What is 2D Rotation in computer graphics? Derive the 3 * 3 rotation matrix…

1. Definition of 2D Rotation

2D Rotation is a geometric transformation that rotates an object about a fixed point called the Pivot Point or Center of Rotation. In most standard derivations, the origin (0, 0) is used as the pivot. Rotation changes the orientation of an object while preserving its shape and size, making it a rigid-body transformation.

1. Mathematical Representation

In computer graphics, we use Homogeneous Coordinates to represent rotation as a 3 * 3$ matrix. This allows us to unify rotation with translation and scaling operations. For a counter-clockwise rotation by an angle theta, the transformation matrix R is:

1. Numerical Calculation

To find the new coordinates P'(x', y') of point P(3, 4) rotated by theta = 90o counter-clockwise:

• Determine Trigonometric Values:

  cos(90o) = 0

  sin(90o) = 1

• Set up the Matrix Equation (P' = R.P):

  

• Solve for x' and y':

  x' = (0*3) + (-1*4) + (0*1) = -4

  y' = (1*3) + (0*4) + (0*1) = 3

Final Result: After a 90o counter-clockwise rotation, the new coordinates of the point are P'(-4, 3).