||From String Art to Caustic Curves: Envelopes in Symbolic Geometry
In this paper, we use the symbolic geometry program Geometry Expressions to analyze three problems involving envelope curves. First we examine the envelopes of families of lines passing through points which are equally spaced on a pair of line segments. We use a combination of symbolic geometry and algebra to develop an expression for the area of the void in a popular string art figure consisting of 3 parabolas inscribed in a triangle. We use an envelope approach to reduce a popular calculus problem — that of finding the longest ladder which fits around an asymmetric corner — to an algebra problem which is readily solved using CAS. Finally we study the caustic curves generated by reflecting a point light source in a shiny cylinder. We analyze these both experimentally and theoretically, and focus on determining the parametric and Cartesian locations of the cusps. These examples illustrate how symbolic geometry technology can be used to make mathematics fun, accessible and challenging.