Chapter 3 describes how complex numbers work with matrices, and shows how to compute complex eigenvalues and eigenvectors. Chapter 6 provides a brief introduction to geometric algebra, which possesses many of the imaginary qualities of quaternions, but works in space of any dimension.
The second half of the book is devoted to applications of complex numbers, quaternions and geometric algebra. John Vince explains how complex numbers simplify trigonometric identities, wave combinations and phase differences in circuit analysis, and how geometric algebra resolves geometric problems, and quaternions rotate 3D vectors. There are two short chapters on the Riemann hypothesis and the Mandelbrot set, both of which use complex numbers.
Filled with lots of clear examples and useful illustrations, this compact book provides an excellent introduction to imaginary mathematics for computer science. This book provides a bridge between algebraic geometry and geometric modelling algorithms, formulated within a computer science framework. Apart from the algebraic geometry topics covered, the entire book is based on the unifying concept of using algebraic techniques — properly specialized to solve geometric problems — to seriously improve accuracy, robustness and efficiency of CAD-systems.
It provides new approaches as well as industrial applications to deform surfaces when animating virtual characters, to automatically compare images of handwritten signatures and to improve control of NC machines. This book further introduces a noteworthy representation based on 2D contours, which is essential to model the metal sheet in industrial processes. It additionally reviews applications of numerical algebraic geometry to differential equations systems with multiple solutions and bifurcations.
Future Vision and Trends on Shapes, Geometry and Algebra is aimed specialists in the area of mathematics and computer science on the one hand and on the other hand at those who want to become familiar with the practical application of algebraic geometry and geometric modelling such as students, researchers and doctorates. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra.
Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups.
Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.
Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford. It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision. Unlike similar texts.
The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the mathematical structure of geometric algebra together with its descriptive power will result in intuitive and more robust algorithms.
This book examines all aspects essential for a successful application of geometric algebra: the theoretical foundations, the representation of geometric constraints, and the numerical estimation from uncertain data.
Formally, the book consists of two parts: theoretical foundations and applications. The first part includes chapters on random variables in geometric algebra, linear estimation methods that incorporate the uncertainty of algebraic elements, and the representation of geometry in Euclidean, projective, conformal and conic space. The second part is dedicated to applications of geometric algebra, which include uncertain geometry and transformations, a generalized camera model, and pose estimation.
Graduate students, scientists, researchers and practitioners will benefit from this book. The examples given in the text are mostly recent research results, so practitioners can see how to apply geometric algebra to real tasks, while researchers note starting points for future investigations. Gibbs — made a special selection of useful techniques for the 3D geometry of engineering, and this limited framework is basically what we have been using ever since in the geometrical applications of linear algebra.
Linear algebra and matrices, with their coordinate representations, became the mainstay of doing geometry, both in practice and in mathematical development. He rescued the half-forgotten geometric algebra by now called Clifford algebra and developed in nongeometric directions , developed it into an alternative to the classical linear algebra—based representations, and started advocating its universal use. In the s, his voice was heard, and with the implementation of geometric algebra into interactive computer programs its practical applicability is becoming more apparent.
Gibbs was wrong in assuming that computing with the geometry of 3D space requires only representations of 3D points, although he did give us a powerful system to compute with those. This book will demonstrate that allowing more extended quantities in higher-dimensional representational spaces provides a more convenient executable language for geometry.
Maybe we could have had this all along; but perhaps we indeed needed to wait for the arrival of computers to appreciate the effectiveness of this approach. It was originally developed as a teaching tool, and a web tutorial is available, using GAViewer to explain the basics of geometric algebra.
You can use GAViewer when reading the book to type in algebraic formulas and have them act on geometrical elements interactively. This interaction should aid your understanding of the correspondence between geometry and algebra considerably.
The GA sandbox source code package used for the programming examples and exercises in this book is built on top of Gaigen 2. To compile and run the programming examples in Part I and Part II, you only have to download the sandbox package from the web site.
It is written in Java and intended to show only the essential structure; we do not deem it usable for anything that is computationally intensive, since it can easily be 10 to times slower than Gaigen 2. PREFACE xxxv If you are serious about implementing further applications, you can start with the GA sandbox package, or other available implementations of geometric algebra, or even write your own package. We are grateful to Joan Lasenby of Cambridge University for her detailed comments on the early chapters, and for providing some of the applied examples.
We are also indebted to Timaeus Bouma for his keen insights that allowed our software to be well-founded in mathematical fact. Daniel Fontijne owes many thanks to Yvonne for providing the fun and artistic reasons to study geometric algebra, and to Femke and Tijmen for the many refreshing breaks while working at home. Stephen Mann would like to thank Mei and Lilly for their support during the writing of this book.
This book is about geometric algebra, a powerful computational system to describe and solve geometrical problems. You will see that it covers familiar ground—lines, planes, spheres, rotations, linear transformations, and more—but in an unfamiliar way. Our intention is to show you how basic operations on basic geometrical objects can be done differently, and better, using this new framework. It helps us discuss some of the important properties of the computational framework.
You should of course read between the lines: you will be able to understand this example fully only at the end of Part II, but the principles should be clear enough now. This is depicted in Figure 1. Here is how geometric algebra encodes this in its conformal model of Euclidean geometry: 1.
The three points are denoted by three elements c1 , c2 , and c3. The geometric product multiplies transformations. Moreover, any element, not just a circle, is rotated by the same rotor-based formula.
An oriented line L is also an element of geometric algebra. Line Rotation. This shows that geometric algebra contains an exponentiation that can make elements into rotation operators. Dualization is an operation that takes the geometric complement.
For the line L, its dual can be visualized as the nest of cylinders surrounding it. Other transformations, such as general rigid body motions, have logarithms as well in geometric algebra and can therefore be interpolated. Again, there are alternatives. The duality operation above is a special case of the contraction. The change from p to p in the equation is not a typo: p denotes a point, p is its location vector relative to the arbitrary origin.
The two entities are clearly distinct elements of geometric algebra, though computationally related. In total, the scene of Figure 1. The outcome is plotted immediately through the calls to the multivector drawing function draw. The computations are fast enough to do this and much more involved calculations in real time; the rendering is typically the slowest component. Although the language is still unfamiliar, we hope you can see that this is geometric programming at a very desirable level, in terms of quantities that have a direct geometrical meaning.
Each item occurring in any of the computations can be visualized. Coordinates are only needed when entering the data, to specify precisely which points and lines are to be operated upon. It is much more than that: all expressions are quantitative prescriptions of computations, and can be executed directly.
Geometric algebra is a programming language, especially tailored to handle geometry. The constant improves readability: the reader immediately knows how many coordinates are returned, and what basis blades they refer to. Someone could decide to reorder the coordinates of vector e. That is of course preferable to compiling but producing nonsense results at run-time.
This is not yet automated in Gaigen 2, so we do it by copying the coordinates one by one. There are algebraic ways of writing coordinates as components of projections involving scalar products or contractions.
Therefore we permit you to to retrieve them more directly. On the right, the same torus with back-face culling. In the sample code, use the left mouse button to rotate the model. The middle and right mouse buttons pop up a menu that allows you to select another model. If you wonder where the 2-D projected vertices originally came from, we used the OpenGL feedback mechanism to obtain the 2-D viewport coordinates of standard GLUT models.
These models can be rendered using a simple call to glutSolid GLUT provides functions for rendering teapots, cubes, spheres, cones, tori, dodecahedrons, octahedrons, tetrahedrons, and icosahedrons. On the other hand, if we place a box around a region of space that does not contain a singularity and repeat the above process, then the tips of the vectors only form part of a sphere.
Further, almost all points on this partial sphere have two vectors pointing to it. This trivector Ti has the same volume as the tetrahedron formed by the center of the sphere and the three normalized vectors. If we sum the trivectors formed by the normalized vectors of all the triangles of the sample points, the magnitude of the resulting trivector T will approximately be the volume of the sphere if there is a singularity inside the cube that we sampled.
If there is no singularity inside the sampling cube, then roughly speaking, each trivector appears twice in the sum, but with opposite sign, and thus T will have a magnitude 0. Figure 2. Typically, we normalize T by the volume of the unit sphere, so the magnitude of T should be close to 1 if there is a singularity inside the cube, and close to 0 otherwise.
The SumFace procedure computes the sum of the trivectors spanned by two triangles made up of the vertices of one square on the face of the cube. The TestSingularity procedure calls the SumFace procedure for each face of the cube, and sums the value computed for each face.
It then normalizes the result by the volume of a unit sphere. For more details, see [46]. A complex example is shown in Figure 2.
The resulting trivector volume is returned. The code was edited for readability. For the unedited source code, see the GA sandbox source code package. It also enables us to compare lengths on a line, areas in the same plane, and volumes in the same space. We clearly have a need to compare lengths on different lines and areas in different planes. The nonmetrical outer product cannot do that, so in this chapter we extend our subspace algebra with a real-valued scalar product to serve this geo metric need.
It generalizes the familiar dot product between vectors to act between blades of the same grade. Then we carry the algebra further, and investigate how the scalar product and the outer product interact. This automatically leads to an inner product between subspaces of different dimensionality that we call the contraction.
The contraction of subspace A onto subspace B computes the part of B least like A. That also gives us a dual way to characterize subspaces, through blades denoting their orthogonal complement. With these metric products, we can easily compute geometrically useful operations like the orthogonal projection and determine reciprocal frames for nonorthonormal coordinate systems.
We can even use them to embed the 3-D cross product, although we provide strong arguments for using geometric algebra constructs instead. Such a metric space is just a vector space with a way to compute the norm of an arbitrary vector. The mathematically preferred method is to use a bilinear form Q, which is a scalar-valued function of vectors. They may seem to be all you need to do Euclidean geometry. But as we already indicated in Chapter 1, there are useful models of Euclidean geometry that use vector spaces with non-Euclidean metrics in their representation of elements in physical space.
More detail at this point would distract from our main goal in this chapter, which is to extend the dot product to blades, so we provide Appendix A as additional explanation. As a notation for these metric spaces, we use Rp,q for a space with p positive dimensions and q negative dimensions. A space Rn,0 is then an n-dimensional metric space that is effectively a space with a Euclidean metric.
We will only start using degenerate metrics in Chapter We provide for this geometric need algebraically by introducing a real-valued scalar product between blades of the same grade. As we have seen in the previous chapter, that is essentially the outer product. The inner product of vectors is a special case of this scalar product, as applied to vectors.
When applied to k-blades, it should at least be backwards compatible with that vector inner product in the case of 1-blades. So indeed, for k-blades, we do compute the squared k-dimensional hypervolume. So the formula works for 2-blades, giving the angle we expect. The 2-blade example also motivates the result for general k-blades.
Of the remainder, there are three possibilities: 1. Only scalars are left. The blades were then multiples of each other, so their angle is 0, the cosine equals 1, and the scalar product is the product of their norms.
We are left with one vector in each term. We saw that in the 2-blade example above. We are left with totally disjoint subblades of at least grade 2. We then need at least two rotations to bring the subblades into alignment. This exercise will give you a good feeling for this case in higher-dimensional space.
Reinterpreting a zero cosine within this larger context, it either means that two blades are geometrically perpendicular in the usual sense it takes a right turn to align them ; or that they are algebraically orthogonal in the sense of being independent, i. It provides a rich source of examples of how to work with this formalism. Clifford or geometric algebra shows strong unifying aspects and turned out in the s to be a most adequate formalism for describing different geometry-related algebraic systems as specializations of one "mother algebra" in various subfields of physics and engineering.
Recent work shows that Clifford algebra provides a universal and powerful algebraic framework for an elegant and coherent representation of various problems occurring in computer science, signal processing, neural computing, image processing, pattern recognition, computer vision, and robotics.
The 30 revised full papers presented were rigorously reviewed and selected from 65 presentations given at the two workshops. The papers are devoted to topics such as applications of computer algebra in celestial and engineering multibody systems, differential equations, computer vision, computer graphics, and the theory and applications of geometric algebra in geometric reasoning, robot vision, and computer graphics. By addressing a wide spectrum of problems in a common language, it offers both fresh insights and new solutions that are useful to scientists and engineers working in areas related with robotics.
It introduces non-specialists to Clifford and geometric algebra, and provides ex- amples to help readers learn how to compute using geometric entities and geomet- ric formulations.
It also includes an in-depth study of applications of Lie group theory, Lie algebra, spinors and versors and the algebra of incidence using the universal geometric algebra generated by reciprocal null cones.
Featuring a detailed study of kinematics, differential kinematics and dynamics using geometric algebra, the book also develops Euler Lagrange and Hamiltoni- ans equations for dynamics using conformal geometric algebra, and the recursive Newton-Euler using screw theory in the motor algebra framework.
Further, it comprehensively explores robot modeling and nonlinear controllers, and discusses several applications in computer vision, graphics, neurocomputing, quantum com- puting, robotics and control engineering using the geometric algebra framework.
Lastly, it shows how program code can be optimized for real-time computations. An essential resource for applied physicists, computer scientists, AI researchers, roboticists and mechanical and electrical engineers, the book clarifies and demon- strates the importance of geometric computing for building autonomous systems to advance cognitive systems research.
In spite of the aura of mystery that surrounds the subject, John Vince makes the subject accessible and very readable. The first two chapters cover the imaginary unit and its integration with real numbers. Chapter 3 describes how complex numbers work with matrices, and shows how to compute complex eigenvalues and eigenvectors. Chapter 6 provides a brief introduction to geometric algebra, which possesses many of the imaginary qualities of quaternions, but works in space of any dimension.
The second half of the book is devoted to applications of complex numbers, quaternions and geometric algebra.
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