# A Cupful of Limacons

### Michael L. Treuden

Department of Mathematics and Computing
University of Wisconsin-Stevens Point
Stevens Point, WI 54481-3897
USA
Phone: (715) 346-4607

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## ABSTRACT

Have you ever examined the light patterns on the bottom of a cup or bowl with reflective sides and wondered what shapes you see? If a light source is placed somewhat off center, rather than directly above, you might conjecture that the pattern resembles a cardioid. The purpose of this paper is to discuss how students may furnish some answers by using elementary vector algebra and orthogonal projections with the aid of a computer algebra system (CAS). For instance, if the cup is a simple right circular cylinder, then it will become readily apparent that the image is generated by infinitely many limacons.

We suppose the unit disk x^2+y^2<=1 is the bottom of the cup and the sides are formed by revolving the graph of a smooth function x=f(z) with f(0)=1 and f'(z)>=0 about the z-axis. For example, f(z)=1 when the cup is a cylinder. A CAS such as Mathematica, for example, makes it easy to model the reflected light rays formally, without even specifying exactly what f(z) is. By then making particular choices for f(z), we may use the plotting capabilities a CAS provides to assess the accuracy of the model by comparing the graphs with what is seen in reality.

Keyword(s): computer algebra systems, Mathematica, geometry