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2011 | Buch

Rigid Body Dynamics and Conformal Geometric Algebra Kapitel lesen Erstes Kapitel lesen

Guide to Geometric Algebra in Practice

herausgegeben von: Leo Dorst, Joan Lasenby

Verlag: Springer London

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

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Über dieses Buch

This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software and hardware tools. Topics and features: provides hands-on review exercises throughout the book, together with helpful chapter summaries; presents a concise introductory tutorial to conformal geometric algebra (CGA) in the appendices; examines the application of CGA for the description of rigid body motion, interpolation and tracking, and image processing; reviews the employment of GA in theorem proving and combinatorics; discusses the geometric algebra of lines, lower-dimensional algebras, and other alternatives to 5-dimensional CGA; proposes applications of coordinate-free methods of GA for differential geometry.

Inhaltsverzeichnis

Frontmatter

Rigid Body Motion

Frontmatter
1. Rigid Body Dynamics and Conformal Geometric Algebra
Abstract
We discuss a fully covariant Lagrangian-based description of 3D rigid body motion, employing spinors in 5D conformal space. The use of this space enables the translational and rotational degrees of freedom of the body to be expressed via a unified rotor structure, and the equations of motion in terms of a generalised ‘moment of inertia tensor’ are given. The development includes the effects of external forces and torques on the body. To illustrate its practical applications, we give a brief overview of a prototype multi-rigid-body physics engine implemented using 5D conformal objects as the variables.
Anthony Lasenby, Robert Lasenby, Chris Doran
2. Estimating Motors from a Variety of Geometric Data in 3D Conformal Geometric Algebra
Abstract
The motion rotors, or motors, are used to model Euclidean motion in 3D conformal geometric algebra. In this chapter we present a technique for estimating the motor which best transforms one set of noisy geometric objects onto another. The technique reduces to an eigenrotator problem and has some advantages over matrix formulations. It allows motors to be estimated from a variety of geometric data such as points, spheres, circles, lines, planes, directions, and tangents; and the different types of geometric data are combined naturally in a single framework. Also, it excludes the possibility of a reflection unlike some matrix formulations. It returns the motor with the smallest translation and rotation angle when the optimal motor is not unique.
Robert Valkenburg, Leo Dorst
3. Inverse Kinematics Solutions Using Conformal Geometric Algebra
Abstract
This paper describes a novel iterative Inverse Kinematics (IK) solver, FABRIK, that is implemented using Conformal Geometric Algebra (CGA). FABRIK uses a forward and backward iterative approach, finding each joint position via locating a point on a line. We use the IK of a human hand as an example of implementation where a constrained version of FABRIK was employed for pose tracking. The hand is modelled using CGA, taking advantage of CGA’s compact and geometrically intuitive framework and that basic entities in CGA, such as spheres, lines, planes and circles, are simply represented by algebraic objects. This approach can be used in a wide range of computer animation applications and is not limited to the specific problem discussed here. The proposed hand pose tracker is real-time implementable and exploits the advantages of CGA for applications in computer vision, graphics and robotics.
Andreas Aristidou, Joan Lasenby
4. Reconstructing Rotations and Rigid Body Motions from Exact Point Correspondences Through Reflections
Abstract
We describe a new algorithm to reconstruct a rigid body motion from point correspondences. The algorithm works by constructing a series of reflections which align the points with their correspondences one by one. This is naturally and efficiently implemented in the conformal model of geometric algebra, where the resulting transformation is represented by a versor. As a direct result of this algorithm, we also present a very compact and fast formula to compute a quaternion from two vector correspondences, a surprisingly elementary result which appears to be new.
Daniel Fontijne, Leo Dorst

Interpolation and Tracking

Frontmatter
5. Square Root and Logarithm of Rotors in 3D Conformal Geometric Algebra Using Polar Decomposition
Abstract
Conformal transformations are described by rotors in the conformal model of geometric algebra (CGA). In applications there is a need for interpolation of such transformations, especially for the subclass of 3D rigid body motions. This chapter gives explicit formulas for the square root and the logarithm of rotors in 3D CGA. It also classifies the types of conformal transformations and their orbits. To derive the results, we employ a novel polar decomposition for the even subalgebra of 3D CGA and an associated norm-like expression.
Leo Dorst, Robert Valkenburg
6. Attitude and Position Tracking
Abstract
Several applications require the tracking of attitude and position of a body based on velocity data. It is tempting to use direction cosine matrices (DCM), for example, to track attitude based on angular velocity data, and to integrate the linear velocity data separately in a suitable frame. In this chapter we make the case for using bivectors as the attitude tracking method of choice since several features make their performance and flexibility superior to that of DCMs, Euler angles or even rotors. We also discuss potential advantages in using CGA to combine the integration of angular and linear velocities in one step, as the features that make bivectors attractive for tracking rotations extend to bivectors that represent general displacements.
Liam Candy, Joan Lasenby
7. Calibration of Target Positions Using Conformal Geometric Algebra
Abstract
This chapter describes an algorithm for calibrating the 3D positions of multiple stationary point targets which form part of an optical positioning system. A group of rigidly co-located calibrated cameras are moved to several positions and images of the targets acquired. The target pixel coordinates are extracted and transformed into 3D lines which are used as input data to the algorithm. A nonlinear solution is developed using geometric algebra and geometric calculus and expressed in the conformal model of Euclidean 3D space. A coordinate free approach to differentiating rotors is developed and used in the algorithm to differentiate motion rotors. Experiments are performed to evaluate the algorithm, and the results show that it performs well and is robust in the presence of noise.
Robert Valkenburg, Nawar Alwesh

Image Processing

Frontmatter
8. Quaternion Atomic Function for Image Processing
Abstract
In this work we introduce a new kernel for image processing called the atomic function. This kernel is compact in the spatial domain, and it can be adapted to the behavior of the input signal by broadening or narrowing its band ensuring a maximum signal-to-noise ratio. It can be used for smooth differentiation of images in the quaternion algebra framework. We discuss the role of the quaternion atomic function with respect to monogenic signals. We then propose a steerable quaternion wavelet scheme for image structure and contour detection. Making use of the generalized Radon transform and images processed with the quaternion wavelet atomic function transform, we detect shape contours in color images. We believe that the atomic function is a promising kernel for image processing and scene analysis.
Eduardo Bayro-Corrochano, Eduardo Ulises Moya-Sánchez
9. Color Object Recognition Based on a Clifford Fourier Transform
Abstract
The aim of this chapter is to propose two different approaches for color object recognition, both using the recently defined color Clifford Fourier transform. The first one deals with so-called Generalized Fourier Descriptors, the definition of which relies on plane motion group actions. The proposed color extension leads to more compact descriptors, with lower complexity and better recognition rates, than the already existing descriptors based on the processing of the r, g and b channels separately. The second approach concerns color phase correlation for color images. The idea here is to generalize in the Clifford framework the usual means of measuring correlation from the well-known shift theorem. Both methods necessitate to choose a 2-blade B of ℝ4 which corresponds to an analysis plane in the color space. The relevance of proposed methods for classification purposes is discussed on several color image databases. In particular, the influence of parameter B is studied regarding the type of images.
Jose Mennesson, Christophe Saint-Jean, Laurent Mascarilla

Theorem Proving and Combinatorics

Frontmatter
10. On Geometric Theorem Proving with Null Geometric Algebra
Abstract
The bottleneck in symbolic geometric computation is middle expression swell. Another embarrassing problem is geometric explanation of algebraic results, which is often impossible because the results are not invariant under coordinate transformations. In classical invariant-theoretical methods, the two difficulties are more or less alleviated but stay, while new difficulties arise.
In this chapter, we introduce a new framework for symbolic geometric computing based on conformal geometric algebra: the algebra for describing geometric configuration is null Grassmann–Cayley algebra, the algebra for advanced invariant manipulation is null bracket algebra, and the algebra underlying both algebras is null geometric algebra. When used in geometric computing, the new approach not only brings about amazing simplifications in algebraic manipulation, but can be used to extend and generalize existing theorems by removing some geometric constraints from the hypotheses.
Hongbo Li, Yuanhao Cao
11. On the Use of Conformal Geometric Algebra in Geometric Constraint Solving
Abstract
To model a geometrical part in Computer Aided Design systems, declarative modeling is a well-adapted solution to declare and specify geometric objects and constraints. In this chapter, we are interested in the representation of geometric objects and constraints using a new language of description and representation, Geometric Algebra (GA). GA is used here in association with the conformal model of Euclidean geometry (CGA) which requires two extra dimensions comparing to the usual vector space model. Topologically and Technologically Related Surfaces (TTRS) Theory is introduced here as a unified framework for geometric objects representation and geometric constraints solving. Based on TTRS, this chapter shows the capability of the CGA to represent geometric objects and geometric constraints through symbolic geometric constraints solving and algebraic classification.
Philippe Serré, Nabil Anwer, JianXin Yang
12. On the Complexity of Cycle Enumeration for Simple Graphs
Abstract
We show how a number of combinatorial problems, such as determining the number of cycles in graphs, can be recast using a graded commutative algebra constructed within a real Grassmann exterior algebra. The computational complexity of this approach is then measured by considering operations at the basis blade level of the algebra. In particular, the worst-case time complexity of counting arbitrary length cycles in simple n-vertex graphs via nilpotent adjacency matrix methods is shown to be https://static-content.springer.com/image/chp%3A10.1007%2F978-0-85729-811-9_12/MediaObjects/213575_1_En_12_IEq1_HTML.gif , where α≤3 is the exponent representing the complexity of matrix multiplication. The storage complexity of the nilpotent adjacency matrix approach is https://static-content.springer.com/image/chp%3A10.1007%2F978-0-85729-811-9_12/MediaObjects/213575_1_En_12_IEq2_HTML.gif . A probabilistic model is used to describe a class of graphs in which the average-case time complexity of cycle enumeration is https://static-content.springer.com/image/chp%3A10.1007%2F978-0-85729-811-9_12/MediaObjects/213575_1_En_12_IEq3_HTML.gif for fixed 0<q<1. For reference, experimental results detailing computation times (in seconds) are compared with algorithms based on the approaches of Bax and Tarjan.
René Schott, G. Stacey Staples

Applications of Line Geometry

Frontmatter
13. Line Geometry in Terms of the Null Geometric Algebra over ℝ3,3, and Application to the Inverse Singularity Analysis of Generalized Stewart Platforms
Abstract
In this chapter, the classical line geometry is modeled in ℝ3,3, where lines are represented by null vectors, and points and planes by null 3-blades. The group of 3D special projective transformations SL(4) when acting upon points in space induces a Lie group isomorphism, with SO(3,3) acting upon lines.
As an application of the use of the ℝ3,3 model of line geometry, this chapter analyzes the inverse singularity analysis of generalized Stewart platforms, using vectors of ℝ3,3 to encode the force and torque wrenches to classify their singular configurations.
Hongbo Li, Lixian Zhang
14. A Framework for n-Dimensional Visibility Computations
Abstract
This chapter introduces global visibility computation using Grassmann Algebra. Visibility computation is a fundamental task in computer graphics, as in many other scientific domains. While it is well understood in two dimensions, this does not remain true in higher-dimensional spaces.
Grassmann Algebra allows to think about visibility at a high level of abstraction and to design a framework for solving visibility problems in any n-dimensional space for n≥2. Contrary to Stolfi’s framework which allows only the representation of geometric lines, its algebraic nature deals means general applicability, with no exceptional cases.
This chapter shows how the space of lines can be defined as a projective space over the bivector vector space. Then line classification, a key point for the visibility computation, is achieved using the exterior product. Actually, line classification turns out to be equivalent to point vs. hyperplane classification relative to a nondegenerate bilinear form. This ensures it is well defined and computationally robust.
Using this, the lines stabbing an n-dimensional convex face are characterized. This set of lines appears to be the intersection of the decomposable bivectors set (i.e., bivectors that represent a line) and a convex polytope. Moreover, this convex polytope is proved to be minimal. This property allows useful algorithmic improvements.
To illustrate the use of our framework in practice, we present the computation of soft shadows for three-dimensional illuminated scenes.
Lilian Aveneau, Sylvain Charneau, Laurent Fuchs, Frederic Mora

Alternatives to Conformal Geometric Algebra

Frontmatter
15. On the Homogeneous Model of Euclidean Geometry
Abstract
We attach the degenerate signature (n,0,1) to the dual Grassmann algebra of projective space to obtain a real Clifford algebra which provides a powerful, efficient model for Euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between the Grassmann algebra and its dual that yields non-metric meet and join operators. We focus on the cases of n=2 and n=3 in detail, enumerating the geometric products between k-blades and m-blades. We identify sandwich operators in the algebra that provide all Euclidean isometries, both direct and indirect. We locate the spin group, a double cover of the direct Euclidean group, inside the even subalgebra of the Clifford algebra, and provide a simple algorithm for calculating the logarithm of group elements. We conclude with an elementary account of Euclidean kinematics and rigid body motion within this framework.
Charles Gunn
16. A Homogeneous Model for Three-Dimensional Computer Graphics Based on the Clifford Algebra for ℝ3
Abstract
We construct a homogeneous model for Computer Graphics using the Clifford Algebra for ℝ3. To incorporate points as well as vectors within this model, we employ the odd-dimensional elements of this graded eight-dimensional algebra to represent mass-points by exploiting the pseudoscalars to represent mass. The even-dimensional elements of this Clifford Algebra are isomorphic to the quaternions, which operate on the odd-dimensional elements by sandwiching. Along with the standard sandwiching formulas for rotations and reflections, this paradigm allows us to use sandwiching to compute perspective projections.
Ron Goldman
17. Rigid-Body Transforms Using Symbolic Infinitesimals
Abstract
There is a requirement to be able to represent three-dimensional objects and their transforms in many applications, including computer graphics and mechanism and machine design. A geometric algebra is constructed which can model three-dimensional geometry and rigid-body transforms. The representation is exact since the square of one of the basis vectors is treated symbolically as being infinite. The non-zero, even-grade elements of the algebra represent precisely all rigid-body transforms. By allowing the transform to vary, smooth motions are obtained. This can be achieved using Bézier and B-spline combinations of even-grade elements.
Glen Mullineux, Leon Simpson
18. Rigid Body Dynamics in a Constant Curvature Space and the ‘1D-up’ Approach to Conformal Geometric Algebra
Abstract
We discuss a ‘1D up’ approach to Conformal Geometric Algebra, which treats the dynamics of rigid bodies in 3D spaces with constant curvature via a 4D conformal representation. All equations are derived covariantly from a 4D Lagrangian, and definitions of energy and momentum in the curved space are given. Some novel features of the dynamics of rigid bodies in these spaces are pointed out, including a simple non-relativistic version of the Papapetrou force in General Relativity. The final view of ordinary translational motion that emerges is perhaps surprising, in that it is shown to correspond to precession in the 1D up conformal space. We discuss the alternative approaches to Euclidean motions and rigid body dynamics outlined by Gunn in Chap. 15 and Mullineux and Simpson in Chap. 17 of this volume, which also use only one extra dimension, and compare these with the Euclidean space limit of the current approach.
Anthony Lasenby

Towards Coordinate-Free Differential Geometry

Frontmatter
19. The Shape of Differential Geometry in Geometric Calculus
Abstract
We review the foundations for coordinate-free differential geometry in Geometric Calculus. In particular, we see how both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the Shape Operator. The challenge is to adapt this formalism to Conformal Geometric Algebra for wide application in computer science and engineering.
David Hestenes
20. On the Modern Notion of a Moving Frame
Abstract
A tutorial on the modern definition and application of moving frames, with a variety of examples and exercises, is given. We show three types of invariants; differential, joint and integral, and the running example is the linear action of SL(2) on smooth surfaces, on sets of points in the plane, and path integrals over curves in the plane. We also give details of moving frames for the group of rotations and translations acting on smooth curves, and on discrete sets of points, in the conformal geometric algebra.
Elizabeth Mansfield, Jun Zhao
21. Tutorial Appendix: Structure Preserving Representation of Euclidean Motions Through Conformal Geometric Algebra
Abstract
Using conformal geometric algebra, Euclidean motions in n-D are represented as orthogonal transformations of a representational space of two extra dimensions, and a well-chosen metric. Orthogonal transformations are representable as multiple reflections, and by means of the geometric product this takes an efficient and structure preserving form as a ‘sandwiching product’. The antisymmetric part of the geometric product produces a spanning operation that permits the construction of lines, planes, spheres and tangents from vectors, and since the sandwiching operation distributes over this construction, ‘objects’ are fully integrated with ‘motions’. Duality and the logarithms complete the computational techniques.
The resulting geometric algebra incorporates general conformal transformations and can be implemented to run almost as efficiently as classical homogeneous coordinates. It thus becomes a high-level programming language which naturally integrates quantitative computation with the automatic administration of geometric data structures.
This appendix provides a concise introduction to these ideas and techniques. Editorial note: This appendix is a slightly improved version of (Dorst in: Bayro-Corrochano, E., Scheuermann, G. (eds.) Geometric Algebra Computing for Engineering and Computer Science, pp. 457–476, [2011]). We provide it to make this book more self-contained.
Leo Dorst
Backmatter
Metadaten
Titel
Guide to Geometric Algebra in Practice
herausgegeben von
Leo Dorst
Joan Lasenby
Copyright-Jahr
2011
Verlag
Springer London
Electronic ISBN
978-0-85729-811-9
Print ISBN
978-0-85729-810-2
DOI
https://doi.org/10.1007/978-0-85729-811-9

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