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An IBL Introduction to Geometries

Mark A. Fitch

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.
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Definition 4.2.1. Homogeneous coordinates.

A point with normal coordinates \((x,y)\) is written \((x,y,1)\) in homogeneous coordinates.
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Table 4.2.2. Linear Transformations for Isometries
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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.
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  1. Find a transformation that moves \([x_0,y_0]^T\) to the origin.
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  2. Find a transformation that moves \([x_0,y_0]^T\) to the origin then rotates by \(\phi\text{.}\)
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  3. Find a transformation that moves \([x_0,y_0]^T\) to the origin, rotates by \(\phi,\) then returns the origin to \([x_0,y_0]^T\text{.}\)
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  4. State, using matrix notation, a transformation that rotates the plane about a point \([x_0,y_0]^T\) by \(\phi\text{.}\)
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Checkpoint 4.2.4.

Goal: develop a reflection about a vertical line given by \(x=a\) using the following steps.
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  1. Find a transformation that move the line \(x=a\) to the \(y\)-axis.
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  2. Find a transformation that move the line \(x=a\) to the \(y\)-axis, then reflects the plane over the \(y\)-axis.
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  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 \(x=a.\)
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  4. State, using matrix notation, a transformation that reflects about an arbitrary vertical line \(x=a.\)
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Checkpoint 4.2.5.

Goal: develop a reflection about a horizontal line given by \(y=b\) using the following steps.
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  1. Find a transformation that move the line \(y=b\) to the \(x\)-axis.
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  2. Find a transformation that move the line \(y=b\) to the \(x\)-axis, then reflects the plane over the \(x\)-axis.
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  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 \(y=b.\)
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  4. State, using matrix notation, a transformation that reflects about an arbitrary horizontal line \(y=b.\)
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Checkpoint 4.2.6.

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