2. Surface modeling through geodesic Reporter: Hongyan Zhao Date: Apr. 18th Email: Hongyanzhao_cn@yahoo.com.cn

3. Surface modeling through 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.

4. 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

5. Background  Applications of geodesics(1) Geodesic Dome  tent manufacturing

6. Background  Applications of geodesics(2) Shoe-making industry Garment industry

7. Surface modeling through 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. Parametric representation of a surface pencil with a common spatial geodesic  Marco Paluszny. Cubic Polynomial Patches though Geodesics.Cubic Polynomial Patches though Geodesics  ***, Wenping Wang, ***. Geodesic-Controlled Developable Surfaces for Modeling Paper Bending.Geodesic-Controlled Developable Surfaces for Modeling Paper Bending

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

9. 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.

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 Representation of a surface pencil through the given curve Representation of a surface pencil through the given curve

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

12. Representation of a surface pencil through the given curve

13. Isoparametric and geodesic requirements IIsoparametric requirements G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.

14.  The representation with isoparametric and geodesic requirements

15. Cubic Polynomial Patches though Geodesics Marco Paluszny

16. Author introduction Marco Paluszny Professor Universidad Central de Venezuela

17. 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.

18. 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 G 1 joining of geodesic curves Patch through one geodesic Patch through one geodesic

19. Cubic Polynomial Patches though Geodesics  Patch through one geodesic Patch through one geodesic Representation  Ribbon (ruled patch)  Non ruled patch Developable patches  Patch through pairs of geodesics Patch through pairs of geodesics Using Hermite polynomials Joining two cubic ribbons  G 1 joining of geodesic curves G 1 joining of geodesic curves

20. Patch through one geodesic  Representation Ribbon (ruled surface) Non ruled surface

21. Patch through one geodesic  Developable patches Then the surface patch is developable.

22. Patch through pairs of geodesics

23.  Using Hermite polynomials

24. Patch through pairs of geodesics  Joining two cubic ribbons X 00 X 02 X 01 X 03 X 11 X 10 X 12 X 13 Y 00 Y 02 Y 01 Y 03 Y 11 Y 10 Y 12 Y 13 X 00 X 02 X 01 X 03 X 11 X 10 X 12 X 13 Y 10 Y 12 Y 11 Y 13 Y 01 Y 00 Y 02 Y 03

25. G 1 joining of geodesic curves  G 1 connection of two ribbons containing G 1 abutting geodesics(1)

26. G 1 joining of geodesic curves  G 1 connection of two ribbons containing G 1 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.

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

28. Author introduction Wenping Wang Associate Professor B.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

29. Geodesic-Controlled Developable Surfaces  Goal: modeling paper bending

30. 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 Propose a representation of developable surface

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

32. 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.

33. Rectifying developable  Representation or where is arc length. The surface possesses as a directrix as well as a geodesic !

34. 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

35. 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

36. Rectifying developable  Keep singularities out of region of interest

37. 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.

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

39. Interactive modifying  Move control points

40. 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

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

42. Interactive modifying  Preserve curve length

43. Composite developable  Boundary planar region

44. Composite developable  Control a composite developable

45. Application  Texture mapping The algorithm computing paper boundary.  Surface approximation

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

47. The end Thank you!