Applying the 2D matrix to points

Suppose we place two points $ (1,0)$ and $ (0,1)$ in the plane. They lie on the $ x$ and $ y$ axes, respectively, at one unit of distance from the origin $ (0,0)$. Using vector spaces, these two points would be the standard unit basis vectors (sometimes written as $ \hat\imath$ and $ \hat\jmath$). Watch what happens if we substitute them into (3.5):

$\displaystyle \begin{bmatrix}m_{11} & m_{12}  m_{21} & m_{22} \end{bmatrix} \begin{bmatrix}1  0 \end{bmatrix} = \begin{bmatrix}m_{11}  m_{21} \end{bmatrix}$ (3.7)

and

$\displaystyle \begin{bmatrix}m_{11} & m_{12}  m_{21} & m_{22} \end{bmatrix} \...
...{bmatrix}0  1 \end{bmatrix} = \begin{bmatrix}m_{12}  m_{22} \end{bmatrix} .$ (3.8)

These special points simply select the column vectors on $ M$. What does this mean? If $ M$ is applied to transform a model, then each column of $ M$ indicates precisely how each coordinate axis is changed.

Figure 3.5: Eight different matrices applied to transform a square face. These examples nicely cover all of the possible cases, in a qualitative sense.
\begin{figure}\begin{center}
\begin{tabular}{cc\vert cc}
$\begin{bmatrix}
1 & 0 ...
...n} \\
& $y$-shear & & Singular  \hline
\end{tabular}\end{center}
\end{figure}

Figure 3.5 illustrates the effect of applying various matrices $ M$ to a model. Starting with the upper right, the identity matrix does not cause the coordinates to change: $ (x,y) \mapsto (x,y)$. The second example causes a flip as if a mirror were placed at the $ y$ axis. In this case, $ (x,y) \mapsto (-x,y)$. The second row shows examples of scaling. The matrix on the left produces $ (x,y) \mapsto (2x, 2y)$, which doubles the size. The matrix on the right only stretches the model in the $ y$ 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 $ x$ and $ y$ axes. This is true, except that the mirror image of a mirror image restores the original. Thus, this corresponds to the case of a $ 180$-degree ($ \pi$ radians) rotation, rather than a mirror image. The matrix on the right produces a shear along the $ x$ direction: $ (x,y) \mapsto (x + y, y)$. The amount of displacement is proportional to $ y$. In the bottom row, the matrix on the left shows a skew in the $ y$ 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 $ M$ is applied: $ (x,y) \mapsto (x+y,x+y)$. 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 2016-12-31