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Extra resources for 3D Computer Graphics. Mathem. Intro with OpenGL
1 Transformations in 2-Space 21 Deﬁne i = 1, 0 and j = 0, 1 . The two vectors i and j are the unit vectors aligned with the x-axis and y-axis, respectively. Any vector x = x1 , x2 can be uniquely expressed as a linear combination of i and j, namely, as x = x1 i + x2 j. Let A be a linear transformation. Let u = u 1 , u 2 = A(i) and v = v1 , v2 = A(j). Then, by linearity, for any x ∈ R2 , A(x) = A(x1 i + x2 j) = x1 A(i) + x2 A(j) = x1 u + x2 v = u 1 x 1 + v1 x 2 , u 2 x 1 + v2 x 2 . Let M be the matrix M x1 x2 = u 1 v1 u 2 v2 .
From the ﬁgure, it is obvious that rotating v2 through an angle of θ around u results in the vector (cos θ)v2 + (sin θ )v3 .
Yet another way of mathematically understanding the two-dimensional projective space is to view it as the space of linear subspaces of three-dimensional Euclidean space. To understand this, let x = x1 , x2 , x3 be a homogeneous representation of a point in the projective plane. This point is equivalent to the points αx for all nonzero α ∈ R; these points Team LRN 34 Transformations and Viewing plus the origin form a line through the origin in R3 . A line through the origin is of course a one-dimensional subspace, and we identify this one-dimensional subspace of R3 with the point x.