Reflecting across the x-axis sends $(x,y) ightarrow(x,-y)$, while the y-axis sends $(x,y) ightarrow(-x,y)$.

Those special cases are both captured by the same matrix formula for a line through the origin.

The reflection matrix uses $cos(2 heta)$ and $sin(2 heta)$ because the vector’s direction is mirrored relative to the line angle $heta$.

This makes the matrix compact while still handling any origin-based mirror line.