Math and Computational Geometry Problems From
The MapTools Company

This objective of this problem is to prove the Bertrand-Chebyshev theorem, which says that there is always at least one prime between any two integers, n and 2n. The material provided explores a proof outline originally put forth by Paul Erdos in 1934. Erdos's proof greatly simplified the rather complicated proof by Chebyshev that was then available.

The Bertrand-Chebyshev Theorem problem is available here

2. Intersections of Bivariate Quadratics.

This problem shows how to calculate the intersection(s) (if any) of a pair of planar bivariate quadratics (a planar bivariate quadratic is an ellipse, parabola or hyperbola defined in a plane). The paper provided takes a student through the process of defining the symbolic discriminant of the linear combination of the two quadratics, showing that the zeros of this quantity define a cubic equation from which the coordiates of any intersections of the quadratics can be obtained. The material has example calculations and displays of problems presented.

The Bivariate Quadratics problem is available here

3. Skeleton of a simple polygon.

At left is an example of a skeletion inside a simple polygon. Skeletons (also known as medial axes) are useful in a variety of graphics problems. David Eppstein has collected a number of applications for this geometric concept here. Points along the path of the skeleton define the "spine" of a polygon - the points furthest away from the outline of the polygon.

The heart of this project is a new, straightforward algorithm for generating the skeleton of a simple polygon. First, a student is shown how to calculate the skeleton of a convex polygon, then is introduced to the generalization to extend the method to a polygon with reflex angles. Also available with this project are computer implementations of the algorithm in both Delphi and C++. (Free versions of Delphi and C++ are available from Embarcadero Technologies.)

A paper describing this problem is available here.

If you would like more information about these problems or have any comments send me an EMail containing your name and affiliation.