What is rotation matrix?

A rotation matrix is a transformation matrix that rotates a vector in a Euclidean space. It is a real square matrix whose determinant is 1 and whose transpose is its inverse. This means that a rotation matrix, denoted by R, satisfies the following conditions:

• R is a square matrix.
• R is a real matrix (its elements are real numbers).
• det(R) = 1 (its determinant is 1).
• Rᵀ = R⁻¹ (its transpose is its inverse). Therefore, RᵀR = RRᵀ = I, where I is the identity matrix.

Rotation matrices are used extensively in various fields, including:

• Computer graphics: Used to rotate objects in 2D and 3D space.
• Robotics: Used to represent the orientation of robots and their joints.
• Physics: Used to describe rotations in classical mechanics and quantum mechanics.
• Linear Algebra: Provides a matrix representation for rotational transformations.

Properties of Rotation Matrices:

• Orthogonal: As mentioned above, Rᵀ = R⁻¹, meaning rotation matrices are orthogonal matrices. This property is crucial for preserving lengths and angles during rotation.
• Preserve Lengths and Angles: Applying a rotation matrix to a vector does not change its length or the angles between vectors.
• Determinant of 1: The determinant of a rotation matrix is always 1. This distinguishes rotations from reflections, which have a determinant of -1. Matrices with determinant of +1 are called special orthogonal matrices.
• Composition of Rotations: Multiplying two rotation matrices results in another rotation matrix. This corresponds to performing two rotations in sequence. The order of multiplication matters, as rotations are generally not commutative.
• Inverse of Rotation: The inverse of a rotation matrix (its transpose) represents the inverse rotation – the rotation that undoes the original rotation.

Examples:

• 2D Rotation Matrix: A 2D rotation matrix that rotates a vector counterclockwise by an angle θ is given by:

  R = | cos(θ)  -sin(θ) |
      | sin(θ)   cos(θ) |
  

• 3D Rotation Matrices: In 3D, rotations can be performed around the x, y, or z axes. These are represented by the following matrices:

  • Rotation around the x-axis by an angle α:

    Rx = | 1      0       0     |
         | 0   cos(α)  -sin(α) |
         | 0   sin(α)   cos(α) |
    

  • Rotation around the y-axis by an angle β:

    Ry = |  cos(β)  0   sin(β) |
         |    0     1      0    |
         | -sin(β)  0   cos(β) |
    

  • Rotation around the z-axis by an angle γ:

    Rz = | cos(γ)  -sin(γ)  0 |
         | sin(γ)   cos(γ)  0 |
         |    0       0      1 |
    

Any arbitrary 3D rotation can be represented as a sequence of rotations about these three axes (Euler angles) or using a quaternion representation which avoids gimbal lock, a problem that Euler angles can suffer from.

Understanding rotation matrices is fundamental for working with transformations in multiple dimensions.