By Dorst L., Fontijne D., Mann S.

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3. 1 PROPERTIES OF VOLUMES We now consider representations of oriented volumes. By analogy with homogeneous lines and planes, volumes can be treated as 3-D homogeneous subspaces of a vector space Rn . 1(d). • Although it is somewhat hard to visualize for us 3-D beings, volumes have an attitude in spaces of more than three dimensions, denoting the 3-D subspace that contains them. In a 3-D space there is, of course, only one choice—all volumes are proportional to the volume of the unit cube. 5: Bivector addition in 3-D space: orientation matters.

This is an automatic consequence of the outer product properties, and easily shown. In R3 , only three vectors can be independent, and therefore the fourth (d) must be expressible as a weighted sum of the other three: d = α a + β b + γ c. Associativity, distributivity, and antisymmetry then make the outer product of these four vectors zero: a ∧ b ∧ c ∧ d = a ∧ b ∧ c ∧ (α a + β b + γ c) = a ∧ b ∧ c ∧ (α a) + a ∧ b ∧ c ∧ (β b) + a ∧ b ∧ c ∧ (γ c) = 0. So the highest-order element that can exist in the subspace algebra of R3 is a trivector.

Na¨ıve implementations run slow, because of the size of the algebra (32-D for the basis of the conformal model of a 3-D Euclidean space). 5 THE STRUCTURE OF THE CHAPTERS 19 In the third part of this book, we give a computer scientist’s view of the algebraic structure and describe aspects that are relevant to any efficient implementation, using its multiplicative and sparse nature. We end with a simple ray tracer to enable comparison of computational speeds of the various methods in a computer graphics application.