3D rotation group

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In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} under the operation of composition, which combines two rotations by performing one after the other. A rotation about a point is a transformation that preserves that point, while also preserving the Euclidean distance between any two points (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation. Owing to the above properties (along composite rotations' associative property), the set of all rotations is a group under composition. Every non-trivial rotation is determined by its axis of rotation (a line through the...

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