4.2: Analytic Transformational Geometry

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Mar 8, 2022
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- 4.1: Transformation
- 4.3: Algebra of Transformations

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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	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_x(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	R_φ(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.

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Definition: Homogeneous coordinates

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