By Gomes, Jonas; Velho, Luiz; Costa Sousa, Mario
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Additional resources for Computer Graphics : Theory and Practice
An1 an2 ··· ann As L is linear, it follows that L(x) = Le x for any vector x ∈ Rn , where, on the righthand side, the juxtaposition denotes the product of matrices and x is written as a column vector, that is, an n × 1 matrix. (We will use this convention throughout: a vector written to the right of a matrix is a column vector. ) 24 2. Geometry We have shown that the value of a linear transformation at a point can be obtained by left-multiplying the point (vector) by the matrix associated to the transformation.
6. Representation and reconstruction. 1 that, given a representation in R of an object in M , the result of reconstruction is generally some other object in M that merely approximates the object of origin. 6. Exact representations are rare; most representations are approximate or lossy. 4 we introduced the sampling representation, in connection with terrain and 2D image models. The reconstruction operation in this case is called interpolation, and it amounts to constructing a continuous function U → R given its values zij at finite number of points of U .
The canonical basis of RPn is given by e1 = (1, 0, . . , 0, 0), e2 = (0, 1, . . , 0, 0), .. en = (0, 0, . . , 1, 0), en+1 = (0, 0, . . , 0, 1), en+2 = (1, 1, . . , 1, 1) = e1 + · · · + en+1 . 5. Given any projective basis a1 , . . , an+1 , there exists a projective transformation T : RPn → RPn such that T (ei ) = λi ai , i = 1, . . 18) where the λi , i = 1, . . , n+2, are nonzero scalars.