Geometric Algebra Computing in Engineering and Computer Science pdf.
Geometric Algebra Computing in Engineering and Computer Science free pdf download.
This book presents new results on applications of geometric algebra. The time when researchers and engineers were starting to realize the potential of quaternions for applications in electrical, mechanic, and control engineering passed a long time ago.
In the history of science, theories would have not been developed at all without essential mathematical concepts. In various periods of the history of mathematics and physics, there is clear evidence of stagnation, and it is only thanks to new mathematical developments that astonishing progress has taken place. Furthermore, researchers unavoidably cause fragmented knowledge in their various attempts to combine different mathematical systems. We realize that each mathematical system brings about some parts of geometry; however, together, they constitute a system that is highly redundant due to an unnecessary multiplicity of representations for geometric concepts. In contrast, in the geometric algebra language, most of the standard matter taught in engineering and computer science can be advantageously reformulated without redundancies and in a highly condensed fashion.
This book presents a selection of articles about the theory and applications of the advanced mathematical language geometric algebra which greatly helps to express the ideas and concepts and to develop algorithms in the broad domains of computer science and engineering. The contributions are organized in seven parts.
The first part presents screw theory in geometric algebra, the parameterization of 3D conformal transformations in conformal geometric algebra, and an overview of applications of geometric algebra. The second part includes thorough studies on Cliffor–Fourier transforms: the two-dimensional Clifford windowed Fourier transform; the cylindrical Fourier transform; applications of the 3D geometric algebra Fourier transform in graphics engineering; the 4D Clifford–Fourier transform for color image processing; and the use of the Hilbert transforms in Clifford analysis for signal processing. In the third part, self-organizing geometric neural networks are utilized for 2D contour and 3D surface reconstruction in medical image processing. The clustering and classification are handled using geometric neural networks and associative memories designed in the conformal geometric algebra. This part concludes with a retrospective of the quaternion wavelet transform, including an application for stereo vision. The fourth part for computer vision starts with a new cone-pixel camera using a convex hull and twists in conformal geometric algebra. The next work introduces a model-based approach for global self-localization using active stereo vision and Gaussian spheres. In the fifth part, the geometric characterization of M-conformal mappings is discussed, and a study of fluid flow problems is carried out in depth using quaternionic analysis. The sixth part shows the impressive space group visualizer for all 230 3D groups using the software packet for geometric algebra computations CLUCalc. The second author studies geometric algebra formalism as an alternative to distributed representation models; here convolutions are replaced by geometric products, and, as a result, a natural language for visualization of higher concepts is proposed. Another author studies computational complexity reductions using Clifford algebras and shows that graph problems of complexity class NP are polynomial in the number of Clifford operations required. The seventh part includes new developments in efficient geometric algebra computing: The first author presents an efficient blade factorization algorithm to produce faster implementations of the Join; with the software packet GALOOP, the second author symbolically reduces involved formulas of conformal geometric algebra, generating suitable code for computing using hardware accelerators. Another chapter shows applications of Grobner bases in robotics, formulated in the language of Clifford algebras, in engineering to the theory of curves, including Fermat and Bezier cubics, and in the interpolation of functions used in finite element theory.
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