Spring 2021

Undergraduate researchers: Sarah Bruce, Maxwell Nakos, John Teague, Anabel T. To

Mentors: Florian Stecker, Neža Žager Korenjak
Faculty advisor: Jeff Danciger

The goal of this project is to tour the fundamentals of projective geometry, as well as to explore some of its relevant theorems experimentally. Additionally, we hoped to create intuitive visualizations for these concepts. In this project, we answered a number of interesting questions: given a triangle in 3D space, a fixed point-light source, and a plane onto which the triangle's shadow is projected, what different triangles can we create with the shadow? If we were to mark a point on the original triangle, can we move its projection on the triangle's image on the plane? For some arbitrary convex polygon, can we use projective transformations to maximize its "roundness", as well as manipulate a point contained within the polygon to a given position? The last question is meant to informally explore Benzécri's compactness theorem on \(\mathbb{R}\mathbb{P}^2\) and to make exact its claim in this context. In order to answer these questions, we developed models of each scenario in Mathematica and p5.js, a JavaScript library for 2D and 3D graphics on the web, which can be explored on this website.