Using Elimination to Describe Cartesian Ovals

• Main Author: Beverlin, Lucas P
• Other Authors: James Madden
• Format: Others
• Language: en
• Published: LSU 2006
• Subjects: Mathematics
• Online Access: http://etd.lsu.edu/docs/available/etd-06122006-192002/

Cartesian ovals are curves in the plane that have been studied for hundreds of years. A Cartesian oval is the set of points whose distances from two fixed points called foci satisfy the property that a linear combination of these distances is a fixed constant. These ovals are a special case of what we call Maxwell curves. A Maxwell curve is the set of points with the property that a specific weighted sum of the distances to n foci is constant. We shall describe these curves geometrically. We will then examine Maxwell curves with two foci and a special case with three foci by deriving a system of equations that describe each of them. Since their solution spaces have too many dimensions, we will eliminate all but two variables from these systems in order to study the curves in xy-space. We will show how to do this first by hand. Then, after some background from algebraic geometry, we will discuss two other methods of eliminating variables, Groebner bases and resultants. Finally we will find the same elimination polynomials with these two methods and study them.