A Fast Algorithm for Incremental Distance Calculation

1. A Fast Algorithm for Incremental Distance Calculation Paper by Ming C. Ling and John F. Canny Presented by Denise Jones

2. Algorithm Concept • A method for calculating the closest features on two convex polyhedra • Algorithm is complete (will always find closest features between 2 polyhedra) • Can be used for: • Collision detection • Motion planning • Distance between objects in 3-D space

3. Algorithm Concept • Apply applicability criteria to features (vertices, edges, faces) of each polyhedron • Without any initialization, running time of the algorithm is linear for the number of vertices • With initialization, running time is constant • Can detect collision • Returns an error and features that have collided or intersected

4. Efficiency • Once the closest features are determined, these will change infrequently • When a change does occur, the new closest features will usually be on a boundary of the previous closest features.

5. Efficiency Exceptions • Initial features are parallel and on opposite sides (only on initialization) • Exceptions after initialization Parallel faces Before Rotation After Rotation

6. Applicability Criterion: Point-Vertex • Determine planes that are perpendicular to coboundary (edges) of the vertex • Point must be contained within the boundary of these planes in order to be the closest point • If outside one of the boundaries, indicates that edge is closer and will perform the test on that edge.

7. Applicability Criterion: Point-Vertex Voronoi Region Point Vertex

8. Applicability Criterion: Point-Edge • Determine region created by planes perpendicular to the head and tail of the edge and perpendicular to the coboundaries (faces) • Point must be contained within the boundary of these planes in order for edge to be the closest feature • If outside one of the boundaries, indicates corresponding feature is closer and will “walk” to the next feature and apply the appropriate test.

9. Applicability Criterion: Point-Edge Voronoi Space Point Edge

10. Applicability Criterion: Point-Face • Determine planes that are perpendicular to each edge of the face • Point must be contained within the applicability prism (region comprised of these planes and the edges of the face) • If outside one of the boundaries, indicates that edge is closer and performs the test for the corresponding edge. • If point lies below the face, two possibilities • Collision • Another feature is closer than face or any of edges • Algorithm will return closest feature

11. Applicability Criterion: Point-Face Point Voronoi Space Face

12. Algorithm Cases • Vertex-Vertex • Vertex-Edge • Vertex-Face • Edge-Edge • Edge-Face • Face-Face

13. Algorithm Example • Determine closest features on the following polyhedra • Randomly choose 2 features

14. Algorithm Example: Vertex-Vertex • Both vertices must meet point-vertex applicability criteria • If either fails, will return vertex and corresponding edge

15. Algorithm Example: Vertex-Vertex Voronoi Region Vertex • Test fails Point

16. Algorithm Example: Vertex-Edge • Edge must meet point-edge criterion • Vertex must meet point-vertex criterion for closest point on edge to vertex • If either fails, will return new feature pair based on failed test

17. Algorithm Example: Vertex-Edge Point Voronoi Space Edge • Test fails Closest point to vertex on edge

18. Algorithm Example: Vertex-Face • Face must meet point-face criterion • Vertex must meet point-vertex criterion for closest point on face to vertex • If either fails, will return new feature pair based on failed test

19. Algorithm Example: Vertex-Face Vertex Voronoi Space Closest point to vertex on face • Test passes for both criteria • As polyhedra continue along their trajectories, the necessary tests are reapplied starting with the last one performed. Usually only one test will be required. Face

20. Motion Planning • Discretization a factor in collision-avoidance. Location 2 Location 1

21. Conclusion • Algorithm calculates closest features on 2 convex polyhedra • Algorithm is relatively simple • Algorithm is efficient • Runs in constant time once initialized • Runs linearly in proportion to number of vertices when initializing • Algorithm is complete

22. Data Structures and Concavity • Polyhedron: faces, edges, vertices, position, and orientation • Face: outward normal, distance from origin, vertices, edges, and coboundary • Edge: head, tail, right face, and left face • Vertex: x, y, z, and coboundary • Polyhedron must be convex • A concave polyhedron must be converted to multiple convex polyhedra • May effect efficiency (quadratic)

23. Algorithm: Edge-Face • Determine if they are parallel • If parallel, they are closest features if both: • The edge passes through the applicability prism created by the face • The normal of the face being evaluated is between the normals of each face bounding the edge being evaluated

24. Algorithm: Edge-Face • If not parallel • One of the vertices of the edge is closer to the face if it meets the point-face applicability criterion and the edge points into the face, and this pair is returned • If this criterion is not met, the edge-edge component of the algorithm is applied to the edge bounding the face that is closest to the edge and the edge under examination.

25. Algorithm: Edge-Face Applicability Prism Edge (Parallel to Face)

26. Algorithm: Edge-Face • Test Failure • Vertex closer to face than edge Edge (Not Parallel to Face)

27. Algorithm: Edge-FaceTest Failure • Test Failure • Edge closer to edge than face Edge (Not Parallel to Face)

28. Algorithm: Edge-Edge • Determine closest points on the edges • Apply point-edge applicability criteria to each pair • If either fails, will return new feature pair based on failed test

29. Algorithm: Edge-Edge Nearest Points

30. Algorithm: Face-Face • Determine if faces are parallel • If parallel • Check to see if overlapping, if so, they are closest features • If not parallel or not overlapping • Return first face and nearest edge of second face to first face

31. Algorithm: Face-Face Parallel Faces