Rotation Of Two-Dimensional Transformation

Rotation on a plane is determined by:

• Rotation center point
• Large angle of rotation
• Direction of rotation angle

The direction of rotation is said to be positive if it is anticlockwise and the direction of rotation is negative if it is handled with the hands of the clock.

1. Rotation with Center O (0, 0)

Rotation formula with the center O(0, 0)

x' = x cos θ - y sin θy' = x sin θ + y cos θ

2. Rotation with center P (a, b)

Rotational formula with center P (a, b)

x' - a = (x - a) cos θ - (y - b) sin θy' - b = (x - a) sin θ + (y - b) cos θ

Example:
Please determine the shadow from the point A(2, 3) when rotated by the 90° corner counterclockwise with the center P (1, -6)!

Answer:
Rotation 90° counterclockwise means θ = 90°
x' - a = (x - a) cos θ - (y - b) sin θ
x' - 1 = (2 - 1) cos 90°- (-3 - (-6)) sin 90°
x' - 1 = cos 90° - 3 sin 90°
x' - 1 = 0 - 3
x' - 1 + 1 = -3 + 1
x' = -2

y' - b = (x - a) sin θ + (y - b) cos θ
y' - (-6) = (2 - 1) sin 90° + (-3 - (-6)) cos 90°
y' + 6 = sin 90° + 3 cos 90°
y' + 6 = 1 + 0
y' + 6 - 6 = 1 - 6
y' = -5

So the shadow coordinates are A'(- 2, -5).

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Referensi :

• To'Ali's book math group accounting and sales

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