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

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

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

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

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.

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.

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

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.

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

1 st Meeting, Industrial Geometry, 2005 Examples in 3D Approximated Model – Balls – Balls – Balls – Balls – 200 Balls

1 st Meeting, Industrial Geometry, 2005 Examples in 3D Approximated Model – Balls – Balls – Balls – Balls – 200 Balls – 100 Balls

1 st Meeting, Industrial Geometry, 2005 Examples in 3D Approximated Model – Balls – Balls – Balls – Balls – 200 Balls – 100 Balls – 50 Balls

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

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 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

1 st Meeting, Industrial Geometry, 2005 References 1.Powercrust, developed by Amenta, Choi and Kolluri

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