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4.2: Analytic Transformational Geometry

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    The goal is to develop matrix formulas for arbitrary isometries using the basic isometry formulas given below as building blocks.

    Definition: Homogeneous coordinates

    A point with normal coordinates (x, y) in homogeneous coordinates is written (x, y, 1).

    Table \(\PageIndex{1}\): Linear Transformations for Isometries
    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.

    1. Find a transformation that moves [x0, y0]T to the origin.
    2. Find a transformation that moves [x0, y0]T to the origin then rotates by φ.
    3. Find a transformation that moves [x0, y0]T to the origin, rotates by φ, then returns the origin to [x0, y0]T.
    4. 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.

    1. Find a transformation that move the line x=a to the y-axis.
    2. Find a transformation that move the line x=a to the y-axis, then reflects the plane over the y-axis.
    3. 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.
    4. 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.

    1. Find a transformation that move the line y=b to the -axis.
    2. Find a transformation that move the line y=b to the x-axis, then reflects the plane over the x-axis.
    3. 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.
    4. State, using matrix notation, a transformation that reflects about an arbitrary horizontal line y=b.

    Develop a reflection about an arbitrary (non-vertical) line.


    This page titled 4.2: Analytic Transformational Geometry is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Mark A. Fitch via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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