction • Kai Tang http://ihome.ust.hk/~mektang/ Department of Mechanical Engineering, Hong Kong University of Science & Technology. • Chiew-Lan Tai http://www.cs.ust.hk/~taicl/ Department of Computer Science & Engineering, Hong Kong University of Science & Technology.

9. Parametric representation of a surface pencil with a common spatial geodesic • Basic idea • Representation of a surface pencil through the given curve • Isoparametric and geodesic requirements Representation of a surface pencil through the given curve

10. Parametric representation of a surface pencil with a common spatial geodesic • Basic idea • Representation of a surface pencil through the given curve • Isoparametric and geodesic requirements

11. Representation of a surface pencil through the given curve Return

12. Isoparametric and geodesic requirements • Isoparametric requirements • Geodesic requirements • At any point on the curve, the principal normal to the curve and the normal to the surface are parallel to each other.

13. Isoparametric and geodesic requirements • The representation with isoparametric and geodesic requirements Return

14. Cubic Polynomial Patches though Geodesics Marco Paluszny

15. Author introduction Marco Paluszny Professor Universidad Central de Venezuela

16. Cubic Polynomial Patches though Geodesics • Goal • Exhibit a simple method to create low degree and in particular cubic polynomial surface patches that contain given curves as geodesics.

17. Cubic Polynomial Patches though Geodesics • Outline • Patch through one geodesic • Representation • Ribbon (ruled patch) • Non ruled patch • Developable patches • Patch through pairs of geodesics • Using Hermite polynomials • Joining two cubic ribbons • G1 joining of geodesic curves Patch through one geodesic

18. Cubic Polynomial Patches though Geodesics • Patch through one geodesic • Representation • Ribbon (ruled patch) • Non ruled patch • Developable patches • Patch through pairs of geodesics • Using Hermite polynomials • Joining two cubic ribbons • G1 joining of geodesic curves

19. Patch through one geodesic • Representation • Ribbon (ruled surface) • Non ruled surface

20. Patch through one geodesic • Developable patches Then the surface patch is developable. Return

21. Patch through pairs of geodesics

22. Patch through pairs of geodesics • Using Hermite polynomials

23. Y12 X02 X02 Y02 Y11 X01 X01 Y01 X03 Y13 X03 Y03 Y00 X00 X00 Y10 X12 Y02 X12 Y12 Y01 X11 X11 Y11 Y03 X13 X13 Y13 Y10 X10 X10 Y00 Patch through pairs of geodesics • Joining two cubic ribbons Return

24. G1 joining of geodesic curves • G1 connection of two ribbons containing G1 abutting geodesics(1)

25. G1 joining of geodesic curves • G1 connection of two ribbons containing G1 abutting geodesics(2) • The tangent vectors and are parallel. • The ribbons share a common ruling segment at . • The tangent planes at each point of the com-mon segment are equal for both patches. Return

26. Geodesic-Controlled Developable Surfaces for Modeling Paper Bending ***, Wenping Wang, ***

27. Author introduction Wenping Wang Associate ProfessorB.Sc. and M.Eng, Shandong University, 1983, 1986; Ph.D., University of Alberta, 1992. Department of Computer Science,The University of Hong Kong. Email: wenping@cs.hku.hk

28. Geodesic-Controlled Developable Surfaces • Goal: modeling paper bending

29. Geodesic-Controlled Developable Surfaces • Outline • Propose a representation of developable surface • Rectifying developable (geodesic-controlled developable) • Composite developable • Modify the surface by modifying the geodesic • Move control points • Move control handles • Preserve the curve length Propose a representation of developable surface

30. Geodesic-Controlled Developable Surfaces • Outline • Propose a representation of developable surface • Rectifying developable (a geodesic-controlled developable) • Composite developable • Modify the surface by modifying the geodesic • Move control points • Move control handles • Preserve the curve length

31. Rectifying developable • Definition • Rectifying plane: The plane spanned by the tangent vector and binormal vector • Given a 3D curve with non-vanishing curvature, the envelope of its rectifying planes is a developable surface, called rectifying developable.

32. Rectifying developable • Representation or where is arc length. The surface possesses as adirectrixas well as ageodesic!

33. Rectifying developable • Curve of regression • Why? • A general developable surface is singular along the curve of regression. • Goal • Keep singularities out of region of interest • Definition: limit intersection of rulings

34. Rectifying developable • Compute Paper boundary • Goal • Keep singularities out of region of interest • Keep the paper shape when bending • Method • Compute the ruling length of each curve point

35. Rectifying developable • Keep singularities out of region of interest

36. Composite developable • Why? • A piece of paper consists of several parts which cannot be represented by a one-parameter family of rulings from a single developable.

37. Composite developable • Definition • A composite developable surface is made of a union of curved developables joined together by transition planar regions. Return

38. Interactive modifying • Move control points

39. Interactive modifying • Move control handles(1) • Why? • Users usually bend a piece of paper by holding to two positions on it. • Give: • positions and orientation vectors at the two ends. • Want: • a control geodesic meeting those conditions

40. Interactive modifying • Move control handles(2) • When the constraints are not enough, minimize

41. Interactive modifying • Preserve curve length

42. Composite developable • Boundary planar region

43. Composite developable • Control a composite developable Return

44. Application • Texture mapping • The algorithm computing paper boundary. • Surface approximation VIDEO

45. Future work • Investigate the representation of the control geodesic curve with length preserving property. • 3D PH curve

46. The end Thank you!