# Overview: A Pisces Slideshow

Pisces is a
Platform for Implicit Surfaces and Curves and the
Exploration of Singularities.
The Pisces software computes curves
and surfaces that are defined implicitly as the level set of a
function.

## LEVEL CURVES

The next image shows the graph of a cubic polynomial of two variables
being sliced by a plane. The plane is decomposed according to one
algorithm's attempt to compute the intersection of the graph with the
plane. Graphically animating an algorithm as it runs gives
considerable insight into the algorithm's strengths and weaknesses.

## SINGULARITIES

Sometimes level sets are not manifolds, but instead contain ``singular
points'' at which the tangent space is not well-defined. Most
algorithms fail in a neighborhood of a singular points; the Pisces
Project is creating algorithms that perform better.

## LEVEL SURFACES

The Pisces Project will provide users with a general-purpose algorithm
for computing two-dimensional surfaces in arbitrary codimension, and
is investigating how to detect and compute singular structures
on level surfaces.

## BIFURCATIONS IN DYNAMICAL SYSTEMS

Pisces contains algorithms that compute bifurcation curves and
surfaces for dynamical systems. Shown is a
fixed-point surface and saddle-node curves for a mapping of the circle
to itself.

**Next: **The Main Idea: Level Sets

*The Pisces Home Page*

Comments to: pisces@geom.umn.edu

Last modified: Mon Nov 27 15:22:08 1995

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