Manual Space-Time Algebra (2nd Edition)

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Lecture notes and tutorials
  1. Motivation
  2. Algebra -- from Wolfram MathWorld
  3. Introduction to the Lorentz transformation
  4. Introduction

  • Geometric Algebra.
  • Space-Time Algebra!
  • What Stalin Knew: The Enigma of Barbarossa.
  • Computer algebra in gravity research;
  • Applied Geometric Algebra in Computer Science and Engineering 2015.

The discussion of how to apply geometric algebra to euclidean n-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from 19th century mathematics. We compare euclidean PGA and the popular 2-up model CGA conformal geometric algebra , restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set.

We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include euclidean sphere primitives. We conclude that euclidean PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA.


In the last years, Geometric Algebra with its Euclidean, Homogeneous and Conformal models attract the research interest in many areas of Computer Science and Engineering and particularly in Computer Graphics as it is shown that they can produce more efficient and smooth results than other algebras. In this paper, we present an all-inclusive algorithm for real-time animation interpolation and GPU-based geometric skinning of animated, deformable virtual characters using the Conformal model of Geometric Algebra CGA.

We compare our method with standard quaternions, linear algebra matrices and dual-quaternions blending and skinning algorithms and we illustrate how our CGA-GPU inclusive skinning algorithm can provide as smooth and more efficient results as state-of-the-art previous methods. Furthermore, the elements of CGA that handle transformations CGA motors can support translation, rotation and dilation uniform scaling of joints under a single, GPU-supported mathematical framework and avoid conversion between different mathematical representations in contrast to quaternions and dual-quaternions that support only rotation and rotation-translation respectively.

The algebraic formulation in this 5D space is then implemented in GPU to allow faster parallel computation queries.

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  4. Results show expected orders of magnitude improvements computing collisions among known mesh models, allowing interactive rates without using optimizations and bounding volume hierarchies. This is the first paper in a series of four designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field.

    In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl V,GE and the theory of its deformations leading to metric geometric algebras Cl V,G and some special types of extensors.

    This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel presentation of several topics of the differential geometry of smooth manifolds of arbitrary global topology.

    A key tool for the development of our program is the mastering of the euclidean geometrical algebra of multivectors that is detailed in the present paper. In this paper we introduce the concept of euclidean Clifford algebra Cl V,GE for a given euclidean structure on V , i. In this paper we give a comparison between the formulation of the concept of metric for a real vector space of finite dimension in terms of tensors and extensors.

    A nice property of metric extensors is that they have inverses which are also themselves metric extensors. This property is not shared by metric tensors because tensors do not have inverses. This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar to the usual theory of vector functions of a real variable, has some subtle issues which make its presentation worthwhile.

    We refer in particular to the derivative rules involving exterior and Clifford products, and also to the rule for derivation of a composition of an ordinary scalar function with a multivector function of a real variable. In this paper we develop with considerable details a theory of multivector functions of a p-vector variable. The concepts of limit, continuity and differentiability are rigorously studied. In this paper we introduce the concept of multivector functionals. We study some possible kinds of derivative operators that can act in interesting ways on these objects such as, e.

    We present an introduction to the mathematical theory of the Lagrangian formalism for multiform fields on Minkowski spacetime based on the multiform and extensor calculus.

    Algebra -- from Wolfram MathWorld

    Our formulation gives a unified mathematical description for the main relativistic field theories including the gravitational field. This paper, the third in a series of eight introduces some of the basic concepts of the theory of extensors needed for our formulation of the differential geometry of smooth manifolds. Key notions such as the extension and generalization operators of a given linear operator a 1, 1 -extensor acting on a real vector space V are introduced and studied in details.

    This paper, sixth in a series of eight, uses the geometric calculus on manifolds developed in the previous papers of the series to introduce through the concept of a metric extensor field g a metric structure for a smooth manifold M. The associated metric compatible connection extensor field, the associated Christoffel operators and a notable decomposition of those objects are given.

    In this paper we study in details the properties of the duality product of multivectors and multiforms used in the definition of the hyperbolic Clifford algebra of multivefors and introduce the theory of the k multivector and l multiform variables multivector or multiform extensors over V studying their properties with considerable detail. A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.

    A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of extensor fields is present using algebraic and analytical tools developed in previous papers. Several important formulas are derived. The real number system is geometrically extended to include three new anticommuting square roots of plus one, each such root representing the direction of a unit vector along the orthonormal coordinate axes of Euclidean 3-space. The resulting geometric Clifford algebra provides a geometric basis for the famous Pauli matrices which, in turn, proves the consistency of the rules of geometric algebra.

    The flexibility of the concept of geometric numbers opens the door to new understanding of the nature of space-time, and of Pauli and Dirac spinors as points on the Riemann sphere, including Lorentz boosts. This article explores group manifolds which are efficiently expressed in lower dimensional Clifford geometric algebras. The spectral basis of a geometric algebra allows the insightful transition between a geometric algebra of multivectors and its representation as a matrix over the real or complex numbers, or over the quaternions or split quaternions.

    This paper presents a tutorial of geometric algebra, a very useful but generally unappreciated extension of vector algebra. The emphasis is on physical interpretation of the algebra and motives for developing this extension, and not on mathematical rigor. The description of rotations is developed and compared with descriptions using vector and matrix algebra.

    The Geometry of Causality - Space Time

    The use of geometric algebra in physics is illustrated by solving an elementary problem in classical mechanics, the motion of a freely spinning axially symmetric rigid body. A tutorial of geometric calculus is presented as a continuation of the development of geometric algebra in a previous paper. The geometric derivative is defined in a natural way that maintains the close correspondence between geometric algebra and the algebra of real numbers. The use of geometric calculus in physics is illustrated by expressing some basic results of electrodynamics.

    This thesis addresses the computational and implementational aspects of geometric algebra, and shows that its mathematical promise can be made into programming reality: geometric algebra provides a modular, structured specification language for geometry whose implementations can be automatically generated, leading to an efficiency that is competitive with the hand- optimized code based on the traditional linear algebra approach. We discuss three applications of a gauge theory of gravity to rotating astrophysical systems.

    Introduction to the Lorentz transformation

    The theory employs gauge fields in a flat Minkowski background spacetime to describe gravitational interactions. The iron fluorescence line observed in AGN is discussed, assuming that the line originates from matter in an accretion disk around a Kerr rotating black hole. Gauge-theory gravity, expressed in the language of Geometric Algebra, allows very efficient numerical calculation of photon paths. From these paths we are able to infer the line shape of the iron line. Comparison with observational data allows us to constrain the black hole parameters, and, for the first time, infer an emissivity profile for the accretion disk.

    Brief advertisement for Gauge Theory Gravity, and a recent extension to scale invariance. How the Cosmic Microwave Background constrains these, and some other areas in fundamental physics. We focus on inverse kinematics applications in computer graphics and robotics based on Conformal Geometric Algebra. Here, geometric objects like spheres and circles that are often needed in inverse kinematics algorithms are simply represented by algebraic objects.

    We present algorithms for the inverse kinematics of a human arm like kinematic chain and for the grasping of robots and virtual humans. The main benefits of using geometric algebra in the virtual reality software Avalon are the easy, compact and geometrically intuitive formulation of the algorithms and the immediate computation of quaternions.

    This paper presents a very efficient approach for algorithms developed based on conformal geometric algebra using reconfigurable hardware. We use the inverse kinematics of the arm of a virtual human as an example, but we are convinced that this approach can be used in a wide field of computer animation applications.

    Department of Mathematics

    We describe the original algorithm on a very high geometrically intuitive level as well as the resulting optimized algorithm based on symbolic calculations of a computer algebra system. The main focus then is to demonstrate our approach for the hardware implementation of this algorithm leading to a very efficient implementation. FABRIK uses a forward and backward iterative approach, finding each joint position via locating a point on a line.


    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. In this paper, we present an efficient method based on geometric algebra for computing the solutions to the inverse kinematics problem IKP of the 6R robot manipulators with offset wrist. Due to the fact that there exist some difficulties to solve the inverse kinematics problem when the kinematics equations are complex, highly nonlinear, coupled and multiple solutions in terms of these robot manipulators stated mathematically, we apply the theory of Geometric Algebra to the kinematic modeling of 6R robot manipulators simply and generate closed-form kinematics equations, reformulate the problem as a generalized eigenvalue problem with symbolic elimination technique, and then yield 16 solutions.

    Finally, a spray painting robot, which conforms to the type of robot manipulators, is used as an example of implementation for the effectiveness and real-time of this method. This paper shows how the recently developed formulation of conformal geometric algebra can be used for analytic inverse kinematics of two six-link industrial manipulators with revolute joints.