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2005 | Buch

On Wintner’s Conjecture About Central Configurations Kapitel lesen Erstes Kapitel lesen

Computer Algebra and Geometric Algebra with Applications

6th International Workshop, IWMM 2004, Shanghai, China, May 19-21, 2004 and International Workshop, GIAE 2004, Xian, China, May 24-28, 2004, Revised Selected Papers

herausgegeben von: Hongbo Li, Peter J. Olver, Gerald Sommer

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Computer Science

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

MathematicsMechanization consistsoftheory,softwareandapplicationofc- puterized mathematical activities such as computing, reasoning and discovering. ItsuniquefeaturecanbesuccinctlydescribedasAAA(Algebraization,Algori- mization, Application). The name “Mathematics Mechanization” has its origin in the work of Hao Wang (1960s), one of the pioneers in using computers to do research in mathematics, particularly in automated theorem proving. Since the 1970s, this research direction has been actively pursued and extensively dev- oped by Prof. Wen-tsun Wu and his followers. It di?ers from the closely related disciplines like Computer Mathematics, Symbolic Computation and Automated Reasoning in that its goal is to make algorithmic studies and applications of mathematics the major trend of mathematics development in the information age. The International Workshop on Mathematics Mechanization (IWMM) was initiated by Prof. Wu in 1992, and has ever since been held by the Key L- oratory of Mathematics Mechanization (KLMM) of the Chinese Academy of Sciences. There have been seven workshops of the series up to now. At each workshop, several experts are invited to deliver plenary lectures on cutting-edge methods and algorithms of the selected theme. The workshop is also a forum for people working on related subjects to meet, collaborate and exchange ideas.

Inhaltsverzeichnis

Frontmatter

Computer Algebra and Applications

On Wintner’s Conjecture About Central Configurations
Abstract
According to Wintner, the study of central configurations in celestial mechanics may be reduced to an extremality problem. Wintner’s Conjecture amounts to saying that the corresponding extremal zeroes for each fixed number n of different masses is finite. By the author’s Finite Kernel Theorem it follows that the corresponding number of extremal values is finite for each fixed n. Thus, Wintner’s Conjecture will be true or false according to whether there will be only a finite number of extremal zeroes or not. This gives thus a new way of attacking Wintner’s Conjecture.
Wen-tsun Wu
Polynomial General Solutions for First Order Autonomous ODEs
Abstract
For a first order autonomous ODE, we give a polynomial time algorithm to decide whether it has a polynomial general solution and to compute one if it exists. Experiments show that this algorithm is quite effective in solving ODEs with high degrees and a large number of terms.
Ruyong Feng, Xiao-Shan Gao
The Newton Polygon Method for Differential Equations
Abstract
We prove that a first order ordinary differential equation (ODE) with a dicritical singularity at the origin has a one-parameter family of convergent fractional power series solutions. The notion of a dicritical singularity is extended from the class of first order and first degree ODE’s to the class of first order ODE’s. An analogous result for series with real exponents is given.
The main tool used in this paper is the Newton polygon method for ODE. We give a description of this method and some elementary applications such as an algorithm for finding polynomial solutions.
José Cano
Implicit Reduced Involutive Forms and Their Application to Engineering Multibody Systems
Abstract
The RifSimp package in Maple transforms a set of differential equations to Reduced Involutive Form. This paper describes the application of RifSimp to challenging real-world problems found in engineering design and modelling. RifSimp was applied to sets of equations arising in the dynamical studies of multibody systems. The equations were generated by the Maple package Dynaflex, which takes as input a graph-like description of a multibody mechanical system and generates a set of differential equations with algebraic constraints. Application of the standard RifSimp procedure to such Differential Algebraic Equations can require large amounts of computer memory and time, and can fail to finish its computations on larger problems.
We discuss the origin of these difficulties and propose an Implicit Reduced Involutive Form to assist in alleviating such problems. This form is related to RifSimp form by the symbolic inversion of a matrix. For many applications such as numerically integrating the multibody dynamical equations, the extra cost of symbolically inverting the matrix to obtain explicit RifSimp form can be impractical while Implicit Reduced Involutive Form is sufficient.
An approach to alleviating expression swell involving a hybrid analytic polynomial computation is discussed. This can avoid the excessive expression swell due to the usual method of transforming the entire input analytic differential system to polynomial form, by only applying this method in intermediate computations when it is required.
Wenqin Zhou, David J. Jeffrey, Gregory J. Reid, Chad Schmitke, John McPhee
Hybrid Method for Solving New Pose Estimation Equation System
Abstract
Camera pose estimation is the problem of determining the position and orientation of an internally calibrated camera from known 3D reference points and their images. We introduce a new polynomial equation system for 4-point pose estimation and apply our symbolic-numeric method to solve it stably and efficiently. In particular, our algorithm can also recognize the points near critical configurations and deal with these near critical cases carefully. Numerical experiments are given to show the performance of the hybrid algorithm.
Greg Reid, Jianliang Tang, Jianping Yu, Lihong Zhi
Some Necessary Conditions on the Number of Solutions for the P4P Problem
Abstract
The perspective-n-point (PnP) problem is to find the position and orientation of a camera with respect to a scene object from n correspondence points and is a widely used technique for pose determination in the computer vision community. Finding out geometric conditions of multiple solutions is the ultimate and most desirable goal of the multi-solution analysis, a key research issue of the problem in the literature. In this paper, we study the multi-solution phenomenon of the P4P problem and give some necessary conditions under which there are five positive solutions for the P4P problem. Moreover, we give a geometric configuration for the five solutions.
Jianliang Tang
A Generalization of Xie-Nie Stability Criterion
Abstract
In this paper, we obtain a sufficient condition for a polynomial with real positive coefficients to be stable, which generalizes Xie-Nie stability criterion.
Xiaorong Hou, Xuemin Wang
Formal Power Series and Loose Entry Formulas for the Dixon Matrix
Abstract
Formal power series are used to derive four entry formulas for the Dixon matrix. These entry formulas have uniform and simple summation bounds for the entire Dixon matrix. When corner cutting is applied to the monomial support, each of the four loose entry formulas simplifies greatly for some rows and columns associated with a particular corner, but still maintains the uniform and simple summation bounds. Uniform summation bounds make the entry formulas loose because redundant brackets that eventually vanish are produced. On the other hand, uniform summation bounds reveal valuable information about the properties of the Dixon matrix for a corner-cut monomial support.
Wei Xiao, Eng-Wee Chionh
Constructive Theory and Algorithm for Blending Several Implicit Algebraic Surfaces
Abstract
Blending implicit algebraic surfaces has long been a hard work due to the complex calculations and the lack of effective algorithms. In this paper, we present a recursive method to derive the existence conditions and expressions of the blending surface for an arbitrary number of quadratic surfaces based on the blending surface for few quadratic surfaces. The existence conditions can be described in terms of geometric parameters of the given quadratic surfaces, which makes them easy to check. This greatly simplifies the calculations. Finally, some examples are presented.
Yurong Li, Na Lei, Shugong Zhang, Guochen Feng
Minimum-Cost Optimization in Multicommodity Logistic Chain Network
Abstract
This paper presents a method of modeling and solving a very complicated real logistic problem in the management of transportation and sales. The problem to be addressed is a large-scale multicommodity, multi-source and multi-sink network flow optimization, of 12 types of coal from 29 mines, through over 200 railway stations along 5 railroad arteries, in Chongqing Coal Industry Company of China. A minimum-cost flow model is established for the network system, and several maximal-flow algorithms are implemented to produce an optimal scheme.
Hongxia Li, Shuicheng Tian, Yuan Pan, Xiao Zhang, Xiaochen Yu
A Survey of Moving Frames
Abstract
This paper surveys the new, algorithmic theory of moving frames developed by the author and M. Fels. Applications in geometry, computer vision, classical invariant theory, the calculus of variations, and numerical analysis are indicated.
Peter J. Olver
Invariant Geometric Motions of Space Curves
Abstract
Motions of space curves in similarity and centro-affine geometries are studied. It is shown that the geometric motions of inextensible space curves in similarity and centro-affine geometries are closely related to integrable equations. Several gauge equivalent integrable equations are derived through the relationship between the curvature and graph of the curves. Motion of space curves corresponding to travelling wave solutions to the KdV flow in centro-affine geometry is discussed in detail.
Changzheng Qu
Classification of Signature Curves Using Latent Semantic Analysis
Abstract
In this paper we describe the Euclidean signature curves for two dimensional closed curves in the plane and will give a discrete numerical method for finding such invariant curves. Further we describe an analog of Latent Semantic Analysis (LSA) and present data and noise reduction techniques as well as an optimal combination of normalizing transformations to categorize signature curves. We will then introduce a system for determining the correct category for a new object from a pre-existing database of information on objects and give an example for sorting out leaves of two types of trees regardless of their orientation using their signature curves.
Cheri Shakiban, Ryan Lloyd
Hamiltonian System and Algebro-Geometric Solution Associated with Dispersive Long Wave Equation
Abstract
By using an iterative algebraic method, we derive from a spectral problem a hierarchy of nonlinear evolution equations associated with dispersive long wave equation. It is shown that the hierarchy is integrable in Liouville sense and possesses bi-Hamiltonian structure. Under a Bargmann constraint the spectral is nonlinearized to a completely integrable finite dimensional Hamiltonian system. By introducing the Abel-Jacobi coordinates, an algebro-geometric solution for the dispersive long wave equation is derived by resorting to the Riemann theta function.
Engui Fan
The Painlevé Test of Nonlinear Partial Differential Equations and Its Implementation Using Maple
Abstract
The so-called WTC-Kruskal algorithm is presented in order to study the Painlevé integrability of nonlinear partial differential equations, which combines the WTC algorithm and Kruskal’s simplification algorithm. Based on the WTC, Kruskal and WTC-Kruskal algorithms, we give an implementation in Maple called PDEPtest. The applications of PDEPtest to several nonlinear partial differential equations are also presented and some new results are reported.
Gui-qiong Xu, Zhi-bin Li

Geometric Algebra and Applications

Hybrid Matrix Geometric Algebra
Abstract
The structures of matrix algebra and geometric algebra are completely compatible and in many ways complimentary, each having their own advantages and disadvantages. We present a detailed study of the hybrid 2 × 2 matrix geometric algebra M(2,IG) with elements in the 8 dimensional geometric algebra IG=IG 3 of Euclidean space. The resulting hybrid structure, isomorphic to the geometric algebra IG 4,1 of de Sitter space, combines the simplicity of 2× 2 matrices and the clear geometric interpretation of the elements of IG. It is well known that the geometric algebra IG(4,1) contains the 3-dimensional affine, projective, and conformal spaces of Möbius transformations, together with the 3-dimensional horosphere which has attracted the attention of computer scientists and engineers as well as mathematicians and physicists. In the last section, we describe a sophisticated computer software package, based on Wolfram’s Mathematica, designed specifically to facilitate computations in the hybrid algebra.
Garret Sobczyk, Gordon Erlebacher
Intrinsic Differential Geometry with Geometric Calculus
Abstract
Setting up a symbolic algebraic system is the first step in mathematics mechanization of any branch of mathematics. In this paper, we establish a compact symbolic algebraic framework for local geometric computing in intrinsic differential geometry, by choosing only the Lie derivative and the covariant derivative as basic local differential operators. In this framework, not only geometric entities such as the curvature and torsion of an affine connection have elegant representations, but their involved local geometric computing can be simplified.
Hongbo Li, Lina Cao, Nanbin Cao, Weikun Sun
On Miquel’s Five-Circle Theorem
Abstract
Miquel’s Five-Circle Theorem is difficult to prove algebraically. In this paper, the details of the first algebraic proof of this theorem is provided. The proof is based on conformal geometric algebra and its accompanying invariant algebra called null bracket algebra, and is the outcome of the powerful computational techniques of null bracket algebra embodying the novel idea breefs.
Hongbo Li, Ronghua Xu, Ning Zhang
On Averaging in Clifford Groups
Abstract
Averaging measured data is an important issue in computer vision and robotics. Integrating the pose of an object measured with multiple cameras into a single mean pose is one such example. In many applications data does not belong to a vector space. Instead, data often belongs to a non-linear group manifold as it is the case for orientation data and the group of three-dimensional rotations SO(3). Averaging on the manifold requires the utilization of the associated Riemannian metric resulting in a rather complicated task. Therefore the Euclidean mean with best orthogonal projection is often used as an approximation. In SO(3) this can be done by rotation matrices or quaternions. Clifford algebra as a generalization of quaternions allows a general treatment of such approximated averaging for all classical groups. Results for the two-dimensional Lorentz group SO(1,2) and the related groups SL(2,ℝ) and SU(1,1) are presented. The advantage of the proposed Clifford framework lies in its compactness and easiness of use.
Sven Buchholz, Gerald Sommer
Combinatorics and Representation Theory of Lie Superalgebras over Letterplace Superalgebras
Abstract
We state three combinatorial lemmas on Young tableaux, and show their role in the proof of the triangularity theorem about the action of Young-Capelli symmetrizers on symmetrized bitableaux. As an application, we describe in detail the way to specialize general results to the representation theory of the symmetric group and to classical invariant theory.
Andrea Brini, Francesco Regonati, Antonio Teolis
Applications of Geometric Algebra in Robot Vision
Abstract
In this tutorial paper we will report on our experience in the use of geometric algebra (GA) in robot vision. The results could be reached in a long term research programme on modelling the perception-action cycle within geometric algebra. We will pick up three important applications from image processing, pattern recognition and computer vision. By presenting the problems and their solutions from an engineering point of view, the intention is to stimulate other applications of GA.
Gerald Sommer
Twists – An Operational Representation of Shape
Abstract
We give a contribution to the representation problem of free-form curves and surfaces. Our proposal is an operational or kinematic approach based on the Lie group SE(3). While in Euclidean space the modelling of shape as an orbit of a point under the action of SE(3) is limited, we are embedding our problem into the conformal geometric algebra ℝ4,1 of the Euclidean space ℝ3. This embedding results in a number of advantages which makes the proposed method a universal and flexible one with respect to applications. It makes possible the robust and fast estimation of the pose of 3D objects from incomplete and noisy image data. Especially advantagous is the equivalence of the proposed shape model to that of the Fourier representations.
Gerald Sommer, Bodo Rosenhahn, Christian Perwass
Recent Applications of Conformal Geometric Algebra
Abstract
We discuss a new covariant approach to geometry, called conformal geometric algebra, concentrating particularly on applications to projective geometry and new hybrid geometries. In addition, a new method of working, which can achieve similar results, but using only one extra dimension instead of two, is also discussed.
Anthony Lasenby
Applications of Conformal Geometric Algebra in Computer Vision and Graphics
Abstract
This paper introduces the mathematical framework of conformal geometric algebra (CGA) as a language for computer graphics and computer vision. Specifically it discusses a new method for pose and position interpolation based on CGA which firstly allows for existing interpolation methods to be cleanly extended to pose and position interpolation, but also allows for this to be extended to higher-dimension spaces and all conformal transforms (including dilations). In addition, we discuss a method of dealing with conics in CGA and the intersection and reflections of rays with such conic surfaces. Possible applications for these algorithms are also discussed.
Rich Wareham, Jonathan Cameron, Joan Lasenby
Conic Sections and Meet Intersections in Geometric Algebra
Abstract
This paper first gives a brief overview over some interesting descriptions of conic sections, showing formulations in the three geometric algebras of Euclidean spaces, projective spaces, and the conformal model of Euclidean space. Second the conformal model descriptions of a subset of conic sections are listed in parametrizations specific for the use in the main part of the paper. In the third main part the meets of lines and circles, and of spheres and planes are calculated for all cases of real and virtual intersections. In the discussion special attention is on the hyperbolic carriers of the virtual intersections.
Eckhard M. S. Hitzer
nD Object Representation and Detection from Single 2D Line Drawing
Abstract
In this paper, we propose the wireframe representation of nD object, which is a single 2D line drawing. A wireframe model of an n D object is composed of a set of edges connecting a set of vertices, a subset of closed r -chains called boundary r -chains which surround the (r +1)D pieces of the object, and a set of filling patterns for the boundary r -chains for 0 < r < n. The wireframe representation is the perspective projection of the wireframe model of the object from its surrounding space to the image plane. Combining the projective geometric constraints of the wireframe representation with the idea of local construction and deletion, we propose an algorithm for high dimensional object detection from single 2D line drawing, under the most general assumption that neither the dimension of the object nor the dimension of the surrounding space is known, two neighboring faces can be coplanar, and whether or not the object is a manifold is unknown. Our algorithm outperforms any other algorithm in 2D face identification in that it generally does not generate redundant cycles that are not assigned as faces, and can handle 3D solids of over 10,000 faces.
Hongbo Li, Quan Wang, Lina Zhao, Ying Chen, Lei Huang
Polyhedral Scene Analysis Combining Parametric Propagation with Calotte Analysis
Abstract
Polyhedral scene analysis studies whether a 2D line drawing of a 3D polyhedron is realizable in the space, and if so, parameterizing the space of all possible realizations. For generic 2D data, symbolic computation with Grassmann-Cayley algebra is necessary. In this paper we propose a general method, called parametric calotte propagation, to solve the realization and parameterization problems in polyhedral scene analysis at the same time. Starting with the fundamental equations of Sugihara in the form of bivector equations, we can parameterize all the bivectors by introducing new parameters. The realization conditions are implied in the scalar equations satisfied by the new parameters, and can be derived by further analysis of the propagation result. The propagation procedure generally does not bifurcate, and the result often contains equations in factored form, thus makes further algebraic manipulation easier. In application, the method can be used to find linear construction sequences for non-spherical polyhedra.
Hongbo Li, Lina Zhao, Ying Chen
A Unified and Complete Framework of Invariance for Six Points
Abstract
Projective geometric invariants play an important role in computer vision. To set up the invariant relationships between spatial points and their images from a single view, at least six pairs of spatial and image points are required. In this paper, we establish a unified and complete framework of the invariant relationships for six points. The framework covers the general case already developed in the literature and two novel cases. The two novel cases describe that six spatial points and the camera optical center lie on a quadric cone or a twisted cubic, called quadric cone case or twisted cubic case. For the general case and quadric cone case, camera parameters can be determined uniquely. For the twisted cubic case, camera parameters cannot be determined completely; this configuration of camera optical center and spatial points is called a critical configuration. The established unified framework may help to effectively identify the type of geometric information appearing in certain vision tasks, in particular critical geometric information. An obvious advantage using this framework to obtain geometric information is that no any explicit estimation on camera projective matrix and optical center is needed.
Yihong Wu, Zhanyi Hu
An Introduction to Logical Animation
Abstract
In this paper the concept logical animation is described in a formal manner. It is abstracted from numerous popular animation applets made by dynamic geometry softwares. The concept is illustrated by a number of examples generated by the logical animation software SuperSketchpad (SSP), and some features and prospects related to applications of it are discussed.
Jingzhong Zhang, Chuanzhong Li
Recent Methods for Reconstructing Surfaces from Multiple Images
Abstract
Many objects can be mathematically represented as smooth surfaces with arbitrary topology, and smooth surface reconstruction from images could be cast into a variational problem. The main difficulties are the intrinsic ill-posedness of the reconstruction, image noise, efficiency and scalability. In this paper, we discuss the reconstruction approaches that use volumetric, graph-cut, and level-set optimization tools; and the objective functionals that use different image information, silhouette, photometry, and texture. Our discussion is accompanied by the implementations of these approaches on real examples.
Gang Zeng, Maxime Lhuillier, Long Quan
Backmatter
Metadaten
Titel
Computer Algebra and Geometric Algebra with Applications
herausgegeben von
Hongbo Li
Peter J. Olver
Gerald Sommer
Copyright-Jahr
2005
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-540-32119-4
Print ISBN
978-3-540-26296-1
DOI
https://doi.org/10.1007/b137294

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