ICS 164 - Computational Geometry Homework 6, 50 Points
Due: Wednesday, November 23, 2005

  1. 10 points. Describe a data structure that allows you to test whether a query point lies inside a 3-D convex polytope with n vertices. What is the running time of performing a query?
  2. 10 points. A simple polytope is a region in 3-D enclosed by a triangulated surface so that there are no holes or handles. Describe a linear-time method for determining if a query point is inside a simple polytope in 3-D space.
  3. 10 points. Design an efficient algorithm for the following problem: Given n lines and n points in the plane, determine for each input point the nearest point of intersection between two of the input lines. What is the running time of your algorithm?
  4. 10 points. Describe in detail how to construct a 3-D convex hull of a set of n points using divide-and-conquer.
  5. 10 points. Let P be a convex polytope with $n$ vertices, stored in a 3-D version of the double-connected edge list data structure. Describe how to compute the shadow of P: given a light source at infinity in a certain direction, the shadow is the polygon that is the projection of P onto a plane perpendicular to that direction.