The Cayley method in computer aided geometric design

A thorough treatment of the Cayley method of elimination is given, with particular emphasis on its implementation to solve problems in computational geometry. An easily programmable formula for the Cayley matrix is derived, it is shown how to extract the birational correspondence from the Cayley matrix, numerous examples are worked, and the connection between the Cayley method and the Euclidean Algorithm is explained.     

有理B样条曲面的区间隐式化

提出有理B样条曲面的区间隐式化方法,即对一个有理B样条曲面,寻求包含给定的曲面的区间隐式B样条曲面,使得区间隐式B样条曲面的"厚度"尽量小,同时尽量避免出现多余分支.该问题等价于求区间隐式B样条曲面的2个边界曲面.针对该问题建立一个最优化模型并求解.     

有理曲线的近似隐式化表示

本文首次提出了曲线近似隐式化的概念，给出了求曲线的近似隐式化表示的有效算法，并以实例说明了算法有效性以及研究这一问题的重要意义。     

平面有理曲线几何连续隐式化的优化方法

文中提出了平面有理曲线隐式化的优化方法,证明了隐式方程的系数实际上是一个这二次型的极小解向量,或是一个齐次线性方程的非平凡解。鉴于隐式方程的复杂性和实际中的近似计算,文中还提出了用低次隐式方程来逼近有理曲线的近似隐式化优化方法,由于加上了端点处的插值性和ＧＣ＾１连续性,最后得到的隐式方程是ＧＣ＾１连续的。数值例子说明这种近似隐式化方法的效果是不错的。     

有理Bézier曲面的区间隐式化

本文依据以往的研究引入了有理Bézier曲面的区间隐式化的概念,即找到一条较低次的区间代数曲面使得给出的有理Bézier曲面落在该区间代数曲面内,并使得该区间代数曲面的宽度达到最小．文中给出了一个通过解一个带有线性限制条件的二次优化问题来计算一有理Bézier曲面的区间代数曲面的算法,并用实例演示了该算法．     

有理曲面的区间隐式化

利用一个低阶多项式区间隐式曲面来包围所给的参数式有理曲面,并构造了一些关于区间隐式曲面厚度和微分张量的目标函数.在最小化这些目标函数的条件下,该区间隐式曲面的中心曲面可以近似地逼近有理曲面,其逼近的误差可以利用区间隐式曲面的区间宽度进行估计.最后提供了具体的算法和一些实例.     

Approximate implicitization of planar curves by piecewise rational approximation of the distance function

We present an approximate implicitization method for planar curves. The computed implicit representation is a piecewise rational approximation of the distance function to the given parametric curve. The proposed method consists of four main steps: quadratic B-spline approximation of the given parametric curve, data reduction, segments-wise implicitization, multiplying with suitable polynomial factors. These segments are joined such that the collection generate a global C ^r spline function which approximates the distance function, for r = 0, 1.     

Approximate implicitization of parametric surfaces by using compactly supported radial basis functions

In this paper, a new approach is proposed to solve the approximate implicitization of parametric surfaces. It is primarily based on multivariate interpolation of scattered data by using compactly supported radial basis functions. Experimental results are provided to illustrate the proposed method is flexible and effective.     

Forward kinematic problem of 5-RPUR parallel mechanisms (3T2R) with identical limb structures

This paper investigates the formulation of the forward kinematic problem of the 5-DOF parallel mechanisms performing three translation and two independent rotations with identical limb structures. The mathematical framework used in this paper is based on algebraic geometry where the forward kinematic problem is explored in a seven-dimensional kinematic space by the means of the so-called Study parameters. In this paper, the algorithms proposed for obtaining the forward kinematic expression is based on two approaches which differ by their concept of eliminating passive variables. The first one is based on the application of the so-called resultant method and the second one, as a novel approach referred to as linear implicitization algorithm, uses an elimination procedure based on solving several systems of linear equations. The forward kinematic problem is solved using the homotopy continuation and it is revealed that it has 1680 finite solutions. Moreover, the forward kinematic problem is studied in three-dimensional Euclidean space where a class of simplified designs are proposed which admit a simpler formulation for the forward kinematic problem. Finally, based on the simplified designs, for a nearly general design a univariate expression of degree 220 is obtained.