Here, _p, q, and 6 are the parameters of the rigid-body model. Note that if the x-axis is defined as horizontal with positive values to the right and the y-axis is defined as vertical with positive values to the top, a positive 6 here defines a clockwise rotation. A 90° rotation maps the positive end of the y-axis onto the positive end of the x-axis. If positive 6 were defined as a counterclockwise rotation, the signs of the two terms involving sin(6) would need to be reversed. These elementary transformations can be rewritten in a homogeneous coordinate matrix formulation as follows:
Note that the results from the two different orders are not the same. Note also that the three independent parameters of the rigid-body model are dispersed across six variable elements of the transformation matrix. The two variables in the third column are free to take on any arbitrary value while the elements of the two-by-two submatrix in the upper left are all constrained to be functions of the rotational angle 6.
For a concrete example, consider a transformation that rotates an object clockwise by 10° around the origin of the coordinate system and then translates it 4 units along x and 9 units along y (see Fig. 1). The net transformation matrix can be derived by computing the appropriately ordered product of the elementary transformation matrices:
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