**P**latform- Pisces strives to provide a computational environment for two categories of users. By far the most common user is the researcher or educator who wants to use Pisces to investigate implicitly-defined curves or surfaces that arise in his or her own work. For this user, Pisces is a tool. The other set of users includes researchers who are developing algorithms that compute implicitly-defined objects. For them, Pisces provides graphics, memory management, a suite of test functions, and other algorithms with which to compare their results.
**I**mplicit- Pisces computes curves and surfaces that are defined
as the set of points that satisfy a given equation. For example,
the set of points that satisfy the equation
*x^2+y^2=1*is a circle with radius*1*. We say that the circle is defined implicitly by this equation. **S**urfaces and**C**urves- Pisces has algorithms to compute one- and two-dimensional objects, although these objects may exist in an arbitrary-dimensional ambient space. For example, Pisces can compute curves in the plane, curves in three-space, and so on. Although it is possible to compute implicitly-defined objects of dimension greater than two, these objects are currently very challenging to visualize and understand.
**E**xploration- Pisces is an
*interactive*tool. It is designed to provide users with the tools to investigate implicitly-defined structures and to experiment with a variety of methods for computing these structures. **S**ingularities- When a curve or surface intersects itself, or when it contains a "pinch point" (for example, a cusp), we say that the surface has a singularity. Numerical algorithms designed to compute implicitly-defined objects typically have great difficulty in the neighborhood of a singularity. A focus of the Pisces Project is to pay particular attention to singularities and to develop algorithms that do not fail near singular points.

Comments to: pisces@geom.umn.edu

Last modified: Mon Nov 27 14:16:32 1995

Copyright © 1995 by The Geometry Center, all rights reserved.