Reflection over a Line

How a Mirror Line Pairs Points

Reflection over a line is a type of geometric transformation that moves every point on a plane to its mirror image position.

The line used as the reference for this reflection is called the mirror line or axis of reflection.

Imagine standing in front of a flat mirror. Your image in the mirror is the result of reflecting yourself over the surface of the mirror.

Mathematically, if we have a point $P(x,y)$ and a line $m$ as the axis of reflection, then the image point $P'(x',y')$ will have the following properties:

1. If point $P$ lies on the mirror line $m$:

   Its image is the point itself.

   $$P' = P$$

2. If point $P$ does not lie on the mirror line $m$:

   The line $m$ is the perpendicular bisector of the line segment $PP'$. Therefore:

   $$\overline{PP'} \perp m$$

   $$d(P,m) = d(P',m)$$

   Consequently, the line $m$ intersects the line segment $PP'$ exactly at its midpoint. We call this intersection point $M$.

Visualization of Point Reflection over a Line

The following example checks both reflection properties at once:

For every line reflection, the segment from the original point to its image is perpendicular to the mirror line, and both points have the same distance from that line. These two conditions determine the image coordinates.