Xiao-Shan Gao

Complete solution classification for the perspective-three-point problem

We use two approaches to solve the perspective-three-point (P3P) problem: the algebraic approach and the geometric approach. In the algebraic approach, we use Wu-Ritts zero decomposition algorithm to give a complete triangular decomposition for the P3P equation system. This decomposition provides the first complete analytical solution to the P3P problem. We also give a complete solution classification for the P3P equation system, i.e., we give explicit criteria for the P3P problem to have one, two, three, and four solutions. Combining the analytical solutions with the criteria, we provide an algorithm, CASSC, which may be used to find complete and robust numerical solutions to the P3P problem. In the geometric approach, we give some pure geometric criteria for the number of real physical solutions.

Evolutionary programming based on non-uniform mutation

A new evolutionary programming using non-uniform mutation instead of Gaussian, Cauchy and Levy mutations is proposed. Evolutionary programming with non-uniform mutation (NEP) has the merits of searching the space uniformly at the early stage and very locally at the later stage during the programming. For a suite of 14 benchmark problems, NEP outperforms the improved evolutionary programming using mutation based on Levy probability distribution (ILEP) for multimodal functions with many local minima while being comparable to ILEP in performance for unimodal and multimodal functions with only a few minima. The detailed theoretical analysis of the executing process of NEP and the expected step size on non-uniform mutation are given.

Geometric constraint satisfaction using optimization methods

Abstract The numerical approach to solving geometric constraint problems is indispensable for building a practical CAD system. The most commonly-used numerical method is the Newton–Raphson method. It is fast, but has the instability problem: the method requires good initial values. To overcome this problem, recently the homotopy method has been proposed and experimented with. According to the report, the homotopy method generally works much better in terms of stability. In this paper we use the numerical optimization method to deal with the geometric constraint solving problem. The experimental results based on our implementation of the method show that this method is also much less sensitive to the initial value. Further, a distinctive advantage of the method is that under- and over-constrained problems can be handled naturally and efficiently. We also give many instructive examples to illustrate the above advantages.

Implicitization of Rational Parametric Equations

Based on the Grobner basis method, we present algorithms for a complete solution of the following problems in the implicitization of a set of rational parametric equations. (1) To find a basis of the implicit prime ideal determined by a set of rational parametric equations. (2) To decide whether the parameters of a set of rational parametric equations are independent. (3) If the parameters of a set of rational parametric equations are not independent, to reparameterize the parametric equations so that the new parametric equations have independent parameters. (4) To compute the inversion maps of parametric equations, and as a consequence, to give a method to decide whether a set of parametric equations is proper. (5) In the case of algebraic curves, to find proper reparameterization for a set of improper parametric equations.

Solving parametric algebraic systems

For a parametric polynomial system: p1 = 0, · · · , pr = 0, d1 6= 0, · · · , ds 6= 0 where pi and di are in K[u1, · · · , um, x1, · · · , xn] and the u are parameters, we present a method for identifying all parametric values for which the system has solutions for the xi and at the same time giving the solutions (for the xi) of the system in an explicit way, i.e., the solutions are given by polynomial sets in triangular form. The algorithm has been implemented and several examples reported in this paper show the algorithm is of practical value.

Generalized Stewart-Gough platforms and their direct kinematics

In this paper, we introduce the generalized Stewart-Gough platform (GSP) consisting of two rigid bodies connected with six distance and/or angular constraints between six pairs of points, lines, and/or planes in the base and the moving platform, respectively. We prove that there exist 3850 possible forms of GSPs. We give the upper bounds for the number of solutions of the direct kinematics for all the GSPs. We also obtain closed-form solutions and the best upper bounds of real solutions of the direct kinematics for a class of 1120 GSPs.

Solving geometric constraint systems. I. A global propagation approach

We describe a geometric constraint solving method/system that takes the declarative description of geometric diagrams or engineering drawings as input and outputs a sequence of steps to construct the diagram with ruler and compass if it succeeds. We extend the local propagation to a global one. Like the local propagation, the global propagation tries to determine the position of a geometric object from the set of geometric objects whose positions are already known. However, our global propagation uses not only the constraints involving this object but also implicit information derived from other constraints. The algorithm can be used to build intelligent CAD and interactive computer graphic systems.

A Deductive Database Approach to Automated Geometry Theorem Proving and Discovering

We report our effort to build a geometry deductive database, which can be used to find the fixpoint for a geometric configuration. The system can find all the properties of the configuration that can be deduced using a fixed set of geometric rules. To control the size of the database, we propose the idea of a structured deductive database. Our experiments show that this technique could reduce the size of the database by one hundred times. We propose the data-based search strategy to improve the efficiency of forward chaining. We also make clear progress in the problems of how to select good geometric rules, how to add auxiliary points, and how to construct numerical diagrams as models automatically. The program is tested with 160 nontrivial geometry configurations. For these geometric configurations, the program not only finds most of their well-known properties but also often gives unexpected results, some of which are possibly new. Also, the proofs generated by the program are generally short and totally geometric.

A C-tree decomposition algorithm for 2D and 3D geometric constraint solving

In this paper, we propose a method which can be used to decompose a 2D or 3D constraint problem into a C-tree. With this decomposition, a geometric constraint problem can be reduced into basic merge patterns, which are the smallest problems we need to solve in order to solve the original problem in certain sense. Based on the C-tree decomposition algorithm, we implemented a software package MMP/Geometer. Experimental results show that MMP/Geometer finds the smallest decomposition for all the testing examples efficiently.

Automated generation of readable proofs with geometric invariants

In this series of papers, we discuss how to use a fixed set of high level geometry lemmas or rules related to geometric invariants, such as area, full-angle, etc., to produce short and human-readable proofs in geometry, especially to produce multiple and shortest proofs of a given geometry theorem. These rules are proved to be much more effective and concise than the rules based on triangle congruence used in related work before. The success of our approach is partially due to a skillful selection of geometric invariants and the related rules. Control and search strategies are proposed and experimented with to enhance the efficiency of the pover. In part I of this series, the high level geometry lemmas are about area and the Ceva-Menelaus configurations.