1. 1 st Meeting, Industrial Geometry, 2005 Approximating Solids by Balls (in collaboration with subproject: "Applications of Higher Geometrics") Bernhard Kornberger Institute for Software Technology, TU Graz

2. 1 st Meeting, Industrial Geometry, 2005 Overview  Motivation:  Definition of the Minkowski addition  An application for Minkowski sums  Approximation  A technique to approximate solids by balls

3. 1 st Meeting, Industrial Geometry, 2005 Definition of the Minkowski Addition The Minkowski sum of two point sets A and B in Euclidean space is the result of adding every point of A to every point of B: A BABAB

4. 1 st Meeting, Industrial Geometry, 2005 One Application of Minkowski Sums  Motion Planning: Find the shortest path for a robot R through the obstacles P.  Solution: Compute the Minkowski sum R  P and thereby fatten the obstacles. This reduces the problem to moving a POINT from the start to the goal using a standard shortest-path algorithm R P P

5. 1 st Meeting, Industrial Geometry, 2005 Minkowski Addition of two Solids  In our case the surface of a solid is given by a dense 3dimensional point cloud which is triangulated.

6. 1 st Meeting, Industrial Geometry, 2005 Minkowski Addition of two Solids  In our case the surface of a solid is given by a dense 3dimensional point cloud which is triangulated.  We replace this representation by an approximation using balls. The Minkowski addition of two primitive elements is easy now: Just the radii and the centers of the balls have to be added.

7. 1 st Meeting, Industrial Geometry, 2005 Our Attempt (first shot which uses parts of the powercrust [1] technology) Minkowski addition using approximations of the models

8. 1 st Meeting, Industrial Geometry, 2005 Our Attempt [1/3] (here in 2D)  The surface of an object is given as a dense cloud of sample points.

9. 1 st Meeting, Industrial Geometry, 2005 Our Attempt [1/3] (here in 2D)  The surface of an object is given as a dense cloud of sample points.  From this input we compute the Voronoi diagram which divides the space into cells, each consisting of all points closest to one particular sample point. Voronoi Cell

10. 1 st Meeting, Industrial Geometry, 2005 Our Attempt [2/3] (here in 2D)  The sampling is dense. Therefore the cells are small, long and approximately normal to the surface. Voronoi Cell

11. 1 st Meeting, Industrial Geometry, 2005 Our Attempt [2/3] (here in 2D)  The sampling is dense. Therefore the cells are small, long and approximately normal to the surface.  A pole is the farthest vertex of a Voronoi cell from the sample point. Each interior pole is part of the approximated medial axis. Voronoi Cell Interior Pole

12. 1 st Meeting, Industrial Geometry, 2005 Our Attempt [2/3] (here in 2D)  The sampling is dense. Therefore the cells are small, long and approximately normal to the surface.  A pole is the farthest vertex of a Voronoi cell from the sample point. Each interior pole is part of the approximated medial axis.  The medial axis consists of all points having more than one nearest point on the surface. Interior Pole

13. 1 st Meeting, Industrial Geometry, 2005 Our Attempt [3/3] (here in 2D)  Each interior pole is used as the center of a circle that touches the surface in at least two points.

14. 1 st Meeting, Industrial Geometry, 2005 Our Attempt [3/3] (here in 2D)  Each interior pole is used as the center of a circle that touches the surface in at least two points.  The centers of all circles approximate the medial axis.  The union of their hulls approximates the hull of the object

15. 1 st Meeting, Industrial Geometry, 2005 Real Examples in 3D (..constructed in joint work with the team in Vienna)

16. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D A CAD model –triangulated with 20 000 points

17. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D A CAD model –triangulated with 20 000 points The approximation –with 20 000 balls

18. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D A CAD model –triangulated with 20 000 points The approximation –with 20 000 balls The approximated medial axis

19. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D Original CAD Model –Triangulation with 20 000 Points

20. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D Approximated Model –20 000 Balls

21. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D Approximated Model –20 000 Balls –10 000 Balls

22. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D Approximated Model –20 000 Balls –10 000 Balls – 4 000 Balls

23. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D Approximated Model –20 000 Balls –10 000 Balls – 4 000 Balls – 1 000 Balls

24. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D Approximated Model –20 000 Balls –10 000 Balls – 4 000 Balls – 1 000 Balls – 200 Balls

25. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D Approximated Model –20 000 Balls –10 000 Balls – 4 000 Balls – 1 000 Balls – 200 Balls – 100 Balls

26. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D Approximated Model –20 000 Balls –10 000 Balls – 4 000 Balls – 1 000 Balls – 200 Balls – 100 Balls – 50 Balls

27. 1 st Meeting, Industrial Geometry, 2005 Examples in 3D The approximated medial axis of the CAD object is not thin and smooth –This leads to smaller balls near the objects surface –Not optimal

28. 1 st Meeting, Industrial Geometry, 2005 Work is ongoing...  Problems of our first attempt:  Extremely strong response of the approximated medial axis to small distortions on the surface.  Balls outside the surface of the original object caused by wrong inner- /outer- pole labeling  A PC with 512 MB RAM can compute no more than 30000 input points.  Further steps  Our first attempt used the Powercrust [1] software which is actually designed for surface reconstruction.  Our own version of the software is planned to use efficient and reliable algorithms from CGAL  A combination with other approaches like octtrees will be investigated

29. 1 st Meeting, Industrial Geometry, 2005 References 1.Powercrust, developed by Amenta, Choi and Kolluri http://www.cs.utexas.edu/users/amenta/powercrust/welcome.html http://www.cs.utexas.edu/users/amenta/powercrust/welcome.html

30. 1 st Meeting, Industrial Geometry, 2005 Thank you for your attention!