NURBS Beyond Fermats Last Theorem

Team: 75


Area of Science: Mathematics and Computer Graphics

Interim: Problem Definition:
For centuries man has used the Pythagorean Theorem
(a*a + b*b = c*c ) to explain the relationship between the sides of any right triangle. The Pythagorean Theorem has many applications in real world problems, but the potential ability for this equation to define any triangle remains unexplored. By replacing the exponential constant(2) with a variable, c^x = a^x + b^x us formed. Since x is in every term of the equation, solving for x analytically is impossibly difficult. Our group intends to solve this problem using computationally intensive methods that will provide an extremely accurate result.

Problem Solution:
The goal of this project is to create a precise model of the modified Pythagorean Theorem using a combination of Newtons method and a 3D medium with a variable level of detail. To find x on a given triangle, we will use Newtons method for approximating solutions of an equation. After finding a sufficient amount of x values we will use NURBS (Non-Uniform, [Non] Rational Bezier Splines) which are mathematically defined curves that are capable of mimicking nearly any shape and have a variable level of detail. Before creating the NURBS matrix, a large set of values will be calculated for all possible triangles. After finding sample values for x, the NURBS curves will then be plotted accordingly. Another goal of this project is to test the accuracy of the NURBS surface created by the matrix. To accomplish this our group will create two surfaces, one with a high level of detail, and the other with a low level of detail. By comparing the x values of the two surfaces, we will be able to determine if low density NURBS surfaces can accurately hold the information without using as much memory and processing power.

Progress so Far:
Currently our group is learning how to program in C++ with the OpenGL API, and is developing various background routines for this problem. Also, we have found an analytical solution for isosceles triangles that will be implemented for testing purposes. Also, we have discovered that an equilateral triangle has no solution for x.

Expected Results:
After producing the graph, our group expects to see a discernible surface with at least one asymptote. There is a possibility that this equation may generate a fractal pattern which would complicate the NURBS matrix. Also, we predict that the low density NURBS matrix will accurately store the information so less storage space and processing power will be needed. The potential of the NURBS matrix to hold and display multi-variable equations without large processing and memory requirements will provide researching scientists with a different and possibly faster way of computing results. Also, the Pythagorean Theorem has been in the eyes of the math community since the mystery of Fermats Last Theorem left mathematicians curious and puzzled for almost 400 years. This project expands on Fermats Last Theorem, and if successful, will show another relationship between the sides of triangles.

Computer Graphics: Principles and Practice??, James D. Foley, Andries van Dam, Steven K. Feiner, and John F. Hughes, Addison-Wesley Publishing Company (1990,1996)
Calculus of a Single Variable: Seventh Edition??, Ron Larson, Robert P. Hostetler, Bruce H. Edwards, and David E. Heyd, Houghton Mifflin Company (2002)

Team Members:

  Kevin Christeson
  Brett Beckett
  Meghan Scott
  Kyle Fitzpatrick

Sponsoring Teacher: Donald Downs