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Rotation Transform

Rotation is a geometric transformation RO, a defined by a point O called the center of rotation, or a rotocenter, and an angle a, known as the angle of rotation. The case a = 0 (mod 2p) leads to a trivial transformation that moves no point. For any point P, its image P' = RO, a(P) lies at the same distance from O as P and, in addition

(1)POP' = a.

(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.)


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


Buy this applet

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

  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 180o is commonly referred to as a half-turn, a rotation through 90o is referred to as a quarter-turn. A half-turn is often referred to as a reflection in point.

Copyright © 1996-2008 Alexander Bogomolny

28695905Page copy protected against web site content infringement by Copyscape


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