TUTORIAL on QUATERNIONS

Part II
Luis Ibáñez

August 13, 2001

This document was created using LYX and the LATEX Seminar style.

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Contents

Differentials of Quaternions
Developments of Functions of Quaternions

2
Differentials of Quaternions

The difficult point in defining Differentials over Quaternions is

Lack
of the Conmutative Property.

P Q Q P

3
Adopted Definition

Following Newton’s definition of Fluxions

Hamilton [2] defined Simultaneous Differentials as

Limits of Equi-multiples of
Simultaneous and Decreasing Differences.

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What it means ?

Given a system of connected Quaternions

q r s

the symbols

q r s

represent their Simultaneous Differences

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What it means ?

The sums

q q r r s s

are a New system of Quaternions

satisfying the Same Laws of connexion as the Old system.

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dq q dq 1
q dq q dq 1
p q
p q
u v
u va What it means ?
r α
z f xy
The differences ∆q ∆r ∆s start at an arbitrary size
f xy
θ

∆q ∆r ∆s

m ∆q 0 ∆r 0 ∆s 0
dq dr ds q r s q ∆q r ∆r s ∆s

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dq q dq 1
q dq q dq 1
p q
p q
u v
u va What it means ?
r α
z f xy
Then, they are simultaneously decreased by a factor
f xy
θ

∆q ∆r ∆s
m ∆q 0 ∆r 0 ∆s 0
dq dr ds q r s q ∆q r ∆r s ∆s

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dq q dq 1
q dq q dq 1
p q
p q
u v
u va
What it means ?
r α
z f Integer
x y multiples of the new differences are considered
f xy
2x 3x
θ

1x
∆q ∆r ∆s
m ∆q 0 ∆r 0 ∆s 0
dq dr ds q r s q ∆q r ∆r s ∆s

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dq q dq 1
q dq q dq 1
p q
p q
u v
u va
What it means ?
r α
z f xy ...the factor is further decreased
f xy
3x
θ
2x

1x
∆q ∆r ∆s
m ∆q 0 ∆r 0 ∆s 0
dq dr ds q r s q ∆q r ∆r s ∆s

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dq q dq 1
q dq q dq 1
p q
p q
u v
u va What it means ?
r α
z f xy ...and decreased...
f xy
θ
3x
2x

1x ∆q ∆r ∆s
m ∆q 0 ∆r 0 ∆s 0
dq dr ds q r s q ∆q r ∆r s ∆s

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dq q dq 1
q dq q dq 1
p q
p q
u v
u va What it means ?
r α
z f xy
...and decreased...
f xy
θ 3x
2x
1x ∆q ∆r ∆s
m ∆q 0 ∆r 0 ∆s 0
dq dr ds q r s q ∆q r ∆r s ∆s

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dq q dq 1
q dq q dq 1
p q
p q
u v
u va What it means ?
r α
z f xy
...and taken to the limit !
f xy
θ
3x
2x
∆q ∆r ∆s 1x dq dr ds
m ∆q 0 ∆r 0 ∆s 0
qrs q ∆q r ∆r s ∆s

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Definition Revisited

If all the multiples n∆ converge to the same value

when the factor ∆ is decreased,
Then the Limit of ∆ ’s exist and they are called

Simultaneous Differentials

dq dr ds

Again....

Limits of Equi-multiples of
Simultaneous and Decreading Differences.

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q q dq 1
q q dq 1
p q
p q Consequence of this Definition
u v
u va The Surface Differentials of this black Rectangle w h
r α
z f xy
f xy
dh
θ
qrs
dS w dh h dw
∆r s ∆s
∆q ∆r ∆s h
0 ∆s 0
dq dr ds w dw

is the sum of shaded rectangles at the sides h dw and w dh

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Lies my Calculus Teacher told me...

Differentials are infinitesimally small...

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The Truth is

What they have to be is linearly related

dS h dw w dh

NOT because
dw dh 0

BUT because
dw dh

is NOT LINEAR with respect to a factor applied

simultaneously to dh, dw and dS

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dq q dq 1
q dq q dq 1
p q
p q
Differentials as Linear Approximations
u v
u va
Differentials don’t need to be SMALL
r α
z f xy They are a LINEAR APPROXIMATION [1].
f xy
θ dy
qrs
q ∆q r ∆r s ∆s
∆q ∆r ∆s
im ∆q 0 ∆r 0 ∆s 0
dq dr ds dx

dy A dx B

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dw
dx
dy
dq 1
q dq
q dq 1 Differentials as Linear Approximations
dq q dq 1
q dq q dq 1
In ap 2D
q function z f xy the Linear Approximation is a Plane.
p q
u v
u va
r α
z f xy
f xy
θ
qrs
q ∆q r ∆r s ∆s
∆q ∆r ∆s
lim ∆q 0 ∆r 0 ∆s 0 y
dq dr ds x

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dq
dw
dx
dy
dq 1
q dq Differentials as Linear Approximations
q dq 1
dq q dq 1
q dq q dq 1
p q dz A dx B dy C
p q
u v
u va
r α
z f xy
f xy
θ
qrs
q ∆q r ∆r s ∆s
∆q ∆r ∆s
lim ∆q 0 ∆r 0 ∆s 0 y
dq dr ds x

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Differentials as Linear Approximations

The Differentials

dx dy dz

Can be as Large as you want

but
They have to be related by a Linear Equation

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Differential of Functions of Quaternions

Let Q be a Function of the Quaternion variables q, r,...

Q F qr

and let

dq dr

be any Simultaneous Differentials of q, r, ...

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Differential of Functions of Quaternions

The Simultaneous Differential of function Q is

dq dr
dQ lim n F q r F qr
n ∞ n n

where n
is an integer multiple of a particular real value.

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Differentials in one Dimension

The well known equation

df x f x h f x
lim
dx h 0 h

Expressed according to the new definition

dx
df x lim n f x f x
n ∞ n

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Example

2
f x x

2
dx 2
df x lim n x x
n ∞ n

2 dx dx 2
df x lim n x 2x x2
n ∞ n n2

dx 2
df x lim 2 x dx 2 x dx
n ∞ n

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Differential of a Function of One Variable

The Differential dx is like another variable

df x g x dx

for example, given

f x x2

the differential is a function of Two Independent Variables x and dx

df x g x dx 2 x dx

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Differentials of Functions of Quaternions

Quaternions composition (multiplication) is NOT commutative

f q q2 q q

The differential

2
dq 2
df q lim n q q
n ∞ n

Results in

df q q dq dq q

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Differentials of Functions of Quaternions

The Quotient

df q 1
df q dq
dq

For the current example f q q2

df q 1 1
q dq dq dq q dq
dq

df q 1
q dq q dq
dq
Which is a function of q and dq

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Comparison with Traditional Differentials

Function Quotient of Differentials

df x
Scalars f x x2 dx 2x
df q
Quaternions f q q2 dq q dq q dq 1

The Quotient of Quaternion Differentials is a new Function

of TWO INDEPENDENT variables :

q and dq

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q dq q dq
p q
p q
u v Geometric Interpretation of the
u va Differential of the Square Function
r α
z f xy
q
f xy q C
q2
θ q dq
B
qrs s
q
∆q r ∆r s ∆s
∆q ∆r ∆s A
0 ∆r 0 ∆s 0
dq dr ds

q A B
q2 A q dq dq
B C C s 1
q q

a variation in q is represented by the Differential dq

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q dq q dq 1
p q
p q
u v Geometric Interpretation of the
u va Differential of the Square Function
r α
z f xy
f xy q
q dq
θ
qrs s s
∆q r ∆r s ∆s
S
∆q ∆r ∆s
∆r 0 ∆s 0
dq dr ds

In order to find the Differential dq, the vector S corresponding to

the chord of Vector-Arc s is taken and shifted to the origin of the sphere.

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q dq q dq
p q
p q
u v Geometric Interpretation of the
u va Differential of the Square Function
r α
z f xy
A A
f xy
dq
θ
qrs s S dq
∆q r ∆r s ∆s
∆q ∆r ∆s S
0 ∆r 0 ∆s 0
dq dr ds

The Quotient between vectors

S
S and A is the Quaternion dq dq
A

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dq q dq 1
p q
p q
Geometric Interpretation of the
u v
Differential of the Square Function
u va
r α
z f xy
q dq
f xy A

dq
θ

1
qrs q
q r ∆r s ∆s dq
q 1
dq
∆q ∆r ∆s 1
∆r 0 ∆s 0
dq q dq 1
dq dr ds

q is composed with dq 1 by The resulting Vector-Arc q dq 1

forming a spherical triangle with can be further composed with dq
their corresponding Vector-Arcs by using another spherical triangle.

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p q
p q Geometric Interpretation of the
u v Differential of the Square Function
u va
r α
z f xy
f xy
θ
qrs 1
1
q r ∆r s ∆s dq dq
q q
∆q ∆r ∆s dq dq
r 0 ∆s 0 q dq q dq 1

dq dr ds q q

The sum of q and dq q dq 1 is performed by finding first a common

denominator vector, then adding the two vector in the numerator.

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Sum of Reciprocals (a property)

1 1
Rq q
q

1 1 1 1
R p Rq p q
p q

1 1 1 1 1 1 1 1
q p p q
p q q p q p q p

1 1 1 1
p q
p q q p

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Differential of the Reciprocal

1
f q Rq q

1
dq 1
df q lim n q q
n ∞ n

1
dq dq 1
df q lim n q q q q
n ∞ n n

1 1 1 1
df q q dq q q dq q

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Comparison with Traditional Differentials

Function Quotient of Differentials

1 df x 2
Scalars f x x dx x
1 df q 1 1 1
Quaternions f q q dq q dq q dq

The Quotient of Quaternion Differentials is a new Function

of TWO INDEPENDENT variables :

q and dq

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Quotient of Differentials

The Quotient between two Differentials can be separated in

Tensor and Versor parts

df q T df q U df q
dq T dq U dq

The Differentials d f q and dq are Equi-Multiples,

so scaling dq will scale d f q by the same factor.
Quotients of Differentials are Invariant to Scale changes in their Tensors

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Quotient of Differentials

In the example f q q2
The Quotient of Differentials

df q 1
q dq q dq
dq
Can be reduced to

df q 1
q U dq q U dq
dq

That only depends on dq ’s Direction represented by the Versor U dq

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Partial Differentials

Given a function of several quaternion variables

Q f qrs

Its Differential satisfies

dQ dq Q dr Q ds Q

each Partial Differential dx Q is obtained by differentiating

with respect to x as if the other variables were constant.

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Succesive Differentials

For example, given the Quaternion function

f q q2 q q

The first Differential is

df q q dq dq q

Taking the Differential of this last expression,

where q and dq are considered as two independent variables

d2 f q dq dq q d2q d2q q dq dq

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Successive Differentials

The Second Differential of

f q q2
is then reduced to

d2 f q q d2q d2q q 2 dq dq

Which is a function of THREE independent Quaternion variables

q dq d 2 q

None of them necessarily SMALL

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Taylor’s Series Extended to Quaternions

Having that

dm f q d dm 1
f q

The Taylor’s Series Expansion can be applied to functions of Quaternions

df q d2 f q d3 f q d4 f q
f q dq f q
1! 2! 3! 4!

Where f q dq will be a function of q dq d 2 q d 3 q Quaternion

variables NOT necessarily SMALL

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Taylor’s Series Approximation

The Tensor of the Quaternion Variables

q dq d 2 q d 3 q

can be scaled by a Scalar factor x to produce an Approximation

x x2 2 x3 3
Fx f q xdq f q df q d f q d f q
1! 2! 3!

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dq q dq
q dq q dq 1

Operations
p q in Versor Space
u v
u va
Composition of Versors is equivalent to
r α
z f xy
f xy
q
θ p q
qrs
q ∆q r ∆r s ∆s
p
∆q ∆r ∆s
lim ∆q 0 ∆r 0 ∆s 0
dq dr ds

sum of their Vector-Arcs on the Unit Sphere Surface.

This is a Non-Commutative operation

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dx
dy
dq 1
q dq
q dq 1 Operations in Versor Space
dq q dq 1
q dq q dq 1
p q Increments of Versor’s Angle is equivalent to Exponentiation
p q
for example, in order to double the angle
u v
u va the versor is applied twice, which is q q q2
r α
z f xy
f xy
θ
qrs
q ∆q r ∆r s ∆s q q

q q q1 2
∆q ∆r ∆s
0 ∆r 0 ∆s 0 q2 q1 5

dq dr ds

Like in Complex numbers

eiθ cosθ isinθ

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q dq 1
dq q dq 1
dq q dq 1
p q Operations in Versor Space

u v Subtraction of Vector-Arcs is equivalent to

u va
a Quotient of Versors
r α
z f xy p 1
p q
f xy q
q
θ p
qrs p 1
q p q q
q r ∆r s ∆s q
p q
∆q ∆r ∆s p q 1
q
∆r 0 ∆s 0
p
dq dr ds

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dq q dq
dq q dq 1
p q
Versor Spherical Linear Interpolation
u v
u va
Sperical Linear Interpolation (Slerp)

z f xy α
p
q 1 r α q
f xy q
q
θ p
qrs p q r 0 q
q r ∆r s ∆s r 1 p
∆q ∆r ∆s r α
α 01
∆r 0 ∆s 0
dq dr ds

The Quotient qp produce the Quaternion that relates p with q.

Exponent α allows to regulate how much of this Quotient is applied

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dq q dq 1
q dq q dq 1
p q
Optimization
p q of Versor Functions
u v
u v a of Quaternions is restricted to Versors
If the Space
r α
z f xy
f xy q
B
θ
qrs
q ∆q r ∆r s ∆s A
∆q ∆r ∆s
lim ∆q 0 ∆r 0 ∆s 0
dq dr ds

The only valid operations are those that keep the end of vectors
in the surface of the Unit Sphere

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dq q dq 1
q dq q dq 1
p q
Optimization
p q of Versor Functions
u v
a
u vThe Versor Space is a 2D Space
r α
z f xy
f xy
q
θ
qrs
q ∆q r ∆r s ∆s
u v
∆q ∆r ∆s
lim ∆q 0 ∆r 0 ∆s 0
dq dr ds

In order to enforce that Variations of a Versor result in another versor,

the only valid operations are compositions with Versors, (e.g. u and v )

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q dq 1
dq q dq 1
dq q dq 1
p q Optimization of Versor Functions
p q
u v
u va Gradient Descent-like Optimization Method
r α
z f xy
f xy q
q q
θ r
qrs
r ∆r s ∆s v v v
u u u
∆q ∆r ∆s
0 ∆s 0 v/u (v/u)^a (v/u)^a
dq dr ds

v α f u q
r u q a
u f u q f v q

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References
[1] C. T. J. Dodson and T. Poston. Tensor Geometry. Graduate Texts in
Mathematics. Springer-Verlag, second edition, 1990.
[2] W.R. Hamilton. Elements of Quaternions, volume I. Chelsea
Publishing Company, third edition, 1969. The original was published
in 1866.

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