Design and Application of a Cubic Hermite Spline for Solving Problems Related to Computational Geometry

Abstract

In this thesis, we present an algorithm of a cubic Hennite spline interpolation (CHSI) and apply it for constructing smooth curves and surfaces. It is a fundamental problem in computer aided geometric design (CAGD). As a result, this approach involves the extraction of the geometrical information in the area of computational geometry. As expected, such study can be extended in other branches of computer science.

In this algorithm, a well-defined parametric curve is designed for representation of a set of discrete data. With the help of a piecewise cubic spline, this avoids occurrence of unphysical oscillations like the so-called wiggly interpolation. Mathematically, it preserves the smoothness of the surface and maintains the continuity with respect to first and second derivatives, respectively. Additionally, the introduction of the Hennite function ensures simplicity and requirements with small memory and less computational time. This leads to fast computations. An essential feature in our algorithm lies that it can directly evaluate the tangent vectors, since such estimate sometimes becomes cumbersome when they tend to infinity. In particular, this is desirable for design of the geometry in the region of interest.

The algorithm has been validated by the problems often encountered in computational geometry. Four types of example demonstrate that our method is robust and simple due to the well-suited function specified; consequently, this provides the possibility for application of this algorithm in broad areas.