A rotation is one of three rigid motions that move a figure in a plane without changing its size or shape. As its name implies, a rotation moves a figure by rotating it around a center somewhere on a plane. This center can be somewhere inside or on the figure, or outside the figure completely. The two other rigid motions are reflections and translations.
Figure 1 illustrates a rotation of 30° around a point C. This rotation is counterclockwise, which is considered positive. Clockwise rotations are negative. The "product" of two rotations, that is, following one rotation with another, is also a rotation. This assumes that the center of rotation is the same for both. When one moves a heavy box across the room by rotating it first on one corner then on the other, that "product" is not a rotation.
Rotations are so commonplace that it is easy to forget how important they are. A person orients a map by rotating it. A clock shows time by the rotation of its hands. A person fits a key in a lock by rotating the key until its grooves match the pattern on the keyhole. Rotating an M 180° changes it into a W; 6s and 9s are alike except for a rotation.
Rotary motions are one of the two basic motions of parts in a machine. An automobile wheel converts rotary motion into translational motion, and propels the car. A drill bores a hole by cutting away material as it turns. The earth rotates on its axis. The earth and the moon rotate around their centers of gravity, and so on.
Astronomy prior to Copernicus was greatly complicated by trying to use the earth as the center of the rotation of the planets. When Kepler and Copernicus made the sun the gravitational center, the motions of the planets became far easier to predict and explain (but even with the sun as the center, planetary motion is not strictly rotational).
When points are represented by coordinates, a rotation can be effected algebraically. How hard this is to do depends upon the location of the center of rotation and on the kind of coordinate system which is used. In the two most commonly employed systems, the rectangular Cartesian coordinate system and the polar coordinate system, the center of choice is the origin or pole.
In either of these systems a rotation can be thought of as moving the points and leaving the axes fixed, or vice versa. The mathematical connection between these alternatives is a simple one: rotating a set of points clockwise is equivalent to rotating the axes, particularly with reflections, it is usually preferable to leave the axes in place and move the points.
When a point or a set of points is represented with polar coordinates, the equations that connect a point (r, θ) with the rotated image (r', θ') are particularly simple. If θ1 is the angle of rotation:
Thus, if the points are rotated 30° counterclockwise, (7,80°) is the image of 7,50°). If the set of points described by the equation r = θ/2 is rotated π units clockwise, its image is described by r = θ - π)/2. Rectangular coordinates are related to polar coordinates by the equations x = r cos θ and y = r sin θ.
Therefore the equations which connect a point (x, y) with its rotated image (x', y') are
Using the trigonometric identities for cos (θ + θ1) and sin (θ + θ1), these can be written x' = x cos θ1 - y sin θ1 and y' = x sin θ1 + y cos θ1 or, after solving for x and y: x = x' cos θ1 + y sin θ1 and y = -x' sin θ1 + y cos θ1.
To use these equations one must resort to a table of sines and cosines, or use a calculator with SIN and COS keys.
One can use the equations for a rotation many ways. One use is to simplify an equation such as x2 - xy + y2 = 5. For any second-degree polynomial equation in x and y there is a rotation which will eliminate the xy term. In this case the rotation is 45°, and the resulting equation, after dropping the primes, is 3x2 + y2 = 10.
Another area in which rotations play an important part is in rotational symmetry. A figure has rotational symmetry if there is a rotation such that the original figure and its image coincide. A square, for example, has rotational symmetry because any rotation about the square's center which is a multiple of 90° will result in a square that coincides with the original. An ordinary gear has rotational symmetry. So do the numerous objects such as vases and bowls which are decorated repetitively around the edges. Actual objects can be checked for rotational symmetry by looking at them. Geometric figures described analytically can be tested using the equations for rotations. For example, the spiral r = 28 has two-fold rotational symmetry. When the spiral is rotated 180°, the image coincides with the original spiral.
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J. Paul Moulton