# Rotation Transform

*Rotation* is a geometric transformation R_{O, α} defined by a point O called the *center of rotation*, or a *rotocenter*, and an angle α, known as the *angle of rotation*. The case _{O, α}(P)

(1) | ∠POP' = α. |

(In the applet below, various rotations are controlled by a hollow blue point - the center of rotation, and a slider that determines the angle of rotation. In the applet, you rotate a pentagon whose shape is defined by draggable vertices.)

The following observations are noteworthy:

Rotation preserves the orientation. For example, if a polygon is traversed clockwise, its rotated image is likewise traversed clockwise.

Rotation is isometry: a rotation preserves distances.

Rotation preserves angles.

Rotation maps parallel lines onto parallel lines.

Except for the trivial rotation through a zero angle which is identical, rotations have a single fixed point - the center of rotation. Except for the trivial case, rotations have no fixed lines. However, all circles centered at the center of rotation are fixed.

Successive rotations result in a rotation or a translation.

The product of rotations is not in general commutative. Two rotations with a common center commute as a matter of course.

Two special rotations have acquired appellations of their own: a rotation through 180° is commonly referred to as a *half-turn*, a rotation through 90° is referred to as a *quarter-turn*. A half-turn is often referred to as a *reflection in point*.

Here is a short list of a problems that are solved with the help of the rotation transform:

- A Problem of Hinged Squares
- About a Line and a Triangle
- Bottema's Theorem
- Bride's Chair
- Equal And Perpendicular Segments in a Square
- Equilateral and 3-4-5 Triangles
- Equilateral Triangle on Parallel Lines
- Equilateral Triangle on Three Lines
- Equilic Quadrilateral I
- Fermat Points and Concurrent Euler Lines II
- Four Construction Problems
- Napoleon on Hinges
- Napoleon's Theorem by Transformation
- On Bottema's Shoulders
- Point in a square
- Similar Triangles on Sides and Diagonals of a Quadrilateral
- Thébault's Problem II
- Two Equilateral Triangles
- When a Triangle is Equilateral?
- Two Equilateral Triangles on Sides of a Square
- Luca Moroni's Problem In Equilateral Triangle
- A Triangle in a Rhombus with a 60 Degrees Angle

### Geometric Transformations

- Plane Isometries
- Plane Isometries As Complex Functions
- Affine Transformations (definition)
- Inversion
- Projective transformation (definition)

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Copyright © 1996-2018 Alexander Bogomolny

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