# Problem Set 4

#### Model of a convex polytope

• Construct a 3d model of a convex polyhedron (your choice, but please do not all do the cube or tetrahedron!).

#### Dual of Convex Hull Algorithms

This assignment is a bit special: it is not just about solving a problem or two, but also about presenting your solution. It is meant to give you some practice, useful for the final presentation of your project at the end of the semester.

#### Topics

In class, I explained the point/line duality and we've seen that questions about configurations of points dualize to questions about line arrangements. The purpose of this homework assignment is to have each of you "implement" these ideas by dualizing one of the 2d convex hulls algorithms we did in class, to an algorithm computing the upper envelope of a set of lines.

 Nr. Convex hull algorithm Textbook ref. page Who 1 Naive algorithms: non-extreme points, extreme edges. p. 66 Ellen, Courtney 2 Gift Wrapping p. 68 Veronica, Jesse 3 Quickhull p.69 Tracy 4 Graham's scan p.72 Beenish 5 Incremental algorithm p. 88 Elif 6 Divide and Conquer p.91 Naomi

#### What to do

Translate the corresponding algorithm into an algorithm for computing the upper envelope of a set of lines. Present the algorithm, in the dual setting, in a short 10 minute presentation on Tuesday, April 4. The presentation should include a very short description of the problem, a dictionaru of dual concept used for dualizing the algorithm, and a presentation of the algorithm in parallel with the original primal one.

Since this is practice for the final presentation, you can use whatever form of presentation you feel more comfortable with. It can be done using just plain blackboard and chalk, or using an overhead projector (just let me know ahead of time so that I bring one to class), or from computer slides (e.g., html files, Powerpoint - whatever). You can include an applet, if you find one on the web illustrating your algorithm.

I also ask you to write the main ideas in the form of an HTML file, and draw the pictures in Cinderella. Link it from your web page. If two students are assigned the same algorithm, they both should do it, separately: this is meant as individual work.