Subdivision, and refinement of non-uniform rational B-spline curves and surfaces in 3-D
Discussed in this paper are several mathematical spline formulations and a history of splines, Bezier's cubic spline, and cubic B- splines. This includes uniform rational B-splines and non-uniform rational B-splines, their construction, pros and cons of the different representations, and subdivision and deformation of curves and surfaces. A 3-D application using NURBS technology was developed to demonstrate the deforming and rendering of objects created using the NURBS representation. The application provides the capability of visualizing what happens to a curve or surface patch when a knot or knot vector and the associated control point or set(s) of control points are inserted or moved. Subdivision and refinement of parametric objects is accomplished using knot insertion and interactive picking. The approach to understanding the more complex and capable parametric curves and surfaces (NURBS) is to start with the simpler cubic spline basis created by Pierre Bezier and an example curve. It should be noted that spline functions can be of any degree, but the paper uses spline functions of order 4 and degree 3, hence the term cubic spline is prominent throughout the paper. Next the concepts are extended to Uniform Rational B-splines. Then the final extension of the theory goes as far as the Non-Uniform Rational B-spline. Last of all a connection between Bezier, URBS, and NURBS is made, for the purpose of ray tracing and shading the final model. The model consists of three-dimensional points and one-dimensional intervals. The 3-D data produced from the model is displayed on the computer monitor using the software developed for the project.
Geometry and Topology, Harmonic Analysis and Representation, Other Mathematics
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