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Rotation in Geometry

In geometry, we often come across with transformations of geometrical shapes. The transformation is said to be the movement of shapes in coordinate plane. There are following four types of basic geometrical transformations - rotation, reflection, translation and resizing. Rotation is an important transformation in geometry. It is referred to a function's mapping rotated from one place to a different place. In this, the position has few geometric formations. The rotation in geometry is rotation of an object from one place to another keeping its structure same. In rotation, the object's position differs, but the shape remains same. The rotation may be of two types : rotation about a vertex of the object and rotation about an axis within the object. In rotating about a vertex, that vertex must be fixed and all other points except that vertex are to be rotated in a circular way. In rotation about an axis, all the points must be shifted. In this section, we shall learn about rotation in detail.


What is Rotation

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Rotation is a types of transformation in which an object is rotated in a circular way. Generally, the rotation has a center as well as an angle of rotation. There must be no change in shape or size of the object. The rotation involves revolving a geometrical object about a preset point which is recognized as the middle of the rotation.
For example :
Have a look at the following diagram in which the axis is rotated. The object originally had Y axis as an axis. During rotation, the axis is shifted to x-axis. The whole object eventually rotates with this shift.Examples for Rotation in Geometry

Take a look at another example shown in figure below. It shows the point rotation. Here, the object has been fixed at the center. Original position of the object is represented by red color shape, while the rotated image is shown by blue. The object is rotated in counterclockwise direction. Since. the center is fixed, its position does not change.

Angle for Rotation

The Angle of Rotation
A half-turn means a rotation of 180 degrees.
The quarter-turn means a rotation of 90 degrees.

Rules of Rotation

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The rotation of an object is performed in angles measured in degrees. There are few rules of rotation for angles are common degrees of rotations simple simple rotations can be performed easily in the coordinate plane using the rules below.
Rule 1 : 90$^{\circ}$ Rotation

(i) Clockwise
(x, y) $\rightarrow$ (-y, x)
i.e. if a point (x, y) is rotated 90$^{\circ}$ in clockwise direction, then it becomes (-y, x).
(ii) Counter Clockwise
(x, y) $\rightarrow$ (y, -x)
i.e. if a point (x, y) is rotated 90$^{\circ}$ in anticlockwise direction, then it becomes (y, -x).
Rule 2 :  180$^{\circ}$ Rotation

(x, y) $\rightarrow$ (-x, -y)
i.e. when a point (x, y) is rotated 180$^{\circ}$ in either clockwise or counter clockwise direction, it becomes (-x, -y) after rotation.
Rule 3 :  270$^{\circ}$ Rotation

Rotating an object 270$^{\circ}$ clockwise is exactly same as rotating it 90$^{\circ}$ counter clockwise. So,
(x, y) $\rightarrow$ (y, -x)
i.e. when a point (x, y) is rotated 270$^{\circ}$ in direction, it becomes (y, -x) after rotation.

In Euclidean Geometry

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Euclidean geometry is said to be a mathematical system which is developed mainly by Greek mathematician Euclid. Euclidean geometry is based upon Euclid methods, axioms and results. The two and three dimensional Euclidean planes are known as Euclidean space. The rotation in Euclidean geometry is very commonly performed. The normal rotation which we spoke in this page is actually a rotation with respect to Euclidean geometry. The motion of Euclidean space is similar to its isometry, so the distance between two given points remains unchanged after any transformation, such as rotation.

In Non-Euclidean Geometry

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The spherical geometry and elliptic geometry are the kinds of non-Euclidean geometry in which motion of the n-sphere is similar to the rotation of Euclidean space of (nā€‰+ā€‰1) dimensions about the origin. When n is odd, the motions generally do not have fixed points on n-sphere. We can say that there exists no rotations of the sphere. Though, In hyperbolic and elliptic geometries, the rotations with respect to a fixed point are similar to that of Euclidean space.

Example of Rotation

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Few examples based on rotation are illustrated below:

Example 1: Take a look at the following figure showing the geometric rotation of whole object, where center of rotation is outside the object.

Math Rotation

Example 2: Rotation about a point

Point Rotation

One more example is shown below.
Center of Rotation

Here, the center of rotation is rotated 90$^{\circ}$ in counterclockwise direction. The next rotation is 180$^{\circ}$. For full rotation (360$^{\circ}$) of the object, there are exactly four positions. The positions of the rotating angle are 90,180,270 and 360 degrees.

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