Soliton Explorer: A Problem Solving Environment for Soliton Geometry

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Soliton Explorer:
A Problem Solving Environment for Soliton Geometry
Santiago V Lombeyda, Raji Venkatesan, Bruce Char, Tom Hewett, Y N Lakshman, Jeremy Johnson, Ron Perline
Department of Mathematics and Computer Science, Drexel University

The Soliton Explorer was developed as the first phase of the development of a small-scale computer-based working environment for the exploration of soliton geometry. Soliton theory refers to an expensive machinery for exat solution of certain partial differential equations; such equations play am improtant role in diverse areas of applied mathematics. Soliton Geomety is the the study of soliton equations governing the geomtry of curves and surfaces is space, particularly curve and surface motion.

	Rather than decelove appropiate software tools from scratch, the philosophy of this software development has been to focus on the integration of existing tools. Soliton Explorer integrates sybolic computation (Maple), numerical computation (NAG Library), and sophisticated visualization tools (NAG's IRIS Explorer). Demonstrations of Soliton Explorer include two-dimensional curves (and their curvatures), and soliton surfaces in sapce, using various rendering options developed as modules within IRIS Explorer. Such options include the ability to pinpoint a specific point on the surface, follow a curvature line in the parametrized surface, or color the surface according to different changes in the parametric variables.

Rather than decelove appropiate software tools from scratch, the philosophy of this software development has been to focus on the integration of existing tools. Soliton Explorer integrates sybolic computation (Maple), numerical computation (NAG Library), and sophisticated visualization tools (NAG's IRIS Explorer).

Demonstrations of Soliton Explorer include two-dimensional curves (and their curvatures), and soliton surfaces in sapce, using various rendering options developed as modules within IRIS Explorer. Such options include the ability to pinpoint a specific point on the surface, follow a curvature line in the parametrized surface, or color the surface according to different changes in the parametric variables.

Rendering options included:

Even Color Shade would simply output a surface whose color is even throughout.
Gradient would actually fade the color of the surface from the color in a start color to an end color. The color will fade either in relation to either parametrization variable, as chosen by the user.
Line would highlight a specific line (all points where one of the parametric variables have a constant value), specified by the user.
Point, worked the same colorwise as line, but will mark the given point in parametric coords with a crosshair drawn on the surface itself.
Cyclic would rotate between two colors with a different frequency in Red, Green, and Blue as specified by user.
Random faded a grandient using two colors using a random factor to add gitter to the colors.

The resulting surfaces where quite interesting aesthetically as well as mathematically; with the added rendering functionality really proving to be as useful as it was novel.

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page by santiago v lombeyda