Module ultraviolet::rotor

A rotor can be thought of in multiple ways, the first of which is that a rotor is the result of the 'geometric product' of two vectors, denoted for two vectors u and v as simply uv. This operation is defined as

uv = u · v + u ∧ v

As can be seen, this operation results in the addition of two different types of values: first, the dot product will result in a scalar, and second, the exterior (wedge) product will result in a bivector. The addition of these two different types is not defined, but can be understood in a similar way as complex numbers, i.e. as a 'bundle' of two different kinds of values.

The reason we call this type of value a 'rotor' is that if you both left- and right-multiply (using the geometric product) a rotor with a vector, you will rotate the sandwiched vector. For example, if you start with two vectors, a and b, and create a rotor ab from them, then rotate a vector u with this rotor by doing ba u ab, you will end up rotating the vector u by in the plane that corresponds to a ∧ b (i.e. the plane which is parallel with both vectors), by twice the angle between a and b, in the opposite direction of the one that would bring a towards b within that plane.

In ultraviolet, the Mul trait is implemented for Rotors such that doing

rotor * vec

will rotate the Vector vec by the Rotor rotor.

To compose rotations, simply left-multiply the rotor by another one in the same way that matrix composition works. For example,

rotor_ab = rotor_b * rotor_a

Will result in the composition of rotor_b and rotor_a such that rotor_ab encodes a rotation as though rotor_a was applied and then rotor_b was applied.

Note that composition of rotors is more efficient than composition of matrices, however, the operation of rotating a vector by a rotor, i.e. the rotor * vec product, is more expensive to compute than the matrix * vec product. So, rotors are excellent for building and interpolating rotations, but it may be preferrable to convert them into matrices before applying them to vectors/points, if the same rotation will be applied to many vectors.

Structs

Rotor2	A Rotor in 2d space.
Rotor3	A Rotor in 3d space.
WRotor2	A Rotor in 2d space.
WRotor3	A Rotor in 3d space.