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.

Cubic

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.

Torus

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.

Surface

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.

Bifurcation

Next: The Main Idea: Level Sets

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