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Suppose that l,m,k,t differnt lines (Like this illustration)
Describe accurately this composition:
$S_k\circ\ S_t\circ\ S_m\circ\ S_l$

Well, composition of isometry is a group and i know that composition of two different reflections is a rotation. meaning, $S_k\circ\ S_t$ is rotation in 180°. same for $S_m\circ\ S_l$.
so $R\circ\ R$? what is the meaning of this composition?
Additionally i know that composition of 3 reflections is glide reflection, so $S_t\circ\ S_m\circ\ S_l$ is glide reflection. Let it be T.
Now i have $S_k\circ\ T$. Again, what is the meaning? hope everything is clear. thanks.

Dan
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1 Answers1

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The composition of two reflections is a rotation, and the composition of two rotations is again a rotation. So you get a rotation. (This reasoning uses the fact that $(A\circ B)\circ (C\circ D) = A\circ B\circ C\circ D$)

In two dimensions, to figure out the angle you just have to check one vector.

Thomas
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  • Thanks! Can you prove it somehow? i didnt understand why the composition of two rotations is rotation. – Dan Jun 09 '17 at 10:37
  • @BOIDEM depends on which tools you have available. If you know what a determinant is then you may know that the determinant of a rotation is $=1$ while the determinant of a reflection is $-1$. And the determinant of the product of two matrices is the product of the determinants. From this the claim trivially follows. If you want a proof with an elementary reasoning you should search the net, you will find explanations like this one: https://math.stackexchange.com/questions/704094/product-of-reflections-is-a-rotation-by-elementary-vector-methods – Thomas Jun 09 '17 at 12:11
  • Can it not be a translation? (Or do you find a translation a particular kind of rotation) – Willemien Jul 27 '17 at 12:17
  • @Willemien rotations and translations are linear maps, and so is their composition. A translation by a vector different from zero is not a linear map, but an affine linear map. Therefore it cannot arise as the composition of rotations and or reflections. – Thomas Jul 27 '17 at 15:57