Rotasjonsformelen

Rotasjonsformelen, often translated as the rotation formula, is a mathematical concept used in geometry and linear algebra to describe the transformation of coordinates when a point or object is rotated around an origin. It is particularly useful in two-dimensional Cartesian coordinate systems. The formula allows us to calculate the new coordinates (x', y') of a point (x, y) after it has been rotated by an angle θ (theta) counterclockwise around the origin (0,0).

The standard rotation formula in two dimensions is given by:

x' = x cos(θ) - y sin(θ)

y' = x sin(θ) + y cos(θ)

Here, x and y are the original coordinates of the point, and x' and y' are the new coordinates after rotation. The angle θ is measured in radians or degrees, and its sign determines the direction of rotation: a positive angle typically denotes a counterclockwise rotation, while a negative angle denotes a clockwise rotation.

This formula can be derived using trigonometry, by considering the trigonometric representations of the original and rotated points. In linear algebra, the rotation can be represented by a rotation matrix. Multiplying the original coordinate vector by this matrix yields the rotated coordinate vector. The rotasjonsformelen has wide applications in computer graphics, robotics, physics, and various engineering disciplines where rotations are a fundamental operation.