What is Rotation in Coordinate Geometry?
Rotation is a geometric transformation that moves every point of a figure around a fixed center by a specified angle. In 2D coordinate geometry, the most common rotation is about the origin. The distance from the origin remains constant while the angular position changes. Rotation is fundamental in computer graphics, robotics, physics simulations, and engineering design.
How to Calculate New Coordinates by Rotation
For a counter-clockwise rotation by angle θ:
[X' = X \cos(\theta) - Y \sin(\theta)]
[Y' = X \sin(\theta) + Y \cos(\theta)]
For a clockwise rotation by angle θ:
[X' = X \cos(\theta) + Y \sin(\theta)]
[Y' = -X \sin(\theta) + Y \cos(\theta)]
Where:
- (X, Y) are the original coordinates.
- (X', Y') are the new coordinates after rotation.
- θ is the angle of rotation (convert degrees to radians if needed).
Calculation Example
Rotate the point (4, 5) by 30 degrees counter-clockwise.
First, convert 30 degrees to radians: 30 × π/180 ≈ 0.5236
[X' = 4 \times \cos(0.5236) - 5 \times \sin(0.5236)]
[X' = 4 \times 0.8660 - 5 \times 0.5 = 3.4641 - 2.5 = 0.9641]
[Y' = 4 \times \sin(0.5236) + 5 \times \cos(0.5236)]
[Y' = 4 \times 0.5 + 5 \times 0.8660 = 2.0 + 4.3301 = 6.3301]
The new coordinates are approximately (0.9641, 6.3301).