Derivation of the 2D Rotation Matrix
Let G be a vector in the x-y plane with a length r and it traces out an angle v with respect to the x-axis. We now rotate G in the counter-clockwise direction by an angle θ. If (x, y) were the original coordinates of the tip of the vector G, then (x’, y’) will be the new coordinates after rotation.
Expressing (x, y) in the polar form we have;
- x = r cos v . . . (1)
- y = r sin v . . . (2)
Similarly, expressing (x’, y’) in polar form
x’ = r cos (v + θ)
y’ = r sin (v + θ)
Expanding the brackets using trigonometric identities we get,
x’ = r (cos v.cos θ – sin v.sin θ)
⇒ x’ = r cos v.cos θ – r sin v.sin θ
From (1) and (2) we have,
x’ = x cos θ – y sin θ . . . (3)
y’ = r (sin v.cos θ + cos v.sin θ)
⇒ y’ = r sin v.cos θ + r cos v.sin θ
⇒ y’ = y cos θ + x sin θ . . . (4)
If we take the help of a 2 × 2 rotation matrix to denote (3) and (4) we get,
[Tex]\begin{bmatrix} x’ \\ y’ \end{bmatrix} = \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix}[/Tex]
Thus, [Tex] R(\theta) = \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix}[/Tex] will be the rotation matrix.
Rotating Points in a 2D Plane
To rotate a point (x, y) in a 2D plane by an angle θ, you can multiply the point vector by the 2D rotation matrix:
[Tex]\begin{bmatrix} x’ \\ y’ \end{bmatrix} = \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}[/Tex]
The resulting vector (x’, y’) represents the rotated point.
Rotation Matrix
Rotation Matrix is a matrix used to perform a rotation in a coordinate space. Rotation matrices are widely used in various fields, including computer graphics, robotics, physics, and navigation systems, to describe and manipulate the orientation of objects in space. In this article, we will discuss Rotation Matrices including 2D and 3D rotation matrices.
Table of Content
- What is a Rotation Matrix?
- Definition of Rotation Matrix
- Example of Rotation using Matrix
- Representation of Rotations in Mathematics
- 2D Rotation Matrix
- Derivation of the 2D Rotation Matrix
- Rotating Points in a 2D Plane
- 3D Rotation Matrix
- Derivation of the 3D Rotation Matrix
- Rotating Points in 3D Coordinates
- Properties of Rotation Matrices
- Euler’s Rotation Theorem
- Applications of Rotation Matrices
- Conclusion