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Geometric Computing: for Wavelet Transforms, Robot Vision, Learning, Control and Action

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Geometric algebra (GA) is a powerful new mathematical system for computational geometry. Although its origins can be traced back to Hermann Grassmann (1844), its development as a language for space–time geometry with applications to all of physics did not begin until 1966. Suddenly, in the year 2000 it was recognized that a specialized version called conformal geometric algebra (CGA) was ideally suited for computational Euclidean geometry. CGA has the great advantage that geometric primitives (point, line plane, circle, sphere) can be directly represented, compared, and manipulated without coordinates, so there is an immediate correspondence between algebraic objects and geometric figures. Moreover, CGA enhances and smoothly integrates the classical methods of projective, affine, and metric geometry with the more specialized methods of quaternions, screw theory, and rigid body mechanics. Applications to computer science and engineering have accumulated rapidly in the last few years. This book assembles diverse aspects of geometric algebra in one place to serve as a general reference for applications to robotics. Then, it demonstrates the power and efficiency of the system with specific applications to a host of problems ranging from computer vision to mechanical control. Perceptive readers will recognize many places where the treatment can be extended or improved. Thus, this book is a work in progress, and its higher purpose will be served if it stimulates further research and development.

This book presents the theory and applications of an advanced mathematical language called geometric algebra that greatly helps to express the ideas and concepts, and to develop algorithms in the broad domain of robot physics.

In the history of science, without essentialmathematical concepts, theorieswould have not been developed at all. We can observe that in various periods of the history of mathematics and physics, certain stagnation occurred; from time to time, thanks to new mathematical developments, astonishing progress took place. In addition, we see that the knowledge became unavoidably fragmented as researchers attempted to combine different mathematical systems. Each mathematical system brings about some parts of geometry; however, together, these systems constitute a highly redundant system due to an unnecessary multiplicity of representations for geometric concepts. The author expects that due to his persistent efforts to bring to the community geometric algebra for applications as a meta-language for geometric reasoning, in the near future tremendous progress in robotics should take place.

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