By Gomes, Jonas; Velho, Luiz; Costa Sousa, Mario
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Additional info for Computer Graphics : Theory and Practice
Sample text
0) plays a special role in the vector space. But in our everyday space there are no special points, nor is the notion of addition of points meaningful. 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.
Xn , 1) | xi ∈ R} of those linear transformations of Rn+1 that map P onto itself. 11) where L is a fixed invertible linear transformation of Rn and t is a fixed vector. 2 Points, Vectors, and Subspaces In both Euclidean and affine spaces, points cannot be added together, nor multiplied by scalars. This is clear in the P model, which has no origin. (Earlier we identified P with Rn , with (0, . . , 0, 1) corresponding to the origin of Rn ; if we had chosen any other point in P for this matter, defining the rest of the correspondence by subtraction, we would have the same result.
Note that if T : RPn → RPn is a projective transformation and λ ∈ R, λ = 0, then by using the linearity of T , we have (λT )P = T (λP ) = T (P ). In other words, projective transformations are defined up to multiplication by a nonzero scalar. 1 Anatomy of a Plane Projective Transformation We now concentrate on the projective plane. A projective transformation RP2 → RP2 is represented by an invertible matrix M of order 3. Our goal now is to understand the anatomy of this transformation: how does it act on projective points?
