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.