Rotation Transform

Rotation is a geometric transformation RO, 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 α = 0 (mod 2p) leads to a trivial transformation that moves no point. For any point P, its image P' = RO, α(P) lies at the same distance from O as P and, in addition

(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:

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

  2. Rotation is isometry: a rotation preserves distances.

  3. Rotation preserves angles.

  4. Rotation maps parallel lines onto parallel lines.

  5. 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.

  6. Successive rotations result in a rotation or a translation.

  7. 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:

  1. A Problem of Hinged Squares
  2. About a Line and a Triangle
  3. Bottema's Theorem
  4. Bride's Chair
  5. Equal And Perpendicular Segments in a Square
  6. Equilateral and 3-4-5 Triangles
  7. Equilateral Triangle on Parallel Lines
  8. Equilateral Triangle on Three Lines
  9. Equilic Quadrilateral I
  10. Fermat Points and Concurrent Euler Lines II
  11. Four Construction Problems
  12. Napoleon on Hinges
  13. Napoleon's Theorem by Transformation
  14. On Bottema's Shoulders
  15. Point in a square
  16. Similar Triangles on Sides and Diagonals of a Quadrilateral
  17. Thébault's Problem II
  18. Two Equilateral Triangles
  19. When a Triangle is Equilateral?
  20. Two Equilateral Triangles on Sides of a Square
  21. Luca Moroni's Problem In Equilateral Triangle
  22. A Triangle in a Rhombus with a 60 Degrees Angle
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