 # Rotation

The meaning of rotation in Maths is the circular motion of an object around a center or an axis. In real-life, we know the earth rotates on its own axis, which is also an example of rotation. In Geometry, there are four basic types of transformations. They are

• Rotation
• Reflection,
• Translation
• Resizing

Here, we will discuss one of the transformation types called “Rotation” in detail along with its definition, formula, rules, rotational symmetry and examples.

## Rotation Definition

Rotation means the circular movement of an object around a center. It is possible to rotate different shapes by an angle around the center point. In three-dimensional shapes, the objects can be rotated about an infinite number of imaginary lines known as rotational axes. The rotations around X, Y and Z axes are known as the principal rotations. The rotations around any axis can be performed by taking the rotation around X-axis, followed by Y-axis and then finally z-axis.

## Rotation Formula

Rotation can be done in both directions like clockwise as well as in counterclockwise. The most common rotation angles are 90°, 180° and 270°. There are certain rules for rotation in the coordinate plane. They are:

 Type of Rotation A point on the Image A point on the Image after Rotation Rotation of 90° (Clockwise) (x, y) (y, -x) Rotation of 90° (CounterClockwise) (x, y) (-y, x) Rotation of 180° (Both Clockwise and Counterclockwise) (x, y) (-x, -y) Rotation of 270° (Clockwise) (x, y) (-y, x) Rotation of 270° (CounterClockwise) (x, y) (y, -x)

## Rotation Matrix

A rotation matrix is a matrix used to perform a rotation in a Euclidean space. In a two-dimensional cartesian coordinate plane system, the matrix R rotates the points in the XY-plane in the counterclockwise through an angle θ about the origin. The matrix R is given as,

$R=\begin{bmatrix} \cos \Theta & -\sin \Theta \\ \sin \Theta & \cos \Theta \end{bmatrix}$

In order to perform the rotation operation using the rotation matrix R, the position of each point in the plane is represented by a column vector “v”, that contains the coordinate point. With the help of matrix multiplication Rv, the rotated vector can be obtained.

### Rotational Symmetry

In geometry, many shapes have rotational symmetry like circles, square, rectangle. All the regular polygons have rotational symmetry. If an object is rotated around its centre, the object appears exactly like before the rotation. Then the object is said to have rotational symmetry. The order of symmetry can be found by counting the number of times the figure coincides with itself when it rotates through 360°.

### Rotation Examples The above example shows the rotation of a rectangle 90° each time. The rectangle has the rotational symmetry of order 2 because when it is rotated twice, we get the original shape at 180° and again when it is rotated twice, the original shape is obtained at 360°. So, the order of rotational symmetry of the rectangle is 2.

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