Computational Geometry II Brian Chen Rice University Computer Science

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

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

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

Who cares, Brian? n Since lines and points are uniquely determined by two values, that means that there exists a bijection between them. n 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.

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

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

What about sets of points? n Sets of points in dual space are very interesting: n 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)

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

Convex Hulls n 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. n The Convex Hull of a set A of points is the smallest convex set of points which contain all the points in A.

Pictures: (on board)

Why do we want to know? n 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 n Convex Hulls also let us do fast visibility calculations (draw pic)

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

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

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

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

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

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

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

Defintion n 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. n A point is on a “Boundary” if the point is equidistant between two or more contellation points.

Pictures, Pictures, Pictures n Simple picture with 2 points n More complicated picture with 3 points n Definition of general position (no three on a line, no four on a circle ) n General picture

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

Delaunay Triangulations n Are the border connectivity graphs of Voronoi diagrams n Picture

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

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

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

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