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 OP' = OP. O is the only point for which O' = O. O is called the center of reflection. The transform itself is a particular case of rotation, viz., rotation through 180°. For this reason, it is often referred to as a half turn.

Double reflection in the same center is an identity transformation.

P' exists for any P. Let's write SO(P) = P'. If P' = SO(P), then P is the reflection in O of P': P = SO(P'). Formally,

SO2 = I,

where I is the identity transform. SO(P) ≠ P, unless P = O, so that O is the only fixed point of SO.

The reflection transform SO 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: SO(S). For any line L through O, SO(L) = L, although the identity does not hold pointwise.

To determine SO(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:

  1. Reflection in point does not change the orientation.

  2. Reflection in point is isometry: half turn preserves distances.

  3. Reflection in point preserves angles.

  4. Reflection in point maps parallel lines onto parallel lines.

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

  6. 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.

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

      SOSQ = -SQSO.

The applet helps experiment with the half turn. Two points A and B (and hence the whole segment AB) are reflected successively in points O1, O2, ...

For any n, AAn = BBn, AAn || BBn. For n even, AAn and BBn have the same orientation so that AB = AnBn and AB || AB = AnBn. For n odd, AAn and BBn meet at a fixed point (cf, Construction of n-gon by midpoints of its edges.)

Geometric Transformations

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