Project: ArangoDB (App Definition and Development / Database)
File: TQE_EA.pdf

The Quaternionic Exponential
(and beyond)
ERRATA & ADDENDA
Hubert HOLIN
23/03/2001
Hubert.Holin@Bigfoot.com
http://www.bigfoot.com/~Hubert.Holin
Errata
• Page 2:
The multiplication is defined in such a way that e=()1000,,, is its neural
element.
• Page 3:
We ask for a norm of a Banach algebra to verify pq p q∗≤ and not just
that there exists some positive k such that pq k p q∗≤ . As noted the case for
real numbers, complex numbers, quaternions and octonions is even better yet,
as we have pq p q∗=.
• Page 3:
Reqq q()=+()1
2 and Urqq q()=−()1
2.
• Page 6:
The formula is not in error, but might be more readable in the form
  ra a uau u a uaurr r r r r r r r r()=()−⋅()[] +()∧()+⋅() cos sinθθ .
• Page 7:
1Instead of qx
y
z=±














cos
sinsinsin
θ
θ
θ
θ2
222, read qx
y
z=±
















cos
sin
sin
sin
θ
θ
θ
θ2
2
2
2.• Page 19, footnote:
Instead of 
  sinc : ,sin
a xx
a
x
aRR→







aπ
, read 
  sinc : ,sin
a xx
a
x
aRR→







aπ
π.
• Page 19, footnote:
Instead of 
  sinhc : ,sinh
a xx
a
x
aRR→







aπ
, read 
  sinhc : ,sinh
a xx
a
x
aRR→







aπ
π.
Addenda
• More structure
It is interesting to note that C, H and O are (left) vector spaces over C.
The basis of H as a left C-vector space is 1,j(), and the basis of O as a
left C-vector space is 1, , ,je j′′(). However, if qi j k=+ + + ∈αβ γ δ H, then
qii j=+()++() αβ γδ , but if oi j k e i j k=+ + + + ′+′+′+′∈ αβ γ δ εζ η θ O then
o i ij ie ij=+()++() ++()′+−()′ αβ γδ εζ η θ  (note the minus sign in the last factor).
If we write qj=+Γ∆, with ∆∈C and Γ∈C, then qj=−Γ∆, and if we also
have pj=+ΑΒ, with Α∈C and Β∈C, then pq j=−() ++() ΑΓ Β∆ Α∆ ΒΓ . In particular,
if z∈C then jz zj=.
Things break down when we want to consider O as a structure over H,
however. Indeed, there is no widely-accepted generalization of vector fieldwhere the role of the scalars is taken by a non-commutative structure, as isthe case with 
H as most interesting properties of vector spaces fail to remain
true in that case, in general (though by requiring the scalars to be merely acommutative ring instead of a full blown field, quite a few properties remaintrue; this structure is known as a module).
However, if 
oi j k e i j k=+ + + + ′+′+′+′∈ αβ γ δ εζ η θ O, then it is also true
that oi j ki j k e=+++() +++ +() ′ αβ γ δ εζ η θ .
• More Geometry
Another interesting way to see H is as RR×3. In this case, if qt V11 13=()∈×,RR
and qt V22 23=()∈×,RR, then the quaternionic product can be expressed as
qq tt V V tV tV V V12 1 2 1 2 12 21 1 2=− ⋅ ++ ∧() , , with “⋅” the scalar product on R3 and “∧” the
vector product in R3.
• Finding the quaternions for a given rotation of R3
If we are given the rotation in term of vector and angle in 0;+[]π, then this
has been solved in the main text (if the angle is in −[]π;0, we just take the
opposite of both the angle and vector; the identity and its opposite are trivial
2to solve). The opposite quaternion is also a solution, of course.If we are simply given a rotation matrix, then, essentially, we first find its
elements (vector and angle), and use the procedure above. To find an invariantvector, we simply solve the linear system which defines them. For the angle,we first find its cosine using the trace. Then we build a vector orthogonal tothe invariant vector we found (always possible starting from one of the canonicalbasis vector and using some classical orthonormalization procedure) to checkthe sign of the angle.
If we are given a succession of rotation, it may be advantageous in applications
to chose among the successions of pairs of opposite solutions that for whichthe distance between successive quaternions is the smallest.
• More rotations
We have considered   HH→[],pp qa and   dp q pq=→[]HH,a in the first
chapter and seen, thru some amount of computation, that they gave rise to
rotations on R4, when q=1.
We present here another take on the same subject, aimed at giving effective
methods of parameterizing SO R4,()
It is also interesting to consider    gp p qq=→[]HH,a, as both gq and dq are
C-linear operators on H, which trivially verify   gg gqq qq ′′=o and   dd dqq qq ′′=o. We
can also very simply verify that gp gp q p p dp dpqq qq()′()() = ′()=()′()()2. Obviously g1
and d1 are both the identity on H. Considering them now as R-linear operators
on H, we see that their determinant in the canonical basis must stay of the
same sign on S3, hence must stay positive (since 13∈S), therefore must be
always equal to 1 on S3.
Hence   SS O R34→()[] ,,qgqa and   SS O R34→()[] ,,qdqa are both group
homeomorphism, sending 1 to the identity.
Using the same kind of topological argument as in the case of SO R3,(), we
get the parameterization of SO R4,() that we announced ([ M. Berger (1990)],
[C. Godbillon (1971)]) simply by considering   pq d gpq,() ao, save for the determination
of the kernel. We will, however, aim here for a more constructive approach to
surjectivity.
Given r a rotation of R4, we seek p∈S3 and q∈S3 such that for any
quaternion s we have psq r s=(). Hence, by applying that to s=1, we find that
necessarily prq=()1 (since qq=−1 as q=1). Therefore we are led to solve
qsq s=()ρ for all s, with ρsrr s()=()()1 (as rt t()= for all quaternion t, since r is a
rotation, and hence r11()=). But then ρµ µ µ µ()=()()=() ()= rr rr11 1  for all µ∈R,
which means R is invariant, and we know that ρ is a rotation R4, as the
composition of r, which is one by hypothesis, and the multiplication on the left
by a unit quaternion, which is one also as we have seen in the main text. Thismeans we are simply back to solving on 
S3 the equation   ρρq=()MC C,, (with
3  qqaρ as presented in the first chapter).Finally, given two unit quaternions pi j k=+ + +αβ γ δ  and qi j k=+ + +εζηθ ,
the rotation matrix on R4 is given explicitly by:
αεβζ γη δθ αζ βεγθ δη αη βθ γεδζ αθ βη γζ δε
αζ βεγθ δη αεβζ γη δθ αθ βη γζ δε αη βθ γεδζ
αη βθ γεδζ αθ βη γζ δ+++ +−−+ ++− − +−+−
−+−+ +−− −++− ++++−++− ++++
εεαεβζ γη δθ αζ βε γθ δη
αθ βη γζ δεαη βθ γε δζ αζ βε γθ δη αε βζ γη δθ−+− −−++
−−++ −+−+ ++++ −−+




Additional Bibliography
4C. Godbillon (1971): Éléments de Topologie Alg ébrique; Collection M éthodes, Hermann, 1971.