#### Applying the 2D matrix to points

Suppose we place two points and in the plane. They lie on the and axes, respectively, at one unit of distance from the origin . Using vector spaces, these two points would be the standard unit basis vectors (sometimes written as and ). Watch what happens if we substitute them into (3.5): (3.7)

and (3.8)

These special points simply select the column vectors on . What does this mean? If is applied to transform a model, then each column of indicates precisely how each coordinate axis is changed. Figure 3.5 illustrates the effect of applying various matrices to a model. Starting with the upper right, the identity matrix does not cause the coordinates to change: . The second example causes a flip as if a mirror were placed at the axis. In this case, . The second row shows examples of scaling. The matrix on the left produces , which doubles the size. The matrix on the right only stretches the model in the direction, causing an aspect ratio distortion. In the third row, it might seem that the matrix on the left produces a mirror image with respect to both and axes. This is true, except that the mirror image of a mirror image restores the original. Thus, this corresponds to the case of a -degree ( radians) rotation, rather than a mirror image. The matrix on the right produces a shear along the direction: . The amount of displacement is proportional to . In the bottom row, the matrix on the left shows a skew in the direction. The final matrix might at first appear to cause more skewing, but it is degenerate. The two-dimensional shape collapses into a single dimension when is applied: . This corresponds to the case of a singular matrix, which means that its columns are not linearly independent (they are in fact identical). A matrix is singular if and only if its determinant is zero.

Steven M LaValle 2020-01-06