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Rotation as Product of Two Reflections

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This Demonstration shows some of the relationships between composition of reflections and rotation. In particular it shows that a composition of two reflections is equivalent to a rotation. The initial graphic consists of a solid blue asymmetric object (upper right) and three translucent transforms of :

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1. a rotation of about the graph origin (green translucency, upper left)

2. a reflection of (magenta translucency, lower right)

3. a reflection of the reflection of (red translucency, lower left)

There are four lines: the axis and action lines of the reflections, colored like their reflections.

By playing with the rotation and reflection sliders you can see that you can align the second reflection with the rotation exactly, showing that a rotation is a composition of two reflections.

You can hide some of the graphical elements to concentrate on certain relationships. You may also note an interesting relationship between the reflection angles and the rotation angle. See the Details section for more information.

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Contributed by: Ted Frazier (February 2012)
Open content licensed under CC BY-NC-SA

Snapshots

Details

Set the "reflection 1" angle to zero. Then set the rotation to an even angle, say 118 degrees (the default). Now change the angle of "reflection 2" until the red figure is exactly over the green figure (the rotation). The reflection angle is exactly half of the rotation angle.

Snapshot 1: shows the "reflection 2" angle is half the rotation angle

Snapshot 2: shows another pair of reflections giving the same rotation; the reflection lines have been rotated

Snapshot 3: shows both reflection lines are the same; their product is the identity

With the identity (or "do nothing" transformation), reflections in lines through a point form a group , and rotations about that point are then a subgroup of .

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