4.2: Analytic Transformational Geometry
- Page ID
- 89853
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
The goal is to develop matrix formulas for arbitrary isometries using the basic isometry formulas given below as building blocks.
A point with normal coordinates (x, y) in homogeneous coordinates is written (x, y, 1).
| Translate | T(x, y, 1)=\(\begin{bmatrix}1 & 0&a \\ 0 & 1&b \\0&0&1 \end{bmatrix}\)\(\begin{bmatrix}x\\ y \\1 \end{bmatrix}\) |
| Reflect over the y-axis | My(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 | Mx(x, y, 1)= \(\begin{bmatrix}1 & 0&0 \\ 0 & -1&0 \\0&0&1 \end{bmatrix}\) (\begin{bmatrix}x\\ y \\1 \end{bmatrix}\) |
| Rotate counterclockwise about the origin | Rφ(x, y, 1)= \(\begin{bmatrix}cosφ & -sinφ &0 \\ sinφ & cosφ &0 \\0&0&1 \end{bmatrix}\) (\begin{bmatrix}x\\ y \\1 \end{bmatrix}\) |
Goal: develop a rotation about a point [x0, y0]T using the following steps.
- Find a transformation that moves [x0, y0]T to the origin.
- Find a transformation that moves [x0, y0]T to the origin then rotates by φ.
- Find a transformation that moves [x0, y0]T to the origin, rotates by φ, then returns the origin to [x0, y0]T.
- State, using matrix notation, a transformation that rotates the plane about a point [x0, y0]T by φ.
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.
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 -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.
Develop a reflection about an arbitrary (non-vertical) line.