Consider the following matrix used for reflection.

|  0  -1 |   
| -1   0 |   

This matrix produces the reflection across y=-x according to [B] = [T]×[A] where T is the above Transformation Matrix. Things are clear till now. Then I was introduced to rotation and taught that every reflection is some sort of rotation according to the following transformation matrix.

| cos(@) -sin(@) |
| sin(@)  cos(@) |   

Where @ is angle of rotation. My question is that for firstly stated transformation matrix that is

|  0   -1 |   
| -1    0 |  

for reflection. Then how this satisfy the transformation matrix for rotation if every reflection (with reflection plane passes through the origin) is some sort of rotation?


Not every reflection is possible to rotate. In this case you'll notice that a 90 degree rotation produces:

| 0 -1 |
| 1  0 |

There is not in fact any value for x that satisfies the equation*

-sin(x) = -1 
 sin(x) = -1

Any two (or multiples thereof 2,4,6 etc...) reflections can be expressed as one rotation tough. See Euclidean Spaces article. Mainly, because twice reflected is right way around which can all be represented as a rotation. So the space of x time mirrored objects is twice as big as the total space of rotations.

* This is easy to experiment, no matter how you turn your hand there's no way your left and right hand aligns. This is also easy to visualize. If we plot both sin(x) and -sin(x), then if their lowest point ever meets that's the solution.

Plot[{Sin[x], -Sin[x]}, {x, -3 Pi , 3 Pi}

Image 1: Plot of Sin(x) and its negative pair never meet at the lowest point. Because the functions are in fact in antiphase.

There is a rotation matrix that is different handed that can describe the mirrored image, it is however eqivalent of a mirror operation so it should not count.

  • Then in which case we can say that a reflection is rotation and vice-versa,and in which case these are not interrelated? – OldSchool Oct 1 '14 at 10:56
  • @Rouftantical Two reflections constitute one rotation. One reflection always gives a a different coordinate system. But if you extend a bit you can say that too is a rotation but with a different basis. – joojaa Oct 1 '14 at 11:34
  • please if you can expand it more.I am just requiring a single example to understand the concept. – OldSchool Oct 14 '14 at 12:28

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