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
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.
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.
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
Next: The Main Idea: Level Sets
The Pisces Home Page
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Last modified: Mon Nov 27 15:22:08 1995
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The Geometry Center,
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