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Pisces

A Platform for Implicit Surface Computation and the Exploration of Singularities

Pisces is a project that is investigating numerical solutions of the following mathematical problem: Given a function from a large dimensional space into a smaller dimensional space, find the set of points mapped onto a particular point in the range. For a simple example, note that the function that maps (x,y) to x^2 + y^2 -1 maps the unit circle to 0, so that the function and the value 0 implicitly define the unit circle. One result of the Pisces project is a software package that provides a variety of state-of-the-art algorithms for computing implicitly-defined curves and surfaces. This software is under development.

The goal of this project is to create software that contains

A set of known continuation algorithms for computing implicitly-defined curves and surfaces. This will include both piecewise-linear methods and also predictor-corrector methods;
A command language for batch processing and high level programming;
A graphical interface for choosing and controlling algorithms;
New algorithms for computing the structure of singular surfaces.

In addition to being a useful tool to researchers and educators who need to compute implicitly-defined surfaces, the Pisces program will serve as an environment that can be used to study the way that algorithms fail and to develop algorithms that perform better.

Pisces will be potentially useful to

Researchers in dynamics (bifurcation surfaces, invariant manifolds), differential geometry (surfaces, curves, singularities), algebraic geometry (real and complex algebraic varieties), differential equations (DEs on manifolds, generation of finite element meshes), mathematical physics (equipotentials, caustics), mathematical biology (nonlinear models in physiology, epidemiology, neurology), etc.;
Educators in ODEs, dynamics (level sets, elementary bifurcations, normal forms), differential geometry (surfaces, curves, envelopes), algebraic geometry (real algebraic varieties).
In addition, we foresee industrial applications in, for example, Computer-Aided Geometric Design (CAGD), and the study of solution manifolds for families of nonlinear equations that include parameters.

In the past year, the Center has sponsored visitors to come to the Center to advise us on the Pisces project. Currently, Pisces has a well-developed user-interface, can compute level curves in the plane (both algebraic varieties and level sets of non-algebraic functions), and implicitly-defined surfaces in three-space.

Pisces can animate a family of level sets as a parameter is varied, and can locate singularities in planar curves. Pisces has five picewise-linear algorithms; two predictor-corrector algorithms are currently under development. This second class of algorithms will allow us to trace curves and surfaces that are imbedded in Euclidean space of arbitrary dimension.

We are currently studying the behavior of algorithms in the neighborhood of singular curves; we are developing algorithms that detect and locally resolve the geometric structure of singular curves in a neighborhood of the singularities. We do not know of any algorithms that can carry out this task for level sets of generic functions. We hope to extend this study to provide users with the first general-purpose algorithm that correctly identifies and depicts singular surfaces.

We are working on a mechanism by which users may derive new equations from existing equations. For example, if the user has specified a function, Pisces should be able to derive a new function which is the determinant of the Jacobian of the original function. The derivable functions should include those functions of interest to the dynamical systems community.

In the fall of 1995 we will begin distributing Pisces to test-sites outside of the University of Minnesota. We are preparing documentation to assist this process.

In the future we will continue our efforts to provide users with high-level mechanisms to compute and graphically display level curves and surfaces by means of a Pisces command language. We will concentrate on communication between algorithms, so that Pisces becomes an extensible platform for computing and visualizing implicitly-defined curves and surfaces.

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