On The Theory And Application Of Pythagorean-hodograph Curve

PH curve, namely Pythagorean-Hodograph curve, plays an important role in CAGD, for it possess positive properties, for example, the arc length of a PH curve can be computed precisely by simply evaluating a polynomial (avoiding the need for numerical integration), and the fixed-distance offsets to PH curves, used as tool paths in numerical-control (NC) machining, are rational curves that have exact representations in CAD systems. Also, in the interpolation of discrete data, PH curves tend to produce "fairer" loci (with more even curvature distributions) than ordinary polynomial curves, and they admit exact evaluation of their bending energy.PH curves not only offer unique computational advantages over "ordinary" polynomial parametric curves in CAD/CAM applications, but also retain full compatibility with standard Bezier/B-spline representations. So study on PH curves is very meaningful. However, research on PH curves is extensive and there exist a lot of work to do, such as distinguishing and construction of PH curves. The theory and application of PH curves are discussed in this dissertation, major works are as follows:Firstly, the problem of interpolating given first-order Hermitian data (end points and derivatives) by the couple of planar PH cubic curves has four distinct formal solutions. Ordinarily, only one of these interpolants is of acceptable shape. Previous interpolation algorithms have relied on explicitly constructing all four solutions. By invoking a suitable measure of shape, e.g., the absolute rotation index or elastic bending energy, a "good" interpolant is selected. A new means to differentiate among the solutions is introduced here, namely, the winding number of the closed loopformed by a union of the hodographs of the couple of PH cubic and of the unique "ordinary" cubic interpolant. It also shows that, for "reasonable" Hermitian data, the "good" couple of PH cubic can be directly constructed, obviating the need to compute and compare all four solutions.Secondly, C1 Hermitian interpolation with spherical PH curve is discussed. Based on the stereographic projection that preserves Pythagorean-Hodographs, a spherical rational PH curve is constructed to solve this problem. Furthermore, a method is also given to determine projecting image of the first derivative vectors on a sphere under the stereographic projection.Thirdly, with the help of optimization methods, several methods to construct the planar C1 cubic PH splines are discussed. Based on this, the spherical C1 PH splines are constructed. Because PH cubics have no inflections they cannot interpolate arbitrary data directly. A method is given that adding medial data to construct cubic PH spline curves to satisfy C1 continuum if the initial data cannot be interpolated directly.