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

            Keywords:

 

            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.