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
Double reflection in the same center is an identity transformation.
P' exists for any P. Let's write
SO2 = I,
where I is the identity transform.
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,
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:
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,
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
Copyright © 1996-2018 Alexander Bogomolny