Research Of The Interval Implicitization Of Rational Curves And Surfaces

Rational curves and rational surfaces, which are a class of important parametric curves and surfaces, are extensive applied in CAD/CAM. However, due to the advantages of implicit curves and surfaces which the parametric curves and surfaces don't have, sometime we need to use the implicit form of a rational curves and surfaces.From classic algebraic geometry, any parametric curve/surface can be converted to implicit form. Previously, a lot of research has focused on the problem on how to find the accurate implicit representations of rational curves and surfaces. However, the practical usefulness of accurate implicitization is hampered by the complex representations of the implicit equations of rational curves and surfaces, so finding approximate implicit representations becomes a practical need.The numerical gap between the parametric equation and the approximate implicit representation may result in problems such as the unreliability in geometrical computation. Base on the practical need, the concepts interval implicitization of rational curves and surfaces are presented. Interval curves and interval surfaces are application and extension of interval analysis method in CAGD, which is important tool for error analysis in numerical analysis field.In this paper, we survey the development of implicitization of rational curves and surfaces and the development of the interval analysis method and state the knowledge of interval curves and surfaces in the first chapter. In the second chapter, interval implicitization of rational curves is introduced. In the third chapter, interval implicitization of rational surfaces is introduced. Using an optimal method, we find an interval algebraic surface with lower degree which bounds a given parametric rational bezier surface such that some objective function involving the width and the tention term of the interval algebraic surface is minimized.