User:Rschwieb/GA Discussion/Comparison of methods: rotations

• Problem A: The matrix b a^T rotates a unit vector a to a unit vector b More complicated than this: matrix needs to be full rank. Without thinking very hard I suspect you may have to construct full before-and-after orthonomal basis sets A and B, then form B A^T similar to Problem B, but there may be some short-cuts for the invariant subspace.
• Problem B: see e.g Triad method for 3d case; in higher dimensions, one could use Gram-Schmidt to enforce orthogonalisation.

In a geometric algebra, a rotation is represented directly by an element of the algebra, called a rotor. A rotor R acts upon any element X of the algebra to produce its rotated equivalent X′:

: $\backslash ,\; X\text{'}\; =\; RXR^\{-1\}$.

In 3 dimensions, this is exactly the quaternion method. (Found at [http://www.euvt.eu/fontijnes-formula-published/ Euvision Technologies]:

“Fontijne’s Formula” (apparently derived from GA): if 3D vector p rotates to p’, and q to q’, the quaternion of the rotation is [(q’+q)·(p’−p), (q’−q)×(p’−p)]. — Quondum^☏✎ 20:34, 6 February 2012 (UTC))

It is customary (but not necessary) to normalize R, so that RR^~ = 1, i.e. R⁻¹ = R^~. R is always an even multivector. Here the notation A^~ (alternately A^†) denotes the reversion of A.

;Solution of Problems A and B in n-D (in 3-D this is no different):

Given two vectors a and b with a² = b² = ±1, the rotation that rotates a into b and keeps their orthogonal complement unchanged is:

:$R=\backslash pm\; \backslash frac\{1+ba\}$

The solution has been normalized. The distinction of sign is topologically significant. The case of (a + b)² = 0 corresponds to no solution.

::This rotation is associated with a plane of rotation given by b∧a. If b = −a, this is not sufficient to define a plane; there are many planes that could achieve such a rotation, but such rotations will also cause some vectors orthogonal to a and b to be transformed.

;Example solution of Problem B:

Given two sets of k vectors {e_i} and {f_i} related by a rotation, find a rotor R that rotates the one into the other (so that f_i = Re_iR⁻¹), leaving all orthogonal vectors unchanged. Doran and Lazenby give the normalized solution for where the vectors form a 3-D basis in Euclidean space:

:$\backslash ,\; R\; =\; \backslash pm\; \backslash frac\{1\; +\; f\_j\; e^j\}$

where {e^k} is the dual basis to {e_k}.

The case of f_ke^k = −1 requires special care.

A solution for n-D and n independent vectors related by a rotation (not verified):

:$\backslash ,\; R\; =\; \backslash pm\; \backslash frac\{1\; +\; f\_k\; e^k\}\{\backslash sqrt\{2\; +\; 2\; f\_k\; \backslash cdot\; e^k\}\}$

;Example 3:

Given a plane and an angle of rotation in n-D, give the bivector describing the rotation and how R is related to it.

Given two vectors a and b spanning the plane and with the magnitude giving angle of rotation of θ in the direction from a to b, the bivector describing the plane and angle of rotation (picture a unit vector sweeping out a pizza slice) is

:$\backslash ,\; B\; =\; \backslash frac\{\backslash theta\; (a\; \backslash wedge\; b)\}$

and the rotor is

:$\backslash ,\; R\; =\; \backslash exp\; (-B/2)$.

This can be easily generalized to independent angles with respect orthogonal planes of rotation in 4-D and above: just add the bivectors.

;Example 4:

Smooth rotation: expression of the SLERP in GA.

;Example "baby" solution in quaternions:

Problem: Demonstrate how to rotate the vector [0,1,1] 90 degrees about the y-axis to lie over [1,1,0] using a quaternion.

:I am inclined to think the problem through in 3D GA, and apply the following mapping (treating quaternions (underlined) as being the even subalgebra of the GA), with the correspondence {{math|1=i = −e₂e₃}}, {{math|1=j = −e₃e₁}}, {{math|1=k = −e₁e₂}}; also {{math|1=I = e₁e₂e₃}}:

::MAPPING : QUATERNION ↔ {{math|1=G_3,0}}

::identity : quaternion scalar ↔ scalar [{{math|1={{underline|a}} = a}}]

::dual : quaternion vector ↔ vector [{{math|1={{underline|a}} = −aI}}, {{math|1=a = {{underline|a}}I}}]

::identity : quaternion pseudovector ↔ bivector [{{math|1={{underline|a}} = a}}]

::identity : quaternion rotor ↔ even grade versor (i.e. proper rotor) [{{math|1={{underline|R}} = R}}]

::dual : quaternion reflection ↔ odd grade versor (i.e. improper rotor)

::dual : quaternion pseudoscalar ↔ pseudoscalar

:In GA, the rotor for the rotation (rotating $\backslash hat\{z\}\; =\; e\_3$ into $\backslash hat\{x\}\; =\; e\_1$ the short way around the {{math|y}}-axis; for the long way negate the rotor) is

::$\backslash underline\{R\}\; =\; R\; =\; \{1\; +\; e\_1e\_3\; \backslash over\; \backslash left|\; e\_1\; +\; e\_2\; \backslash right|\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; +\; \{\; j\; \backslash over\; \backslash sqrt\{2\}\; \}$

::$a\; =\; e\_2\; +\; e\_3\; \backslash qquad\; b\; =\; e\_1\; +\; e\_2$

::$\backslash underline\{a\}\; =\; -aI\; =\; -(e\_2\; +\; e\_3)e\_1e\_2e\_3\; =\; j+k\; \backslash qquad\; \backslash underline\{b\}\; =\; -bI\; =\; -(e\_1\; +\; e\_2)e\_1e\_2e\_3\; =\; i+j$

:So working purely with quaternions,

::$\backslash underline\{a\}\; =\; j+k\; \backslash qquad\; \backslash underline\{b\}\; =\; i+j$

::$\backslash underline\{R\}\; =\; \backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\; +\; \backslash tfrac\{j\}\{\backslash sqrt\{2\}\}\; \backslash qquad\; \backslash underline\{R\}^\{\backslash dagger\}\; =\; \backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\; -\; \backslash tfrac\{j\}\{\backslash sqrt\{2\}\}$

:Checking, we find

::$\backslash underline\{R\}\backslash underline\{a\}\backslash underline\{R\}^\{\backslash dagger\}\; =\; (\backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\; +\; \backslash tfrac\{j\}\{\backslash sqrt\{2\}\})(j+k)(\backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\; -\; \backslash tfrac\{j\}\{\backslash sqrt\{2\}\})\; =\; i+j\; =\; \backslash underline\{b\}$

:as expected.

:Beware the normal convention with quaternions is not the GA-centric approach I have used, though I have deliberately chosen the dual mapping for vectors to have matching signs, so it may be directly equivalent. Note that I have used the GA vectors (not underlined) rather than the corresponding quaternion vectors to calculate the rotor, though finding the expression for $\backslash underline\{R\}$ from the quaternions $\backslash hat\{\backslash underline\{x\}\}\; =\; -\backslash hat\{x\}I\; =\; i$ etc. will be simple enough. Feel free to trim this down to a concise version. Also, no guarantees that this is correct. — Quondum^☏ 21:29, 28 April 2012 (UTC)