Reflection Formulas
Shared 11/27/2025•81 views
Reflection in 2D Point: $(x, y)$ Reflection Matrix: $R = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ Reflected Point: $(x', y') = R \begin{pmatrix} x \\ y \end{pmatrix}$ Reflection about X-axis Transformation: $(x, y) \to (x, -y)$ Matrix: $$ R_x = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ Example: Reflecting $(2, 3)$ about the X-axis gives $(2, -3)$. Reflection about Y-axis Transformation: $(x, y) \to (-x, y)$ Matrix: $$ R_y = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} $$ Example: Reflecting $(2, 3)$ about the Y-axis gives $(-2, 3)$. Reflection about the Origin Transformation: $(x, y) \to (-x, -y)$ Matrix: $$ R_O = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} $$ Note: Equivalent to $R_x$ then $R_y$, or vice versa. Reflection about the line $y=x$ Transformation: $(x, y) \to (y, x)$ Matrix: $$ R_{y=x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$ Example: Reflecting $(2, 3)$ about $y=x$ gives $(3, 2)$. Reflection about the line $y=-x$ Transformation: $(x, y) \to (-y, -x)$ Matrix: $$ R_{y=-x} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} $$ Example: Reflecting $(2, 3)$ about $y=-x$ gives $(-3, -2)$. Reflection about an arbitrary line $y = mx + c$ A more general reflection involves: Translate the line to pass through the origin. Rotate the line to align with an axis. Reflect about that axis. Reverse the rotation. Reverse the translation. For a line through the origin ($c=0$): Angle $\theta$ with X-axis: $m = \tan \theta$ Matrix: $$ R_{\theta} = \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix} $$ This matrix reflects about the line forming angle $\theta$ with the positive x-axis. Reflection about an arbitrary line $ax + by + c = 0$ Point: $(x_0, y_0)$ Reflected Point: $(x', y')$ The formula for the reflected point $(x', y')$ is: $$ \frac{x' - x_0}{a} = \frac{y' - y_0}{b} = -2 \frac{ax_0 + by_0 + c}{a^2 + b^2} $$ This formula directly gives the reflected coordinates without matrix multiplication for a line in general form.