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Extra resources for 3D Computer Graphics. Mathem. Intro with OpenGL
We are continuing to identify afﬁne transformations with homogeneous matrices, and so T 1,3 and R−90◦ can be viewed as 3 × 3 matrices. 11. The results of drawing the triangle with two different model view matrices. The dotted lines are not drawn by the OpenGL program and are present only to indicate the placement. then by the translation. That is to say, the transformations are applied to the drawn vertices in the reverse order of the OpenGL function calls. The reason for this convention is that it makes it easier to transform vertices hierarchically.
The vector v2 being rotated around u. 14 but viewed looking directly down the vector u. 7 for the matrix representing a rotation, Rθ,u , through an angle θ around axis u. 7 , (1 − c)u 1 u 3 − su 2 (1 − c)u 2 u 3 + su 1 (1 − c)u 23 + c 0 0 0 0 1 where c = cos θ and s = sin θ . The vector u must be a unit vector. There is no loss of generality in assuming that u is a unit vector since if not, it may be normalized by dividing by ||u||. To derive the matrix for Rθ,u , let v be an arbitrary point and consider what w = Rθ,u v is equal to.
2. For all x, y ∈ R2 , A(x + y) = A(x) + A(y). Note that A(0) = 0 for any linear transformation A. This follows from condition 1 with α = 0. Examples: Here are ﬁve examples of linear transformations: 1. 2. 3. 4. 5. A1 A2 A3 A4 A5 : : : : : x, y x, y x, y x, y x, y → → → → → −y, x . x, 2y . x + y, y . x, −y . −x, −y . 1 Verify that the preceding ﬁve transformations are linear. 2. We deﬁned transformations as acting on a single point at a time, but of course, a transformation also acts on arbitrary geometric objects since the geometric object can be viewed as a collection of points and, when the transformation is used to map all the points to new locations, this changes the form and position of the geometric object.