# 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).

 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.