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Units, Constants and Useful Formulas
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Last modified: December 16, 2021 ✓
Quaternions 1
-------------
Quaternions are a number system that extends the
complex numbers.
They are represented in the form:
q = a + bi + cj + dk
Where i2 = j2 = k2 = ijk = -1
They are cyclic, i.e.
i ij = k ji = -k
^ \ jk = i ik = -j
/ v ki = j etc.
k <- j
Proof:
ijk = -1
iijk = -i
-jk = -i
jk = 1
Quaternions can be decomposed into scalar and vector
parts:
q = (a,b,c,d) = (a,v)
^ -----
| ^
scalar \
vector
A quaternion (0,v) is called a PURE quaternion.
A UNIT quaternion is a quaternion of modulus one.
(q = q/√(a2 + b2 + c2 + d2)
The Quaternion Group, Q8
------------------------
Quaternions form a group, Q8 = {1,-1,i,-i,j,-j,k,-k}
Operations: + or .
Closure: q1 + q2/ = q3
q1.q2 = q3
Inverse: q1 + (-q1) = 0
q1.q1* = 1
Identity: 0 + q1 = q1
1.q1 = q1
Associativity: q1 + (q2 + q3) = (q1 + q2) + q3)
(q1q2)q3 = q1(q2)q3q2)
Commutativity: q1 + q2 = q2 + q1
q1.q2 ≠ q2.q1
[i,j] = ij - ji = k - (-k) = 2k
Subgroups: {1,-1,i,-i} {1,-1,j,-j} {1,-1,k,-k}
{1,-1} = center (kernel)
{1}
Quaternion Multiplication
-------------------------
q1 = (a,b,c,d)
q2 = (e,f,g,h)
q1q2 = (ae - bf - cg - dh,
af + be + ch - dg,
ag - bh + ce + df,
ah + bg - cf + de)
This can also be written in terms of real matrices
as follows:
- - - -
| a -b -c -d || e |
| b a -d c || f |
| c d a -b || g |
| d -c b a || h |
- - - -
Example:
(1,2,3,6)(0,1,0,0)
- - - - - -
| 1 -2 -3 -6 || 0 | | -2 |
| 2 1 -6 3 || 1 | = | 1 |
| 3 6 1 -2 || 0 | | 6 |
| 6 -3 2 1 || 0 | | -3 |
- - - - - -
Individual elements are given by:
- -
| 1 0 0 0 |
(1,0,0,0) = | 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
- -
- -
| 0 -1 0 0 |
(0,1,0,0) = | 1 0 0 0 |
| 0 0 0 -1 |
| d 0 1 0 |
- -
etc.
Extracting the Dot and Cross Products
-------------------------------------
Let q1 = (a,x) and q2 = (b,y)
q1q2 = (ae - (bf + cg + dh),
af + be + ch - dg,
ag - bh + ce + df,
ah + bg - cf + de)
(ae - x.y,ay + ex + x ^ y)
Vector Rotation (Perpendicular Axis)
------------------------------------
n ^ v
^
|
| v' ☉ means out of screen
| /
| /
| /
|/θ
n ☉-----------> v
v' = vcosθ + (n ^ v)sinθ
Quaternion Rotation (Perpendicular Axis)
----------------------------------------
q = (0,v)
q is the PURE quaternion corresponds to the
vector being rotated.
q' = (0,v')
qn = (0,n)
n is the unit vector coming out of the screen
corresponding to the axis of rotation, and qn
is the correponding pure quaternion. We need
to figure out what n ^ v equals. Try,
qnq = (0,n)(0,v) = (0 - n.v, 0v + 0n + n ^ v)
= (0,n ^ v)
= n ^ v
Therefore, analagous to the vector forn, we can
write:
q' = qcosθ + (qnq)sinθ
= (cosθ + qnsinθ)q
Now,
qn2 = (0,n)(o,n)
= (0 - n.n,n ^ n)
= (-|n|2,0)
= (-1,0)
= -1
So qn is analagous to i so we can write the
quaternion form of Euler's equation as:
exp(qnθ) = (cosθ + qnsinθ) ... 1.
and,
q' = exp(qnθ)q
Example:
q = (0,0,1) = k
qn = (0,1,0) = j
θ = π/2
z
|
^ q
|
|
-------->------- y
/ qn
/
v q'
/
x
q' = exp(θqn)q
= exp(πj/2)k
= jk
= i
= (1,0,0)
Vector Rotation (Arbitrary Axis)
--------------------------------
v = v∥ + v⊥
v' = v∥' + v⊥'
= v∥ + v⊥'
= v∥ + cosθv⊥ + sinθ(n ^ v⊥)
= v∥ + cosθ(v - v∥) + sinθ(n ^ v⊥)
n = nxi + nyj + nzk is a unit vector corresponding
to the arbitrary axis of rotation.
(n ^ v⊥) = n ^ (v - v∥)
= n ^ v - n ^ v∥
= n ^ v
v' = (1 - cosθ)v∥ + cosθv + sinθ(n ^ v)
Now,
v∥ = (v.n)n projection of v onto n
v' = (1 - cosθ)(v.n)n + cosθv + sinθ(n ^ v)
This is the RODRIGUES ROTATION FORMULA.
Example:
n = (0,0,1) v = (1,0,0) θ = π/2
v' = (1 - cosθ)(v.n)n + cosθv + sinθ(n ^ v)
= 0 + 0 + (0,0,1) ^ (1,0,0)
= (0,1,0)
Quaternion Rotation (Arbitrary Axis)
------------------------------------
q = (0,v)
q∥ = (0,v∥)
q⊥ = (0,v⊥)
q⊥' = (0,v⊥')
qn = (0,n)
Where,
qn = (0,nxi,nyj,nzk) is now a pure quaternion
corresponding to the arbitrary axis of rotation,
n. As before,
q' = q∥ + q⊥'
= q∥ + exp(θ.qn)q⊥
= q∥ + (cosθ + sinθnxi + sinθnyj + sinθnkk)q⊥
Now exp(θ.qn)q⊥ = q⊥exp(-θ.qn)
Proof:
(cosθ,sinθn)(0,v⊥) = (0,v⊥)(cosθ,-sinθn)
(0,cosθv⊥ + sinθ(n ^ v⊥)) = (0,cosθv⊥ - sinθ(v⊥ ^ n))
(0,cosθv⊥ + sinθ(n ^ v⊥)) = (0,cosθv⊥ + sinθ(n ^ v⊥)) Q.E.D.
Likewise,
exp(θ.qn)q∥ = q∥exp(θ.qn)
Proof:
[q1,q2] = 2(v1 ^ v2)
q' = exp(θqn/2)exp(-θqn/2)q∥ + exp(θqn/2)exp(θqn/2)q⊥
Using exp(-θqn/2)q∥ = q∥exp(-θqn/2) and exp(θ.qn)q⊥ =
q⊥exp(-θ.qn) from above gives:
q' = exp(θqn/2)q∥exp(-θqn/2) + exp(θqn/2)q⊥exp(-θqn/2)
= exp(θqn/2)(q∥ + q⊥)exp(-θqn/2)
= exp(θqn/2)qexp(-θqn/2)
In terms of sin and cos this is:
g = exp(θqn/2)
= cos(θ/2) + sin(θ/2)nxi + sin(θ/2)nyj + sin(θ/2)nkk
g* = exp(-θqn/2)
= cos(θ/2) - sin(θ/2)nxi - sin(θ/2)nyj - sin(θ/2)nkk
Therefore, we can write:
q' = gqg* = gqg-1
Example:
qn = (0,0,0,1) q = (0,1,0,0) θ = π/2
g = cos(π/4) + sin(π/4)nzk
= √2/2 + (√2/2)k
= (1 + k)/√2
q' = gqg*
= [(1,0,0,1)/√2](0,1,0,0)[(1,0,0,-1)/√2]
= (0,0,1,0)
Why the conjugate? Consider:
q' = gq
= [(1,0,0,1)/√2](0,1,0,0)
= (0,1/√2,1/√2,0)
This is a rotation of 45°. Therefore, the product
gq does not accomplish the full rotation. by itself.
Quaternion Rotation Matrix
--------------------------
A quaternion rotation:
q' = gqg* = gqg-1
With
q = w + xi + yj + zk and qq* = 1 = w2 + x2 + y2 + z2
Can be algebraically manipulated (not proven here)
into a matrix rotation q' = Rq, where R is the
rotation matrix given by:
- -
| 1 - 2y2 - 2z2 2xy - 2zw 2xz + 2yw |
R = | 2xy + 2zw 1 - 2x2 - 2z2 2yz - 2xw |
| 2xz - 2yw 2yz + 2xw 1 - 2x2 - 2y2 |
- -
This is a rotation around the vector n = (x,y,z)
(or equivalently qn = nxi + nyj + nzk) by an angle
2θ. We will demonstrate this using the previous
example where qn = (0,0,0,1) q = (0,1,0,0) θ = π/2
∴ 2θ = π. Plugging the numbers in gives:
- - - - - -
| -1 0 0 || 1 | | -1 |
| 0 -1 0 || 0 | = | 0 |
| 0 0 1 || 0 | | 0 |
- - - - - -
This is indeed a rotation of π!
The 2:1 nature is apparent since both qn and -qn
map to the same R, i.e. the z2 is always positive.
Relation Between SU(2) and SO(3)
--------------------------------
Instead of the 4 x 4 real matrices shown before,
we can also write quaternions as 2 x 2 complex
matrices as follows:
q = a + bi +cj + dk
= (a + bi) + (c + di)j ij = k
= z + wj
We can then write this as a matrix in the same
way that we can write a complex number as a
matrix.
- - - -
| z -w | = | a + bi -(c + di) |
| w* z* | | c + di a - bi |
- - - -
-----------------------------------------------------
Digression:
Complex Numbers as Matrices
---------------------------
Consider the 2 complex numbers.
(a + bi)(c + di) = ac - bd + (ad + bc)i
This can be written as the matrix:
- - - -
| a -b || c |
| b a || d |
- - - -
The matrix on the LHS enables us to write.
- - - -
1 = | 1 0 | and i = | 0 -1 |
| 0 1 | | 1 0 |
- - - -
Euler's formula can then be written as:
exp(iθ) = cosθ + isinθ
- - - -
exp(| 0 -θ |) = | cosθ -sinθ |
| θ 0 | | sinθ cosθ |
- - - -
Differentiating both sides w.r.t. θ gives:
iexp(iθ) = -sinθ + icosθ
- - - - - -
| 0 -1 || cosθ -sinθ | = | -sinθ -cosθ |
| 1 0 || sinθ cosθ | | cosθ -sinθ |
- - - - - -
-----------------------------------------------------
Example:
(1,2,3,6)(0,1,0,0)
- - - -
| (1 + 2i) -(3 + 6i) || (0 + i) -(0 + 0i) |
| (3 - 6i) (1 - 2i) || (0 - 0i) (0 - i) |
- - - -
- -
= | (-2 + i) -(6 - 3i) |
| (6 + 3i) (-2 - i) |
- -
- -
| -2 |
= | 1 |
| 6 |
| -3 |
- -
Which is exactly the same result from before.
The following matrices satisfy these conditions:
- - - - - - - -
I = | 1 0 | i = | 0 -1 | j = | 0 i | k = | -i 0 |
| 0 1 | | 1 0 | | i 0 | | 0 i |
- - - - - - - -
= -iσ2 = -iσ1 = -iσ3
It is easy to check that the matrices satisfy the
quaternion identites. In addition, all have
determinnt = 1 and i† = -i, j† = -j, k† = -k.
These are the Pauli matrices of SU(2) that are
isomorphic to the quaternions of unit norm.
Given the 2:1 correspndence between rotations of
R and qqg-1, it implies that there is a 2:1
homomorphism from SU(2) to SO(3). This has very
important consequences in the physics of spin.
The unit quaternions can also be thought of as
the group corresponding to the 3-sphere, S3, that
gives the group Spin(3), which is isomorphic to
SU(2) and also the double cover of SO(3).