__Electronic Proceedings
of the 12 ^{th} 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.