The Pisces Acronym

Platform

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.

Implicit

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.

Surfaces and Curves

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.

Exploration

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.

Singularities

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.