# Half Turn, Reflection in Point

Given a point O and a point P. *Reflection P' of P in O* is the point collinear with P, O and such that *half turn*.

Double reflection in the same center is an identity transformation.

P' exists for any P. Let's write _{O}(P) = P'._{O}(P), then P is the reflection in O of P': _{O}(P').

S_{O}^{2} = I,

where I is the identity transform. _{O}(P) ≠ P,_{O}.

The reflection transform S_{O} applies to arbitrary shapes point-by-point. Each point of a given shape S is reflected in O, and the collection of these reflections is the symmetric image of S: S_{O}(S). For any line L through O, _{O}(L) = L,

To determine S_{O}(S) when S is a segment, suffice it to reflect its end points. This is exactly what has been done in the applet below. In the applet, you can drag the red arrow as a whole or by its starting; you can drag the centers of reflections, or choose to have three or four of them.

The following observations are noteworthy:

Reflection in point does not change the orientation.

Reflection in point is isometry: half turn preserves distances.

Reflection in point preserves angles.

Reflection in point maps parallel lines onto parallel lines.

All lines through the center of a half turn are fixed.

Successive reflections in several centers result in a translation, if the number of reflections is even, or in another half turn, if the number of reflections is odd.

The order of reflections matters: two reflections do not commute. In fact, for two centers O and Q,

S _{O}S_{Q}= -S_{Q}S_{O}.

The applet helps experiment with the half turn. Two points A and B (and hence the whole segment AB) are reflected successively in points O_{1}, O_{2}, ...

For any n, AA_{n} = BB_{n}, AA_{n} || BB_{n}. For n even, AA_{n} and BB_{n} have the same orientation so that _{n}B_{n}_{n}B_{n}._{n} and BB_{n} meet at a fixed point (cf, Construction of n-gon by midpoints of its edges.)

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