Energy Optimization Methods For Hermite Interpolation

In this paper, quartic and quintic bezier curves to interpolate the given data points are constructed after Geometric hermite interpolation. On each sub-interval, there is one freedom degree in the quartic bezier curve segment which can be used to increase the flexibility of design and construction the control the shape of the curves.Similarly, on each sub-interval, there are two freedom degrees in the quintic bezier curve segment can be used to adjust the shape of the segment. And, the freedom degrees are determined by minimizing the bending energy or the tensile energy of the spline curve. After that, the paper analyzes the error accuracy of the interpolation. As follows, the paper constructs the tensor product bezier surfaces whose boundary lines are quintic bezier curves to interpolate the given data corner points. And there are some freedom inner control points on the surface, which are determined by minimizing the squared mean curvature energy of the surface.The first chapter is the introduction part, simply introducing the origin, development, application of computer aided geometric design. Then this chapter simply introduces the application, research status of energy optimization on computer aided geometric design.The second chapter introduces the professional knowledge which is related to the following sections. The Bernstein basis functions, bezier curve, cubic, quartic and quintic bezier curve interlolation, tensor product bezier surfaces are introduced in this chapter.In the third chapter, the problem of constructing an adjustable quartic bezier curve to interpolate a set of given data points is discussed. Constructing interpolation curve by quartic bezier curves offers additional freedom degrees which are determined by minimizing the bending energy or the tensile energy of the curve. Then the chapter analyze the error accuracy of the interpolation. As follows, the chapter selected the right first derivative of intermediate interpolation points to construct c2interpolation curve by quartic bezier curve.In the fourth chapter, the problem of constructing an adjustable quintic bezier curve to interpolate a set of given data points is discussed. Constructing interpolation curve by quartic bezier curves offers two freedom degrees which are determined by minimizing the bending energy or the tensile energy of the curve. After that chapter analyze the error accuracy of the interpolation. As follows, the chapter selected the right first derivative of intermediate interpolation points to construct c2interpolation curve by quintic bezier curves.In the last chapter, tensor product bezier surfaces whose boundary lines are cubic or quantic bezier curves to interpolate the given data corner points are constructed. And the freedom points of the quanric bezier curves are decided by minimizing the bending energy function or tensile energy function. There are some pending inner control points in the tensor product bezier surfaces. By minimization of the squared mean curvature energy function, the inner control points are decided by boundary curves.