Mathematical Basics of Motion and Deformation in Computer Graphics. Ken Anjyo
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      Note that the angle θ ∊ R is not uniquely determined. To be more precise, two matrices Rθ and Rθ′ give the same rotation if and only if θ – θ′ is an integer multiple of 2π. The compositions of two rotations and the inverse of a rotation are again rotations:

      Figure 2.2: 2D rotation.

Image

      Here Image denotes the inverse of Rθ. The totality of the rotations in 2D forms a group (also recall the definition of group in Chapter 1). It is denoted by

Image

      We also write as

Image

      where M(2, R) is the set of square matrices of size two, I is the identity matrix, and det is the determinant. The transpose1 of a matrix A is denoted by AT. A matrix A ∊ M(2, R) is a rotation matrix if and only if the column vectors u, v ∊ R2 of A form an orthonormal basis and the orientation from u to v is counter-clockwise. This means that a rotation matrix sends any orthonormal basis with the positive orientation to some orthonormal basis with the positive orientation.

      The result of the composition of several rotations in 2D is not affected by the order. This fact comes from the commutativity; Rθ Rθ′ = Rθ′ Rθ. Note that this is never true for 3D or a higher dimensional case.

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