By John Casey
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Extra info for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples.
We describe the geometric concepts with increasing precision, and simultaneously develop the computational tools, culminating in the conformal model for Euclidean geometry. We do so in a style that is not more mathematical than we deem necessary, hopefully without sacrificing exactness of meaning. We believe this approach is more accessible than axiomatizing geometric algebra first, and then having to discover its significance afterwards. The book consists of three parts that should be read in order (though sometimes a specialized chapter could be skipped without missing too much).
3 k-BLADES VERSUS k-VECTORS We have constructed k-blades as the outer product of k vector factors. 3), it is easy to show that the properties of the outer product allow k-blades in Rn to to be decomposed on an nk -dimensional basis.
Having said that, we already have some useful instruments. In an n-dimensional space Rn we can compare arbitrary hypervolumes. If we have n vectors ai (i = 1, . . , n), then the hypervolume of the parallelepiped spanned by them is proportional to a unit hypervolume in Rn by the magnitude of a1 ∧ . . ∧ an . The hypervolume of a simplex in that space, which is the convex body containing the origin and the n endpoints of the vectors ai (i = 1, · · · , n), is a fraction of that: 1 a1 ∧ · · · ∧ an .
A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples. by John Casey