Quaternion

Quaternions possess the following advantages over 3×3 rotation matricies in representing 3D rotations:

• Less storage (just 4 floating-point numbers vs. 9)
• Composition of two rotations is more efficient.
• Interpolation between two rotations is easier using quaternions.

One can easily convert unit-norm quaternions to rotation matricies. Given that q = [ a b c d ] is a unit quaternion (i.e. a² + b² + c² + d² = 1), the rotation matrix representing that quaternion is denoted M(q) and is defined by

       | a2+b2-c2-d2  2bc-2ad    2bd+2ac   |
M(q) = |  2bc+2ad   a2-b2+c2-d2  2cd-2ab   |
       |  2bd-2ac    2cd+2ab   a2-b2-c2+d2 |

M(q)(M(p)v) = (M(q)M(p))v = M(qp)v.

Two interpolate between two quaternions, you must interpolate linearly over the surface of the unit hypersphere in four dimensions. This process is called spherical linear interpolation. The quaternion so interpolated t of the way from p to q is given by the vector r using the formula

r = (psin((1-t)φ) + qsin(tφ)) / sin(φ).

Finally, one can obtain the angle θ and axis u of rotation from the unit norm quaternion q using the following formulae:

cos(θ/2) = a

sin(θ/2) = (b² + c² + d²)^1/2

u = [ b c d ] / (b² + c² + d²)^1/2

They were invented by Hamilton in 1843 who was so pleased that he scratched the defining relations on Brougham Bridge on the Royal Canal in Dublin.

As a real vector space the Quaternions have basis 1,i,j,k and the mutiplication can be deduced from the rules

i2=j2=k2=-1,  ij=-ji=k, jk=-kj=i,  ki=-ik=j.

qb(q)=t2+x2+y2+z2

Note that if q is nonzero, so that one of t,x,y,z is nonzero, then qb(q) is a nonzero real number. It follows that such a q has inverse

b(q)/(t2+x2+y2+z2)

  --         --
 | t+xi   y+zi |
 | -y+zi  t-xi |
  --         --  

The subgroup of the group of units of H consisting of {1,-1,i,-i,j,-j, k,-k} is called the Quaternion group (or Pauli spin group). In this group of order 8 all of the elements except for 1 and -1 have order 4, with -1 having order 2.

Multiplying Quaternions

No-one has really explained how to do this yet. We'll be working with the two quaternions:
A = a + bi + cj + dk
E = e + fi + gj + hk
As has already been mentioned, multiplying quaternions is non-commutative, and multiplying the units is actually anti-commutative. This means we have to be extra careful. So then:

AE = (a + bi + cj + dk)(e + fi + gj + hk)

Now we have to laboriously expand these brackets, being very careful not to change the order of the units as we go:

ae + afi + agj + ahk
+bei + bfi² + bgij + bhik
+cej + cfji + cgj² + chjk
+dek + dfki + dgkj + dhk²

Currently we have a lot of unresolved units all over the place, so let's tidy them up using Hamilton's Laws:

ae + afi + agj + ahk
+bei - bf + bgk - bhj
+cej - cfk - cg + chi
+dek + dfj - dgi - dh

Lovely, right right? Now we just need to group together the terms with the same unit:

ae - bf - cg - dh
+i( af + be + ch - dg )
+j( ag - bh + ce + df )
+k( ah + bg - cf + de )

And there you have it.
Corrections to the normal address.

Qua*ter"ni*on (?), n. [L. quaternio, fr.quaterni four each. See Quaternary.]

1.

The number four.

2.

A set of four parts, things, or person; four things taken collectively; a group of four words, phrases, circumstances, facts, or the like.

Delivered him to four quaternions of soldiers. Acts xii. 4.

Ye elements, the eldest birth Of Nature's womb, that in quaternion run. Milton.

The triads and quaternions with which he loaded his sentences. Sir W. Scott.

3.

A word of four syllables; a quadrisyllable.

4. Math.

The quotient of two vectors, or of two directed right lines in space, considered as depending on four geometrical elements, and as expressible by an algebraic symbol of quadrinomial form.

⇒ The science or calculus of quaternions is a new mathematical method, in which the conception of a quaternion is unfolded and symbolically expressed, and is applied to various classes of algebraical, geometrical, and physical questions, so as to discover theorems, and to arrive at the solution of problems.

Sir W. R. Hamilton.

© Webster 1913.

Qua*ter"ni*on, v. t.

To divide into quaternions, files, or companies.

Milton.

© Webster 1913.