# Foundations of Geometric Algebra Computing (Geometry and Computing)

Language: English

Pages: 196

ISBN: 3642317936

Format: PDF / Kindle (mobi) / ePub

The author defines “Geometric Algebra Computing” as the geometrically intuitive development of algorithms using geometric algebra with a focus on their efficient implementation, and the goal of this book is to lay the foundations for the widespread use of geometric algebra as a powerful, intuitive mathematical language for engineering applications in academia and industry. The related technology is driven by the invention of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and by the recent progress in parallel processing, and with the specific conformal geometric algebra there is a growing community in recent years applying geometric algebra to applications in computer vision, computer graphics, and robotics.

This book is organized into three parts: in Part I the author focuses on the mathematical foundations; in Part II he explains the interactive handling of geometric algebra; and in Part III he deals with computing technology for high-performance implementations based on geometric algebra as a domain-specific language in standard programming languages such as C++ and OpenCL. The book is written in a tutorial style and readers should gain experience with the associated freely available software packages and applications.

The book is suitable for students, engineers, and researchers in computer science, computational engineering, and mathematics.

the property that 2 D 0. In CGA, there are many blades with this property of squaring 2 to zero, since e02 D e1 D 0. We use D e1 ^ e2 ^ e3 ^ e1 ; (4.30) 56 4 Maple and the Identification of Quaternions and Other Algebras Fig. 4.10 Dual numbers in CGA as used in [67], which has this property, to obtain dual numbers and also dual quaternions (see Sect. 4.6). We can write a dual number in CGA as follows: x C y D x C e1 ^ e2 ^ e3 ^ e1 y: (4.31) Figure 4.10 shows the subset of the 32 blades of

definition of quaternion_i: specialization: versor quaternion_i(1.0, e3ˆe2); 9.3 Approaches to Runtime Optimization Table 9.3 Computation of the shoulder quaternion 133 e1 QQ12 yz2 D Q12 c3 D yz2 j swi vel j j yz2 j swi vel sig n Dq yz2 Pwq 3 Q3 D 1Cc C 2 Qs D Q12 Q3 1 c3 k 2 9.3.1.4 Rotation to the Elbow Position The Gaigen 2 implementation of the computation of Q12 is as follows: Sphere p_ze = _Sphere( d1*e3); originPlane pi_m = _originPlane(p_ze-p_e); originPlane pi_e =

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.4 Cascading Multiplications . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.5 Linear Operation Tables . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.6 Multiplication Tables with a Non-Euclidean Metric . . . . . . 10.3.7 Additional Symbolic Optimizations Using Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .

. . . Fig. 13.3 Fig. 13.4 Fig. 14.1 Fig. 14.2 166 166 166 167 Generation of optimized FPGA implementations from Geometric Algebra algorithms . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 180 Pipeline schedule for the coefficient pex of a multivector. All of the computations specified by (14.1) for all of the pipeline stages can be done in parallel . . . . . . . . 180 List of Figures Fig. 14.3 Fig. 14.4 Fig. 14.5 xxiii Parallel dot product of two

r represents the radius of the sphere, and the parameter d the distance from the origin to the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4 Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 2.5 List of the 8 blades of 3D Euclidean Geometric Algebra . . . . . . . . Notations for the geometric algebra products .. . . . . . . . . . . . . . . . . . . . Properties of the outer product ^ . . . . . . . . . . . . . .. . . . . . . . . . . . . .