Geometric Algebra for Computer Science (Revised Edition): An by Leo Dorst

By Leo Dorst

It is a sturdy e-book, however the arithmetic is poorly handled, now not adequate rigorous as will be anticipated.

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Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry (The Morgan Kaufmann Series in Computer Graphics)

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Additional info for Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry (The Morgan Kaufmann Series in Computer Graphics)

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3) = (a1 b2 − a2 b1 ) e1 ∧ e2 + (a2 b3 − a3 b2 ) e2 ∧ e3 + (a3 b1 − a1 b3 ) e3 ∧ e1 This cannot be simplified further. We see that an outer product of two vectors in 3-D space can be written as a scalar-weighted sum of three standard elements e1 ∧ e2 , e2 ∧ e3 , e3 ∧ e1 . Their weighting coefficients are obviously 2-D determinants, which we know represent directed area measures, now of the components of the original plane on the coordinate planes of the basis. The formula is then consistent with the interpretation of these three elements as standard area elements for the coordinate planes of the basis vectors.

For now, we cannot, since we are still working in a nonmetric vector space. We choose to use the term weight (because the term speed does not transfer well to higher-dimensional subspaces). A line with twice the weight could be said to pass through its points twice as fast for the same change in λ. We will allow the weight to be negative, in which case the line is oriented oppositely to a standard direction for that 1-D subspace. Those three properties of an oriented line through the origin are all part of its specification by a vector.

We will indicate these connections at the appropriate places in the book, and in some applications we actually revert to classical linear algebra when we find that it is more efficient or that it provides numerical tools that have not yet been developed for geometric algebra. Yet seeing all these classical techniques in the context of the full algebra enriches them, and emphasizes their specific geometric nature. The geometric algebra framework also exposes their cross-connections and provides universal operators, which can save time and code.

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