Reflection

Definition

• Reflection is a transformation that flips an object over a line (in 2D) or a plane (in 3D), creating a mirror image.
• Reflection can be defined across various axes (e.g., X-axis, Y-axis).
• In 2D, reflection matrices invert one axis’s coordinates while keeping the other the same.
• In matrix form, reflection is represented by using -1 to invert the axis across which reflection occurs.

Reflection across X-axis:

$$\text{Reflect across the X-axis:}\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$$

Reflection across Y-axis:

$$\text{Reflect across the Y-axis:}\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

Example: Reflection across X-axis

$$\text{Reflect a point }(2,3)\text{ across the X-axis:}\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]=\left[\begin{array}{c}2\\ -3\\ 1\end{array}\right]$$

Example: Reflection across Y-axis

$$\text{Reflect a point }(2,3)\text{ across the Y-axis:}\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]=\left[\begin{array}{c}-2\\ 3\\ 1\end{array}\right]$$