Reflection in Line

Given a line L and a point P. Reflection P' of P in L is the point such that PP' is perpendicular to L, and PM = MP', where M is the point of intersection of PP' and L. In other words, P' is located on the other side of L, but at the same distance from L as P. P' is said to be a mirror or symmetric image of P in L. The line L is called the axis of symmetry or axis of reflection.

P' exists for any P. Let's write S_L(P) = P'. If P' = S_L(P), then P is the reflection in L of P': P = S_L(P'). So that repeated reflection does noting: it does not move a point. Formally,

where I is the identity transform. S_L(P) = P, iff P lies on L.

The reflection transform S_L applies to arbitrary shapes point-by-point. Each point of a given shape S is reflected in L, and the collection of these reflections is the symmetric image of S: S_L(S). To determine S_L(S) when S is a polygon, suffice it to reflect its vertices. This is exactly what has been done in the applet below.

On the other hand, if S' is known to be a mirror image of S, then any pair of points P and P' not fixed by the reflection (P ≠ P'), the axes of reflection is uniquely determined as the perpendicular bisector of PP'.

In the applet, you can create polygons with a desired number of vertices, drag the vertices one at a time, or drag the polygon as a whole. Axes of reflection can also be dragged. They rotate if dragged near the applet's border, or translate if dragged nearer their midpoint.

The following observations are noteworthy:

1. Reflection changes the orientation: if a polygon is traversed clockwise, its image is traversed counterclockwise, and vice versa.

2. Reflection is isometry: a reflection preserves distances.

3. Reflection preserves angles.

4. Reflection maps parallel lines onto parallel lines.

5. All points on the axis of refleclation are fixed as are all the lines perpendicular to the axis.

6. Successive reflections in two parallel axes result in a translation in the direction perpendicular to the axes to twice the distance between them.

7. Successive reflections in two axes that meet in a point O is equivalent to a rotation around O through double the angle between them.

8. The order of reflections matters: two reflections do not commute. In fact, for two lines L and M,

   SLSM = -SMSL.

Following is a short list of problems that are solved with the help of the reflection transform:

1. Heron's Problem
2. Optical Property of Ellipse
3. Reflections of a Line Through the Orthocenter
4. Three Congruent Circles by Reflection III
5. Two Congruent Circles by Reflection

And there is a whole lot more.

Geometric Transformations

• Plane Isometries
  • Reflection in Point
  • Reflection in Line
    • Geometric Problems by Reflection
  • Translation Transform
  • Product of Rotations
    • Rotation Transform
  • Glide Reflection
• Affine Transformations (definition)
  • Iterated Function Systems
  • Homothety
    • Geometric Problems by Homothety
  • Shearing
  • Spiral Similarity
    • Construction of the center of spiral similarility
• Inversion
  • Stereographic Projection and Inversion
  • Möbius Transformations
• Projective transformation

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