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| − | When two reflections are performed with respect to a certain axis, an | + | When two reflections are performed with respect to a certain axis, an invariant point with respect to the point(s) being reflected is formed in the intersection of the lines of reflection. An angle is formed as well, at this intersection. It turns out that the double of this angle is the very angle through which an equivalent rotation is performed. We may choose any intersecting orientation for the two lines, that is to say, any angle between the lines of reflection is possible. |
Thinking about it visually, imagine you have a 2-D shape with a front and a back, front side facing up. If only a rotation was performed on the shape, the front side should still be facing up. This is why two reflections are needed: to preserve the cyclic order of the vertices of the shape. | Thinking about it visually, imagine you have a 2-D shape with a front and a back, front side facing up. If only a rotation was performed on the shape, the front side should still be facing up. This is why two reflections are needed: to preserve the cyclic order of the vertices of the shape. | ||
Revision as of 22:47, 24 January 2009
In $ \scriptstyle D_n $, explain geometrically why a reflection followed by a reflection must be a rotation.
When two reflections are performed with respect to a certain axis, an invariant point with respect to the point(s) being reflected is formed in the intersection of the lines of reflection. An angle is formed as well, at this intersection. It turns out that the double of this angle is the very angle through which an equivalent rotation is performed. We may choose any intersecting orientation for the two lines, that is to say, any angle between the lines of reflection is possible.
Thinking about it visually, imagine you have a 2-D shape with a front and a back, front side facing up. If only a rotation was performed on the shape, the front side should still be facing up. This is why two reflections are needed: to preserve the cyclic order of the vertices of the shape.
Mathematically, it's easy to prove that the composition of two reflections is a rotation. With respect to the positive Cartesian x-axis, a reflection of a point in a line at an angle $ \scriptstyle\theta $ can be expressed by a matrix transformation of the form:
$ \begin{bmatrix} \scriptstyle cos(2\theta) & \scriptstyle sin(2\theta) \\ \scriptstyle sin(2\theta) & \scriptstyle-cos(2\theta) \end{bmatrix} $.
And a rotation about the origin through an angle $ \scriptstyle\theta $ (measuring anticlockwise from the positive x-axis) may be represented as:
$ \begin{bmatrix} \scriptstyle cos(\theta) & \scriptstyle-sin(\theta) \\ \scriptstyle sin(\theta) & \scriptstyle cos(\theta) \end{bmatrix} $.
Certainly, when we compose two reflections, we obtain
$ \begin{bmatrix} \scriptstyle cos(2\theta) & \scriptstyle sin(2\theta) \\ \scriptstyle sin(2\theta) & \scriptstyle-cos(2\theta) \end{bmatrix} \begin{bmatrix} \scriptstyle cos(2\phi) & \scriptstyle sin(2\phi) \\ \scriptstyle sin(2\phi) & \scriptstyle-cos(2\phi) \end{bmatrix} $
$ =\begin{bmatrix} \scriptstyle cos(2\theta)cos(2\phi)+sin(2\theta)sin(2\phi) & \scriptstyle cos(2\theta)sin(2\phi)-sin(2\theta)cos(2\phi) \\ \scriptstyle sin(2\theta)cos(2\phi)-cos(2\theta)sin(2\phi) & \scriptstyle sin(2\theta)sin(2\phi)+cos(2\theta)cos(2\phi) \end{bmatrix} =\begin{bmatrix} \scriptstyle cos(2(\theta-\phi)) & \scriptstyle sin(2(\phi-\theta)) \\ \scriptstyle sin(2(\theta-\phi)) & \scriptstyle cos(2(\theta-\phi)) \end{bmatrix} $
$ =\begin{bmatrix} \scriptstyle cos(2(\theta-\phi)) & \scriptstyle-sin(2(\theta-\phi)) \\ \scriptstyle sin(2(\theta-\phi)) & \scriptstyle cos(2(\theta-\phi)) \end{bmatrix} $
Which represents a rotation about the origin through the angle $ \scriptstyle2(\theta-\phi) $.
- --Nick Rupley 03:10, 25 January 2009 (UTC)
