By Dietmar Hildenbrand

The writer defines “Geometric Algebra Computing” because the geometrically intuitive improvement of algorithms utilizing geometric algebra with a spotlight on their effective implementation, and the objective of this publication is to put the rules for the common use of geometric algebra as a strong, intuitive mathematical language for engineering functions in academia and undefined. The comparable know-how is pushed by way of the discovery of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and through the new growth in parallel processing, and with the categorical conformal geometric algebra there's a becoming neighborhood in recent times employing geometric algebra to functions in machine imaginative and prescient, special effects, and robotics. This e-book is equipped into 3 elements: partly I the writer specializes in the mathematical foundations; partially II he explains the interactive dealing with of geometric algebra; and partially III he bargains with computing know-how for high-performance implementations according to geometric algebra as a domain-specific language in ordinary programming languages similar to C++ and OpenCL. The ebook is written in an instructional kind and readers should still achieve event with the linked freely to be had software program programs and functions. The e-book is acceptable for college students, engineers, and researchers in laptop technological know-how, computational engineering, and arithmetic.

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**Extra resources for Foundations of Geometric Algebra Computing (Geometry and Computing)**

**Sample text**

What, for instance, does e0 12 r 2 e1 mean? 31) describing all points at the same distance r from the origin, namely a sphere at the origin. For more details, see Chap. 4 of [81] (especially Sect. 3). 40) with a homogeneous scaling factor of 2. 32) can also be used to obtain the centers of circles and point pairs. Note, that circles and point pairs are specific lower-dimensional spheres in CGA. 41) This is used in the robot example of Chap. 8. 4 Distances and Angles In CGA points, planes and spheres are represented as vectors.

E1 ^ e2 // ^ e3 D . e1 ^ e2 ^ e3 / D 2I: In Sect. 1, we shall see that the geometric meaning of the resulting bivector is a volume element spanned by the three vectors a; b; c, which is equal to 2 multiplied by the 3-blade e1 ^ e2 ^ e3 spanned by the three basis vectors e1 ; e2 ; e3 , which is equal to the pseudoscalar I . 2 The Inner Product For 3D Euclidean space, the inner product of two vectors is the same as the wellknown Euclidean scalar product of two vectors. For perpendicular vectors, the inner product is 0; for instance, e1 e2 D 0.

5. 10) The origin is mapped to e0 by this mapping. 12) and then projected to the inhomogeneous point x D x1 e1 C x2 e2 C x3 e3 . Further details can be found in Sect. 2 of [81]. 5 The 16 blades of 4D projective Geometric Algebra Grade 0 Term Scalar Blades 1 No. 1 1 Vector e 1 ; e2 ; e3 ; e0 4 2 Bivector e1 ^ e 2 ; e 1 ^ e 3 ; e 1 ^ e 0 ; e2 ^ e 3 ; e 2 ^ e 0 ; e 3 ^ e 0 6 3 Trivector e 1 ^ e2 ^ e3 ; e1 ^ e2 ^ e0 ; e1 ^ e 3 ^ e 0 ; e 2 ^ e 3 ^ e 0 4 4 Pseudoscalar e 1 ^ e2 ^ e3 ^ e0 1 Chapter 3 Conformal Geometric Algebra In this book, we focus on 5D Conformal Geometric Algebra (CGA).