James Davis

Usage Cases for a Star-like Envelope of a Family of Curves

Started July 18, 2023

Abstract or project description

By assigning a value n and connecting straight lines from (n, 0) to (0, 1), (n - 1, 0) to (0, 2), and so on until the final line from (1, 0) to (0, n), a curve-like shape is made. Reflecting this curve across y = -x + n and reflecting the new curve across the axes creates a shape that resembles a circle. Our intent is to analyze how adding more line segments to the shape and increasing its size will affect the shape's behavior, and we plan on analyzing the end behavior of this shape as n approaches infinity. Through creating a function that measures the distance of each point on the shape from the origin, we discovered that the point farthest from the origin always lies on the line y = x. We plan to analyze the growth of the distance of the farthest point from the origin in comparison to points of the shape on the axes to determine the end behavior of the shape as n approaches infinity. In analyzing the end behavior of this shape, we hope to discover other properties of this function that may have some potential usages in mathematics or other fields.