**Team:** 75

**School:** ONATE HIGH

**Area of Science:** Mathematics and Computer Graphics

**Abstract:** 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 has not been explored. By replacing the exponential constant(2) with a variable, c^x = a^x + b^x is 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 numerical methods that will provide an extremely accurate result.

The goal of this project is to create an 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 Newton's Method for approximating solutions of an equation. After finding a sufficient amount of x values NURBS (Non-Uniform, [Non] Rational Bezier Splines) are mathematically defined curves that are capable of mimicking nearly any shape and have a variable level of detail which is ideal for our problem. 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 array. 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.

**Team Members:**

Kevin Christeson

Brett Beckett

Meghan Scott

**Sponsoring Teacher:** Donald Downs