Definition 4.2.1. Homogeneous coordinates.
A point with normal coordinates \((x,y)\) is written \((x,y,1)\) in homogeneous coordinates.
Section 4.2 Analytic Transformational Geometry
The goal is to develop matrix formulas for arbitrary isometries using the basic isometry formulas given below as building blocks.
| Translate | \(T(x,y,1) = \begin{bmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \text{.}\) |
| Reflect over the \(y\)-axis | \(M_y(x,y,1)= \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}. \) |
| Reflect over the \(x\)-axis | \(M_y(x,y,1)= \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \text{.}\) |
| Rotate counterclockwise about the origin | \(R_\phi(x,y,1)= \begin{bmatrix} \cos \phi & -\sin \phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \) |
Checkpoint 4.2.3.
Goal: develop a rotation about a point \([x_0,y_0]^T\) using the following steps.
- Find a transformation that moves \([x_0,y_0]^T\) to the origin.
- Find a transformation that moves \([x_0,y_0]^T\) to the origin then rotates by \(\phi\text{.}\)
- Find a transformation that moves \([x_0,y_0]^T\) to the origin, rotates by \(\phi,\) then returns the origin to \([x_0,y_0]^T\text{.}\)
- State, using matrix notation, a transformation that rotates the plane about a point \([x_0,y_0]^T\) by \(\phi\text{.}\)
Solution.
1) Use the “translate” transformation from the given list and adapt it so that the \(a\) and \(b\) values are \(-x_0\) and \(-y_0\) respectively. This is so that the resulting matrix has \(0\) for the \(x\) and \(y\) value places.
\(\begin{bmatrix}
1 & 0 & -x_0 \\
0 & 1 & -y_0 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
x_0 \\
y_0 \\
1 \\
\end{bmatrix}\)
2) Use the “rotate counterclockwise about the origin” transformation from our given list. Put this transformation BEFORE the previous transformation.
\(\begin{bmatrix}
cos\phi & -sin\phi & 0 \\
sin\phi & cos\phi & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
1 & 0 & -x_0 \\
0 & 1 & -y_0 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
x_0 \\
y_0 \\
1 \\
\end{bmatrix}\)
3) Use the “translate” transformation from the given list and adapt it so that the \(a\) and \(b\) values are \(-x_0\) and \(-y_0\) respectively. This is to “undo” the transformation we did in step one.
\(\begin{bmatrix}
1 & 0 & x_0 \\
0 & 1 & y_0 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
cos\phi & -sin\phi & 0 \\
sin\phi & cos\phi & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
1 & 0 & -x_0 \\
0 & 1 & -y_0 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
x_0 \\
y_0 \\
1 \\
\end{bmatrix}\)
If we multiply together the \(3 \times 3\) matrices the result is as follows:
\(\begin{bmatrix}
cos\phi & -sin\phi & (-x_0 cos\phi - y_0 sin\phi + x_0) \\
sin\phi & cos\phi & (-x_0 sin\phi - y_0 cos\phi + y_0) \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
x_0 \\
y_0 \\
1 \\
\end{bmatrix}\)
This simplifies to: \(\begin{bmatrix}
x_0 cos\phi - y_0 sin\phi + (-x_0 cos\phi - y_0 sin\phi + x_0) \\
x_0 sin\phi + y_0 cos\phi + (-x_0 sin\phi - y_0 cos\phi + y_0) \\
1 \\
\end{bmatrix}\) \(\ldots\) which further simplifies to:
\(\begin{bmatrix}
x_0 \\
y_0 \\
1 \\
\end{bmatrix}\)
Checkpoint 4.2.4.
Goal: develop a reflection about a vertical line given by \(x=a\) using the following steps.
- Find a transformation that move the line \(x=a\) to the \(y\)-axis.
- Find a transformation that move the line \(x=a\) to the \(y\)-axis, then reflects the plane over the \(y\)-axis.
- Find a transformation that move the line \(x=a\) to the \(y\)-axis, reflects the plane over the \(y\)-axis, then returns the \(y\)-axis to the line \(x=a.\)
- State, using matrix notation, a transformation that reflects about an arbitrary vertical line \(x=a.\)
Checkpoint 4.2.5.
Goal: develop a reflection about a horizontal line given by \(y=b\) using the following steps.
- Find a transformation that move the line \(y=b\) to the \(x\)-axis.
- Find a transformation that move the line \(y=b\) to the \(x\)-axis, then reflects the plane over the \(x\)-axis.
- Find a transformation that move the line \(y=b\) to the \(x\)-axis, reflects the plane over the \(x\)-axis, then returns the \(x\)-axis to the line \(y=b.\)
- State, using matrix notation, a transformation that reflects about an arbitrary horizontal line \(y=b.\)
Solution.
1. First, let's use a translation matrix to get
\begin{equation*}
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -b \\ 0 & 0& 1 \end{bmatrix} \begin{bmatrix} x_0 \\ y_0 \\ 1 \end{bmatrix}\text{.}
\end{equation*}
2. Next, let's add on a reflection to get
\begin{equation*}
\begin{bmatrix} 1 & 0& 0 \\ 0& -1& 0 \\ 0& 0& 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -b \\ 0 & 0& 1 \end{bmatrix} \begin{bmatrix} x_0 \\ y_0 \\ 1 \end{bmatrix}\text{.}
\end{equation*}
3. Finally, let's use a translation matrix to get
\begin{equation*}
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & b \\ 0 & 0& 1 \end{bmatrix} \begin{bmatrix} 1 & 0& 0 \\ 0& -1& 0 \\ 0& 0& 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -b \\ 0 & 0& 1 \end{bmatrix} \begin{bmatrix} x_0 \\ y_0 \\ 1 \end{bmatrix}\text{.}
\end{equation*}
4. After multiplying the three matrices, we get
\begin{align*}
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & b \\ 0 & 0& 1 \end{bmatrix} \begin{bmatrix} 1 & 0& 0 \\ 0& -1& 0 \\ 0& 0& 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -b \\ 0 & 0& 1 \end{bmatrix} \begin{bmatrix} x_0 \\ y_0 \\ 1 \end{bmatrix} \amp =
\begin{bmatrix} 1 & 0& 0 \\ 0& -1& 2b \\ 0& 0& 1 \end{bmatrix} \begin{bmatrix} x_0 \\ y_0 \\ 1 \end{bmatrix}\\
\amp = \begin{bmatrix} x_0 \\ 2b - y_0 \\ 1 \end{bmatrix}
\end{align*}
Checkpoint 4.2.6.
Develop a reflection about an arbitrary (non-vertical) line.
