Computational Geometry II

1. Computational Geometry II Brian Chen Rice University Computer Science

2. uhh. . . Recap, Please? • Arrangements are sets of Lines in the plane, in general position • Each pair intersects in exactly one point • Can be described by Euclidean Coordinates (y = mx + b and all that)

3. Purposes and Topics • Present a mathematical framework for discussing the topics in the remainder of the course • Duality, Convex Hulls, Envelopes, Voronoi Diagrams, and Delauney Triangulations

4. Duals, duality and dual spaces • Lines can be uniquely designated by the equations which describe them • Example: y = 3x + 4 is uniquely described by the values 3 and 4. • Points can be uniquely designated by their coordinates on the plane • Example: (3, 4) is obviously the only point at x = 3, y = 4.

5. Who cares, Brian? • Since lines and points are uniquely determined by two values, that means that there exists a bijection between them. • A Bijection is a mapping which correlates one member of one set with exactly one member of the other set, and also correlates one member of the other set with one of the first.

6. A new point of view • This bijection means that we can now transform problems about points in space into problems about lines, and vice versa. • It also leads to some very interesting properties on the plane and the dual plane: • definitions: dual of an arrangement, dual of a constellation, dual space.

7. Duality Mappings • There isnt just one mapping between points and lines. • As long as the mapping is one-one and onto (e.g. a bijection) then the mapping is a duality mapping. (Examples) • (note: we ignore pairs with identical x-values, because this results in parallel lines)

8. What about sets of points? • Sets of points in dual space are very interesting: • a Segment is a set of points along a line. Since the segment doesn’t have infinite slope, all the points have lines in the dual space which intersect. In fact; they all intersect at the same point. (Proof)

9. What about more than just 2 points? • A line segment is composed of an infinite number of collinear points • In the dual space, it looks like two wedges

10. Convex Hulls • An object which is Convex is an object where for any pair of points p and q inside the object, the line segment pq is entirely contained within the object. • The Convex Hull of a set A of points is the smallest convex set of points which contain all the points in A.

11. Pictures: (on board)

12. Why do we want to know? • Knowing that we are operating on convex sets lets us write fast collision detection algorithms, and we can use a convex shape to approximate many complicated objects • Convex Hulls also let us do fast visibility calculations (draw pic)

13. Interesting Duality: Envelopes • The Upper Envelope is the intersection of all upper half planes of an arrangement of lines. • The Lower Envelope is the intersection of all lower half planes of the lines.

14. 3d Generalization • Points in 3d turn into planes in the dual space • segments in 3d turn into X-prisms in the dual space (drawing)

15. Putting ideas together • Now we can exactly correlate constellations of points with arrangements of lines • Constellations have the interesting property of Convex Hulls • Lines have the interesting property of Envelopes • *cough*CORRELATION*cough*

16. Are Hulls and Envelopes related? • We have a notion of boxing in points • We have a notion of boxing in lines • Because of duals, EVERY point is a line in dual space, and every line in dual space is a point.

17. The dual of an arrangement • Is a set of points in cartesian space. • Doesn’t APPEAR to have anything to do with the set of lines. (hint: I’m lying)

18. Tell us already! • Upper Convex Hulls Correspond to lower Envelopes, and Lower Convex Hulls Correspond to upper Envelopes.

19. Section II: Voronoi Diagrams • What are Voronoi Diagrams? • A Voronoi Diagram Is a partition of a space defined for an individual constellation of points in the space.

20. Defintion • The space is partitioned into cells, such that a point on the plane is in a cell iff the constellation point in the cell is the closest constellation point. • A point is on a “Boundary” if the point is equidistant between two or more contellation points.

21. Pictures, Pictures, Pictures • Simple picture with 2 points • More complicated picture with 3 points • Definition of general position (no three on a line, no four on a circle <all points on the circle are equidistant from the center>) • General picture

22. Interesting Properties • definition: Voronoi vertex, Voronoi edge, Voronoi boundary. • A point p is a Voronoi vertex iff the largest empty circle C(p) around p contains at least three Voronoi sites. • A point p is on the Voronoi boundary iff the largest empty circle surrounding it contains exactly two sites.

23. Delaunay Triangulations • Are the border connectivity graphs of Voronoi diagrams • Picture

24. Properties • Delaunay Graphs are always planar. • If T is a triangulation of points P, then T is Delaunay iff the circumcircle of any triangle contains no other point of P. • Definition of “Legal Triangulation” (see next slide)

25. General Triangulation • aribitrarily making triangles everywhere. • Triangulations used to describe height maps from point samples. • Need triangulations which don’t do stupid things

26. Last Property • Delauney Triangulations are the only legal triangulation • If a Triangulation T is Delaunay iff it is legal. • Delauney Triangulations are the “best” Triangulations

27. Next Topic: Collision detection • Begin using the structures described this week and last week.