1. Rational surfaces with linear normals and their convolutions with rational surfaces Maria Lucia Sampoli, Martin Peternell, Bert J ü ttler Computer Aided Geometric Design 23 (2006) 179–192 Reporter: Wei Wang Thursday, Dec 21, 2006

2. About the authors Marai Lucia Sampoli, Italy Universit à degli Studi di Siena Dipartimento di Scienze Matematiche ed Informatiche http://www.mat.unisi.it/newsito /docente.php?id=32 http://www.mat.unisi.it/newsito /docente.php?id=32

3. About the authors Martin Peternell, Austria Vienna University of Technology Research Interests Classical Geometry Computer Aided Geometric Design Reconstruction of geometric objects from dense 3D data Geometric Modeling and Industrial Geometry

4. About the authors Bert J ü ttler, Austria J. Kepler Universit ä t Linz Research Interests: Computer Aided Geometric Design (CAGD) Applied Geometry Kinematics, Robotics Differential Geometry

5. Previous related work J ü ttler, B., 1998. Triangular B é zier surface patches with a linear normal vector field. In: The Mathematics of Surfaces VIII. Information Geometers, pp. 431 – 446. J ü ttler, B., Sampoli, M.L., 2000. Hermite interpolation by piecewise polynomial surfaces with rational offsets. CAGD 17, 361 – 385. Peternell, M., Manhart, F., 2003. The convolution of a paraboloid and a parametrized surface. J. Geometry Graph. 7, 157 – 171. Sampoli, M.L., 2005. Computing the convolution and the Minkowski sum of surfaces. In: Proceedings of the Spring Conference on Computer Graphics, Comenius University, Bratislava. ACM Siggraph, in press.

6. Introduction(1) LN surfaces Some geometric properties Its dual representation

7. Introduction(2) Convolution surfaces Computation of convolution surfaces Convolution of LN surfaces and rational surfaces

8. LN surface Linear normal vector field Model free-form surfaces [Juttler and Sampoli 2000] Main advantageous LN surfaces possess exact rational offsets.

9. Definition LN surface a polynomial surface p(u,v) with Linear Normal vector field certain constant coefficient vectors

10. Properties(1) Obviously Assume That is

11. Properties(2) Tangent plane of LN surface p(u, v) where

12. Computation given a system of tangent planes Then,the envelope surface is a LN surface. The normal vector

13. Geometric property Gaussian curvature of the envelope

14. Geometric property K > 0 elliptic points, K < 0 hyperbolic points, If the envelope possesses both, the corresponding domains are separated by the singular curve C.

15. The dual representation A polynomial or rational function f the LN-surfaces p (u,v) the associated graph surface q(u,v) is dual to LN-surface in the sense of projective geometry.

16. The dual representation Since det(H) of q(u,v) So, det(H)>0 elliptic points, det(H)=0 parabolic points, det(H)<0 hyperbolic points.

17. The dual representation Graph surface LN surface q(u,v) p(u,v) elliptic elliptic hyperbolic hyperbolic parabolic singular points dual to

18. Convolution surfaces and Minkowski sums Application Computer Graphics Image Processing Computational Geometry NC tool path generation Robot Motion Planning 何青, 仝明磊, 刘允才. 用卷积曲面生成脸部皱纹的方 法, Computer Applications, June 2006

19. Definition Given two objects P,Q in, then Minkowski sum

20. Definition Given two surfaces A,B in,then Convolution surface

21. Relations between them In general, In particular, if P and Q are convex sets Where, =

22. Kinematic generation(1)

26. ★

27. Kinematic generation(2)

30. Convolution surfaces of general rational surfaces Two surfaces A=a(u,v), B=b(s,t) parameter domains Ω A, Ω B. unit normal vectors,.

31. Reparameterization such that Where,. Convolution of general rational surfaces ∥

32. Then, is a parametric representation of the convolution surface of Convolution surfaces of general rational surfaces

33. Assumed LN-surface A rational surface B Convolution of LN surfaces and rational surfaces

34. If correspond, that is Then, Convolution of LN surfaces and rational surfaces

35. So, That is Where Convolution of LN surfaces and rational surfaces

36. The parametric representation c(s, t) of the convolution C = A ★ B Convolution of LN surfaces and rational surfaces

37. The convolution surface A ★ B of an LN-surface A and a parameterized surface B has an explicit parametric representation. If A and B are rational surfaces, their convolution A ★ B is rational, too. Convolution of LN surfaces and rational surfaces

38. Example

39. Conclusion and further work To our knowledge, this is the first result on rational convolution surfaces of surfaces which are capable of modeling general free-form geometries. This result may serve as the starting point for research on computing Minkowski sums of general free-form objects.

40. Thank you !