Visualizing surfaces in Euclidean 3-space with Visumath

Abstract

As is well-known, surfaces in a 3-dimensional Euclidean Space can be represented locally by means of several types of equations. Though these are mathematically equivalent, the choice of representation can make a difference when it comes to visualizing the surface. More precisely, some representations may produce artifacts in the visualization that are merely a consequence of the visualization process itself. Sometimes, by rewriting equations or by switching to another representation, these can be avoided. In this paper, this phenomenon is explained and demonstrated by means of the new version of the visualization software VisuMath. Moreover, the new capabilities of the software and the geometry behind it are explained. These include the possibility of dynamically changing the shape of surfaces by altering parameters in the representation. This feature can be useful in examining the precise role of parameters in the equation(s) representing the surface.