A rotation equals two reflections

In the picture below, the red line and the blue line are the reflection lines.

two_reflections_equals_rotation_3Image credit

  • The A’B’C’ triangle is a reflection of the ABC triangle in the red line. Let’s call this transformation R1
  • The EDF triangle is a reflection of the A’B’C’ triangle in the blue line. Let’s call this transformation R2
  • If you perform R1 followed by R2 (product of R1 and R2), the combined operation transforms the ABC triangle into the EDF triangle. This is equivalent to a rotation.
  • You have rotated the ABC triangle 120° clockwise about the point where the red line and the blue line intersects.
  • Rotation results in the  EDF triangle.
  • The rotation angle (120° in this example) is twice the angle between the red line and the blue line (60° in this example).
  • Any rotation on a plane can be formed by performing 2 reflections in intersecting lines.
  • Any translation on a plane can be formed by performing 2 reflections in parallel lines.

translationImage credit

Any rotation in 3D space can be formed by 2 reflections as well.  As an example, put two mirrors at 90° to each other and put an object in front of them. The reflection on the second mirror will appear to have been rotated by 180° relative to the original object. The rotation angle 180° is twice the angle between mirrors which is  90° in this example.

rotated150Image credit

  • A product of two reflections in intersecting planes is equivalent to a rotation. The axis of this rotation is the line of intersection of the planes while the angle of rotation is twice the angle between the two planes.

Invariants

Rotations or translations (and also reflections by the logic explained above) preserve the distance (size and shape of the image). All angles are preserved as well. For example, the interior angles of the triangles shown in the figures above do not change under rotation or translation or reflection.

The order of the transformations

From the classic textbook “Quantum Mechanics: Non-Relativistic Theory” by L.D.Landau and E.M Lifshitz

Although the result of two successive transformations in general depends on the order in which they are performed, in some cases the order of operations is immaterial; the transformations commute. This is so for the following transformations:

  1. Two rotations about the same axis
  2. Two reflections in mutually perpendicular planes (equivalent to a rotation through \pi about their line of intersection)
  3. Two rotations through \pi about mutually perpendicular axes (equivalent to a rotation through \pi about the third perpendicular axis)
  4. A rotation and reflection in a plane perpendicular to the axis of rotation
  5. Any rotation or reflection and an inversion with respect to a point lying on the axis of rotation or in the plane of reflection; this follows from (1) and (4)

 

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About Suresh Emre

I have worked as a physicist at the Fermi National Accelerator Laboratory and the Superconducting Super Collider Laboratory. I am a volunteer for the Renaissance Universal movement. My main goal is to inspire the reader to engage in Self-discovery and expansion of consciousness.
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