Question

Describe the general properties of rotations. Include a discussion of the properties of rigid transformations, and line segments connecting corresponding points to each other and to the center of rotation.

Answer 1

Here are the essential geometric properties of a rotation (in the Euclidean plane) and how it fits into the family of rigid transformations.

Definition
- A rotation R about a point O (the center) through angle θ is the map that sends each point P to the unique point P' such that O is fixed and the directed angle ∠POP' = θ (measured in a chosen orientation).

Rotations as rigid transformations (isometries)
- A rotation is a rigid motion (isometry): it preserves all distances. In particular for any points A, B,
AB = A'B' where A', B' are the images under the rotation.
- It preserves angles, lengths of segments, areas, and congruence of figures.
- It is orientation-preserving (unlike a reflection), continuous, bijective, and invertible; the inverse is the rotation about the same center by angle −θ.
- Rotations, together with translations, reflections and glide reflections, form the Euclidean isometries. Composition rules: rotation about O by θ1 followed by rotation about the same O by θ2 is rotation about O by θ1+θ2. Composition of two rotations with different centers is either another rotation (if net angle ≠ 0 mod 360°) or a translation (if the net rotation angle is 0 mod 360°).

Fixed points and invariant sets
- The center O is fixed. If θ is not a multiple of 360°, no other points are fixed.
- Every circle centered at O is invariant (the rotation permutes its points). A rotation maps any circle centered at O to itself.
- Lines: a rotation maps lines to lines. A line through O is sent to the line through O obtained by rotating the original line by θ; a line not through O is sent to a parallel line if θ = 0 or 180°? (More generally it is sent to a different line of the same direction only if θ = 0; however parallelism of pairs is preserved because rotations preserve angles.)

Algebraic/linear viewpoint
- With O as origin, rotation is a linear map represented by the 2×2 orthogonal matrix with determinant +1:
[cos θ −sin θ; sin θ cos θ].
- The rotation group (about a fixed center) is isomorphic to the circle group S^1.

Geometry of segments joining corresponding points (P to P')
For a rotation about O by angle θ and any point P with image P':
- OP = OP' (P and P' lie on the same circle centered at O).
- The triangle OPP' is isosceles with apex angle at O equal to θ: ∠POP' = θ.
- The length of the chord PP' is determined by OP and θ:
PP' = 2·OP·sin(θ/2).
- The perpendicular bisector of PP' passes through O. Equivalently, the midpoint M of PP' lies on the line through O that bisects angle ∠POP', and OM ⟂ PP'.
- The distance from O to the midpoint M of PP' is
OM = OP·cos(θ/2).
- As OP varies, the chords PP' for a fixed θ are chords on concentric circles; for a fixed radius r, all such midpoints M lie on a circle of radius r cos(θ/2).

Other useful consequences
- Corresponding segments are congruent: for any two points P,Q, PQ = P'Q'.
- Collinearity and betweenness are preserved (images of collinear points are collinear and order along a line is preserved).
- Parallelism is preserved (rotation preserves the angle between two lines).
- If θ = 0 (mod 360°) rotation is the identity; if θ = 180° the rotation is a half-turn and every line through O is setwise invariant (each point on such a line goes to the point directly opposite across O).

Summary
A rotation is a distance- and angle-preserving rigid motion fixing a center O and rotating every radius by the same angle. For each point P, OP = OP', the chord PP' has length 2·OP·sin(θ/2), its midpoint lies on the line from O that bisects ∠POP', and O lies on the perpendicular bisector of PP'. Rotations preserve congruence, orientation, areas, and the full Euclidean structure.