本文讨论非PH(Pythagorean hodograph)曲线的一类五次OR曲线(curves with rational offsets)的构造方法.OR曲线是具有有理形式的等距线的一类参数曲线,在CAD中有着广泛的应用.本文采用参数曲线的复数表示形式,根据导矢曲线的因式分解的不同,将非PH曲线的五次OR曲线分为两种类型,并分别给出这两类曲线的构造方法.在给定C1连续的初始条件下,通过指定一个自由实参数来确定构造曲线.本文进一步阐述了这个自由实参数的几何意义.由于五次OR曲线是非正则的曲线,对于第一类曲线,奇异点可以在构造过程中显式地被指定,因此可以有效地避免其在特定曲线段上的出现;而对于第二类曲线,奇异点在曲线中的位置则不易被直接控制.
Unified and extended splines(UE-splines), which unify and extend polynomial, trigonometric, and hyperbolic B-splines, inherit most properties of B-splines and have some advantages over B-splines. The interest of this paper is the degree elevation algorithm of UE-spline curves and its geometric meaning. Our main idea is to elevate the degree of UE-spline curves one knot interval by one knot interval. First, we construct a new class of basis functions, called bi-order UE-spline basis functions which are defined by the integral definition of splines.Then some important properties of bi-order UE-splines are given, especially for the transformation formulae of the basis functions before and after inserting a knot into the knot vector. Finally, we prove that the degree elevation of UE-spline curves can be interpreted as a process of corner cutting on the control polygons, just as in the manner of B-splines. This degree elevation algorithm possesses strong geometric intuition.
We propose an angle-based mesh representation, which is invariant under translation, rotation, and uniform scaling, to encode the geometric details of a triangular mesh. Angle-based mesh representation consists of angle quantities defined on the mesh, from which the mesh can be reconstructed uniquely up to translation, rotation,and uniform scaling. The reconstruction process requires solving three sparse linear systems: the first system encodes the length of edges between vertices on the mesh, the second system encodes the relationship of local frames between two adjacent vertices on the mesh, and the third system defines the position of the vertices via the edge length and the local frames. From this angle-based mesh representation, we propose a quasi-angle-preserving mesh deformation system with the least-squares approach via handle translation, rotation, and uniform scaling. Several detail-preserving mesh editing examples are presented to demonstrate the effectiveness of the proposed method.
Gang XULi-shan DENGWen-bing GEKin-chuen HUIGuo-zhao WANGYi-gang WANG
By using the geometric constraints on the control polygon of a Pythagorean hodo- graph （PH） quartic curve, we propose a sufficient condition for this curve to have monotone curvature and provide the detailed proof. Based on the results, we discuss the construction of spiral PH quartic curves between two given points and formulate the transition curve of a G2 contact between two circles with one circle inside another circle. In particular, we deduce an attainable range of the distance between the centers of the two circles and summarize the algorithm for implementation. Compared with the construction of a PH quintic curve, the complexity of the solution of the equation for obtaining the transition curves is reduced.
Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonai. We construct an orthogonal basis for the n-order（n ≥ 3） algebraic-trigonometric spline space in order to resolve the theo- retical problem that there is not an explicit orthogonai basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions, which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.
