MOTIVATING  GEOMETRY THROUGH
COMPUTATION  AND  VISUALIZATION
funded by NSF-CCLI grant DUE-0126687

The previous summers have concentrated on creating applets for creating Bezier curves (in the plane and in space) and rectangular Bezier patches in space (see applets available at Project Home Page). This summer, we want to extend the construction to triangular Bezier patches and other surface creation methods, such as polyhedral surfaces and subdivision algorithms. In addition, it is desired to add functionality to the applets in order to allow saving of data and the exporting of the data points.

The primary goal is the construction of applets to construct triangular Bezier patches and to construct subdivision surfaces. This will involve the creation and use of an interactive data structure for the creation and manipulation of a triangular grid and the input of polyhedral structures for the use of subdivision surfaces. A triangular grid is a structure as in the diagram below, where each dot is a point in three space and each line is signifies the adjacency of the points.

A polyhedral structure is a generalization of the platonic solids of a tetrahedron, a cube, a hexahedron, etc. Each structure is described topologically as a set of points, a set of edges, and a set of faces. A subdivision algorithm is a method for constructing a new polyhedral surface via a simple rule. Consider the diagrams below in which a cube generates a new surface by cutting corners.