Minimum Distance between curved surfaces

1. Minimum Distance between curved surfaces Li Yajuan Oct.25,2006

2. Computation of the minimum distance between twoobjects is very important • Collision detection • Physical simulation in computer graphics • Animation • Virtual prototyping in haptic rendering • Robot motion planning and path modification • Computer games

3. Computation of the minimum distance between two polyhedra • A polygonal representation is not considered a true restriction since real objects can be approximated arbitrarily precisely by a polyhedron. • The basic algorithms and predicates can be implemented robustly and very efficiently on polygons. • Time: the number of polyhedral faces approximating the given curved surfaces is usually very large

4. Computation of the minimum distance between two curved surfaces • Ellipsoids and degenerate quadrics, such as cylinders and cones, are important primitives for solid modeling systems. • Ellipsoids can be used for efficiently bounding more general solids. • Bischoff and Kobbelt (02) developped techniques for approximating general objects by ellipsoids.

5. References • [1]Kim K-J. Minimum distance between a canal surface and a simple surface. CAD, 2003;35(10):871–9. • [2]Lennerz C, Schomer E. Efficient distance computation for quadratic curves and surfaces. In: Proceeding of Geometric Modeling and Processing. 2002. p. 60–9. • [3]Sohn K-A, Juttler B, Kim M-S, Wang W. Computing distances between surfaces using line geometry. In: Pacific conference on computer graphics and applications. 2002. p. 236–45. • [4]Chen Xiaodiao, etc.Computing minimum distance between two implicit algebraic surfaces. CAD, 2006; 38 1053–1061.

6. Minimum distance between a canal surface and a simple surface[1] • The minimum distance betweentwo parametric surfaces F(u,v) and G(s,t) are described by Piegl(1995):

7. A Canal • A canal surface is defined by the trajectory of the center C(t) and the function determining the radius r(t).

8. A Canal • A canal surface is defined by the trajectory of the center C(t) and the function determining the radius r(t).

9. A general solution • Finding roots of a function of a single parameter of a necessary condition:

10. Distance between a canal surface and a plane

11. Distance between a canal surface and a plane

12. Distance between a canal surface and a sphere

13. Distance between a canal surface and a sphere

14. Distance between a canal surface and a cylinder

15. Distance between a canal surface and a cylinder

16. Distance between a canal surface and a cone

17. Distance between a canal surface and a cone

18. Distance between a canal surface and a torus

19. Efficient distance computation for quadratic curves and surfaces[2].

20. Efficient distance computation for quadratic curves and surfaces[2].

21. Efficient distance computation for quadratic curves and surfaces.

22. Efficient distance computation for quadratic curves and surfaces.

23. Efficient distance computation for quadratic curves and surfaces.

24. Efficient distance computation for quadratic curves and surfaces.

25. Efficient distance computation for quadratic curves and surfaces.

26. Efficient distance computation for quadratic curves and surfaces.

27. Efficient distance computation for quadratic curves and surfaces.

28. Computing Distances Between Surfaces Using Line Geometry[3] • Using line geometry, the distance computation is reformulated as a simple instance of a surface-surface intersection problem, which leads to lowdimensional • root finding in a system of equations.

29. Line Coordinates(Plucker,1846)

30. The normal congruence of a surface:

31. The normal congruence of a surface: • Parameter representation

32. The normal congruence of a surface: • Implicit representation

33. The normal congruence of a surface: • Implicit representation

34. The normal congruence of a surface:

35. Distance Computation

36. Distance Computation • Distance between two ellipsoids; • Distance between an ellipsoid and a cylinder; • Distance between an ellipsoid and a cone; • Distance between an ellipsoid and a torus.

37. Experimental Results

38. Experimental Results

39. Experimental Results

40. Experimental Results

41. Computing minimum distance between two implicit algebraic surfaces[4].

42. Computing minimum distance between two implicit algebraic surfaces.

43. Computing minimum distance between two implicit algebraic surfaces.

44. Resultant method

45. If S1 is an implicit surface Computing minimum distance between two implicit algebraic surfaces. Eliminate λandμ If S1 is a parameter surface

46. Computing minimum distance between two implicit algebraic surfaces.

47. Algorithm

48. Minimum distance between a quadric surface and an implicit algebraic surface. • A cylinder and an implicit algebraic surface.

49. Minimum distance between a quadric surface and an implicit algebraic surface. • A cone and an implicit algebraic surface.

50. Minimum distance between a quadric surface and an implicit algebraic surface. • An elliptic paraboloid and an implicit algebraic surface.