By Gomes, Jonas; Velho, Luiz; Costa Sousa, Mario

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Fewer points (p0 , p1 , . . , pm ) will form an affine basis of the subspace they span—that is, the set of their affine combinations—as long as the differences pi − p0 are linearly independent. An affine transformation between two m-dimensional affine subspaces is any transformation that is affine when expressed as a transformation from Rm to Rm using affine bases for the two subspaces; the choice of bases does not matter. 5 The Geometry of Computer Graphics Based on our discussion of Euclidean and affine geometries up to this point, we may be tempted to conclude that affine geometry is a good answer to our question, what is the right geometry for computer graphics?

In other words, the vector space structure of Rn is not actually part of Euclidean geometry; we just use it as a stepping stone. 2. 1 Linear Transformations Transformations preserving the vector space structure of Rn are called linear. 2) for every u, v ∈ Rn and λ ∈ R. In particular, L must fix the origin. For computations, we use the well-known matrix representation of linear transformations. If L : Rn → Rn is linear and e1 = (1, 0, 0, . . , 0), e2 = (0, 1, 0, . . , 0), . . , en = (0, 0, 0, .

1 s 1 0 0 s In all cases so far, the projective transformation preserves both affine points and points at infinity. Therefore, in essence, the projective transformations do not introduce any novelty. What we showed above can be summarized by saying that the group of projective transformations contains the group of affine transformations (and therefore, the rigid motions of Euclidean geometry). Effect of P . 16). We take A as being the identity matrix, T as null and s = 1. By applying the transformation to an affine point with coordinates (x, y, 1), we get ⎛ ⎞⎛ ⎞ ⎛ ⎞ 1 0 0 x x ⎝ 0 1 0⎠ ⎝ y ⎠ = ⎝ ⎠.