# 2D Rotation

The Rotation of any object depends upon the two points.

**Rotation Point: **It is also called **the Pivot point.**

**Rotation Angle: **It is denoted by **Theta ()**.

We can rotate an object in two ways-

**Clockwise: **An object rotates clockwise if the value of the Rotation angle is negative (-).

**Anti-Clockwise:** An object rotates anti-clockwise if the value of the Rotation angle is positive (+).

We can apply Rotation on following objects-

**Straight Lines****Curved Lines****Polygon****Circle**

**For Example**-

**Rotation of a Point: **If we want to Rotate a point **A (P _{0}, Q_{0}) **that has a Rotation angle

**with distance**

**r**from origin to

**A` (P**that has a Rotation angle

_{1}, Q_{1})**.**Then, we can rotate by following Rotation equation-

We can represent the coordinates of point A (P_{0}, Q_{0}) by using standard trigonometry-

We can also define point A` (P_{1}, Q_{1}) in the same way-

By using equation (1) (2) (3) (4), we will get-

**We can also represent the Rotation in the form of matrix**-

**Homogeneous Coordinates Representation:** The Rotation can also be represented in the form of 3 x 3 Rotation matrix-

**Example**- A line segment with the starting point (0, 0) and ending points (5, 5). Apply 30-degree rotation anticlockwise direction on the line. Find the new coordinates of the line?

**Solution**- We can rotate the straight line by its endpoints with the same angle.

We have,

(P_{0}, Q_{0}) = (0, 0)

Rotation Angle () = 30°

Let the new coordinates of line = (P_{1}, Q_{1})

We can apply the rotation matrix, then,

Thus, the new endpoint coordinates of the line are = (P_{1}, Q_{1}) = (1.83, 6.83)