ThroughReflectLine

ThroughReflectLine is a geometric construction referring to the reflection of a line across a fixed mirror line in the plane. Given a line L and a mirror line R, the ThroughReflectLine of L with respect to R is the image L′ obtained when every point of L is reflected across R. This operation is the action of the plane’s reflection isometry about R, and L′ is the mirror image of L under that symmetry.

Construction and computation can be described in two common ways. One method uses two distinct points on L: reflect each point across R to produce two points on L′, then draw the line through those points. A coordinate approach reflects each point (x,y) of L across R, which has equation ax+by+c=0, using the standard point-reflection formula, and then determines L′ from the reflected points or from transforming the direction vector of L.

Basic geometric properties follow directly from reflection. The mirror line R is the perpendicular bisector of the segment joining any point X on L to its image X′ on L′. If L intersects R at a fixed point P, then P lies on L′ as well. If L is parallel to R, then L′ remains parallel to R, and the angle between L and R equals the angle between L′ and R, on opposite sides.

Notationally, ThroughReflectLine is often described as the image of L under the reflection about R, and it is a specific instance of a general isometry. It preserves distances and angles, making it a useful tool in symmetric constructions, computer graphics, tiling, and geometric proofs.

In practice, ThroughReflectLine can be implemented in geometry software and programming libraries as a routine that takes L and R as inputs and outputs L′, the reflected line.