4.2: Analytic Transformational Geometry
- Page ID
- 89853
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.