Electronic Proceedings of the 12th Asian Technology Conference in Mathematics



Abstract for 13038

From String Art to Caustic Curves: Envelopes in Symbolic Geometry

Authors: Philip Todd

Affiliations: Saltire Software



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 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.