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
|(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
Copyright © 1996-2018 Alexander Bogomolny