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Concept
Plane of rotation
Summary
In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space.
The main use for planes of rotation is in describing more complex rotations in four-dimensional space and higher dimensions, where they can be used to break down the rotations into simpler parts. This can be done using geometric algebra, with the planes of rotations associated with simple bivectors in the algebra.
Planes of rotation are not used much in two and three dimensions, as in two dimensions there is only one plane (so, identifying the plane of rotation is trivial and rarely done), while in three dimensions the axis of rotation serves the same purpose and is the more established approach.
Mathematically such planes can be described in a number of ways. They can be described in terms of planes and angles of rotation. They can be associated with bivectors from geometric algebra. They are related to the eigenvalues and eigenvectors of a rotation matrix. And in particular dimensions they are related to other algebraic and geometric properties, which can then be generalised to other dimensions.
Hyperplane
For this article, all planes are planes through the origin, that is they contain the zero vector. Such a plane in n-dimensional space is a two-dimensional linear subspace of the space. It is completely specified by any two non-zero and non-parallel vectors that lie in the plane, that is by any two vectors a and b, such that
where ∧ is the exterior product from exterior algebra or geometric algebra (in three dimensions the cross product can be used). More precisely, the quantity a ∧ b is the bivector associated with the plane specified by a and b, and has magnitude sin φ, where φ is the angle between the vectors; hence the requirement that the vectors be nonzero and nonparallel.
If the bivector a ∧ b is written B, then the condition that a point lies on the plane associated with B is simply
where is the dot product. This is true in all dimensions, and can be taken as the definition on the plane.
Official source
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