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 S_O(P) = P'. If P' = S_O(P), then P is the reflection in O of P': P = S_O(P'). Formally,

S_O² = I,

where I is the identity transform. S_O(P) ≠ P, unless P = O, so that O is the only fixed point of S_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, S_O(L) = L, although the identity does not hold pointwise.

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:

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

Geometric Transformations

• Plane Isometries
• Plane Isometries As Complex Functions
  • Reflection in Point
  • Reflection in Line
    • Geometric Problems by Reflection
  • Translation Transform
  • Product of Rotations
    • Rotation Transform
  • Glide Reflection
• Affine Transformations (definition)
  • Iterated Function Systems
  • Homothety - an Affine Transform
    • Geometric Problems by Homothety
  • What Is Shear Transform?
  • Spiral Similarity
    • Construction of the center of spiral similarility
    • Spiral Similarity and Collinearity
• Inversion
  • Stereographic Projection and Inversion
  • Möbius Transformations
• Projective transformation (definition)

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