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4. An afﬁne transformation acting on an F. 5. Effect of a rotation through angle θ . The origin 0 is held ﬁxed by the rotation. Conventions on Row and Column Vectors and Transposes. The conventions adopted in this book are that points in space are represented by column vectors, and linear transformations with matrix representation M are computed as Mx. Thus, our matrices multiply on the left. Unfortunately, this convention is not universally followed, and it is also common in computer graphics applications to use row vectors for points and vectors and to use matrix representations that act on the right.
OpenGL follows the same conventions as we do: points and vectors are column vectors, and transformation matrices multiply on the left. However, OpenGL does have some vestiges of the transposed conventions; namely, when specifying matrices with glLoadMatrix and glMultMatrix the entries in the matrix are given in column order. 3 Rigid Transformations and Rotations A rigid transformation is a transformation that only repositions objects, leaving their shape and size unchanged. If the rigid transformation also preserves the notions of “clockwise” versus “counterclockwise,” then it is orientation-preserving.
Deﬁnition Let A be a transformation. A is a linear transformation provided the following two conditions hold: 1. For all α ∈ R and all x ∈ R2 , A(αx) = α A(x). 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.