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Additional info for 3D Computer Graphics. Mathem. Intro with OpenGL
For example, the line at inﬁnity corresponds to the intersection of the 2-sphere with the x y-plane. ] 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 .
Because it is permitted that u = 0, every linear transformation is afﬁne. However, not every afﬁne transformation is linear. 1 is not linear since it does not map 0 to 0. 1 Let A be an afﬁne transformation. The translation vector u and the linear transformation B are uniquely determined by A. Proof First, we see how to determine u from A. We claim that in fact u = A(0). This is proved by the following equalities: A(0) = Tu (B(0)) = Tu (0) = 0 + u = u. Then B = Tu−1 A = T−u A, and so B is also uniquely determined.
The reason for this is that a rotation can be performed about any axis whatsoever. This includes not just rotations around the x-, y- and z-axes but also rotations around an axis pointing in an arbitrary direction. A rotation that ﬁxes the origin can be speciﬁed by giving a rotation axis u and a rotation angle θ, where the axis u can be any nonzero vector. We think of the base of the vector being placed at the origin, and the axis of rotation is the line through the origin parallel to the vector u.