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

(In the applet below, various rotations are controlled by a hollow blue point - the center of rotation, and a dial with a draggable point which determines the angle of rotation. In the applet, you can create polygons with a desired number of vertices, drag the vertices one at a time, or drag the polygon as a whole. You'll see the applet in action after you Add a shape to apply rotations to.)

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. Equilateral and 3-4-5 Triangles
6. Equilateral Triangle on Parallel Lines
7. Equilic Quadrilateral I
8. Fermat Points and Concurrent Euler Lines II
9. Four Construction Problems
10. Napoleon on Hinges
11. Napoleon's Theorem by Transformation
12. On Bottema's Shoulders
13. Point in a square
14. Similar Triangles on Sides and Diagonals of a Quadrilateral
15. Thébault's Problem II
16. Two Equilateral Triangles
17. When a Triangle is Equilateral?

Geometric Transformations

• Plane Isometries
  • 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
    • Geometric Problems by Homothety
  • Shearing
  • Spiral Similarity
    • Construction of the center of spiral similarility
• Inversion
  • Stereographic Projection and Inversion
  • Möbius Transformations
• Projective transformation

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Copyright © 1996-2012 Alexander Bogomolny