Geometric
Algeb
ra:
New
F
oundations,
New
Insights
Three-Dimensional
Geometric
Algeb
ra
and
Rotations
Dr.
Chris
Do
ran
Ca
v
endish
Lab
oratory
Madingley
Road
Cam
bridge
Univ
ersit
y
,
CB3
0HE,
UK
c.doran@mrao.c
am.ac
.uk
Dr.
Joan
Lasenb
y
Departmen
t
of
Engineering
Univ
ersit
y
of
Cam
bridge
T
rumpington
Street
Cam
bridge,
CB2
1PZ,
UK
jl@eng.cam.ac.uk
www.mrao.cam.a
c.uk
/cliff
ord
2
1
Geometric
Algebra
of
3-D
Space
The
geometric
algebra
(GA)
of
3-d
space
is
a
remark
ably
p o
w
erful
to ol
for
solving
prob-
lems
in
geometry
and
classical
mec
hanics.
It
describ es
v
ectors,
planes
and
v
olumes
in
a
single
algebra,
whic
h
con
tains
all
of
the
familiar
v
ector
op erations
for
3-d
space.
These
include
the
v
ector
cross
pro duct,
whic
h
is
rev
ealed
as
a
disguised
form
of
biv
ector.
The
algebra
pro
vides
a
v
ery
clear
and
compact
metho d
for
enco ding
rotations,
whic
h
is
con-
siderably
more
p o
w
erful
than
w
orking
with
matrices.
This
rev
eals
the
true
signi cance
of
Hamilton's
quaternions,
and
resolv
es
man
y
of
the
historical
diÆculties
encoun
tered
with
their
use.
As
a
basis
set
for
the
geometric
algebra
of
3-d
space
w
e
use
c
hose
a
set
of
orthonormal
v
ectors
fe
1
;
e
2
;
e
3
g.
All
three
v
ectors
are
p erp endicular,
so
they
all
antic
ommute,
and
ha
v
e
unit
length
so
they
square
to
+1.
F
rom
these
3
basis
v
ectors
w
e
can
generate
3
indep enden
t
biv
ectors:
e
1
e
2
;
e
2
e
3
;
and
e
3
e
1
:
(1.1)
Eac
h
of
these
enco des
a
distinct
plane,
and
there
are
3
of
them
to
matc
h
the
3
inde-
p enden
t
planes
in
3-d
space.
As
w
ell
as
the
3
biv
ectors
the
algebra
con
tains
one
further
ob
ject.
This
is
the
pro duct
of
3
orthogonal
v
ectors,
resulting
in
(e
1
e
2
)e
3
=
e
1
e
2
e
3
:
(1.2)
This
corresp onds
to
sw
eeping
the
biv
ector
e
1
e
2
along
the
v
ector
e
3
.
The
result
is
a
3-dimensional
v
olume
elemen
t
and
is
called
a
trive
ctor.
This
is
said
to
ha
v
e
gr
ade
-3,
where
the
w
ord
`grade'
refers
to
the
n
um
b er
of
indep enden
t
v
ectors
forming
the
ob
ject.
So
v
ectors
are
grade-1,
biv
ectors
are
grade-2,
and
so
on.
The
term
`grade'
is
preferred
to
`dimension'
as
the
latter
is
reserv
ed
for
the
size
of
a
linear
space.
In
3-d
the
maxim
um
n
um
b er
of
indep enden
t
v
ectors
is
3,
so
the
triv
ector
is
the
highest
grade
ob
ject,
or
multive
ctor,
in
the
algebra.
This
triv
ector
is
unique
up
to
scale
(i.e.
v
olume)
and
handedness
(see
b elo
w).
The
unit
highest-grade
m
ultiv
e
ctor
is
called
the
pseudosc
alar,
or
dir
e
cte
d
volume
element.
The
latter
name
is
more
accurate,
but
the
former
is
seen
more
often.
(Though
b e
careful
with
this
usage
|
pseudoscalar
can
mean
di eren
t
things
in
di eren
t
con
texts).
T
o
simplify
,
w
e
in
tro duce
the
sym
b ol
I
,
I
=
e
1
e
2
e
3
:
(1.3)
Our
3-d
algebra
is
therefore
spanned
b
y
1
fe
i
g
fe
i
^
e
j
g
I
=
e
1
e
2
e
3
1
scalar
3
v
ectors
3
biv
ectors
1
triv
ector
(1.4)
These
de ne
a
linear
space
of
dimension
8
=
2
3
.
W
e
call
this
algebra
G
3
.
Notice
that
the
dimensions
of
eac
h
subspace
are
giv
en
b
y
the
binomial
co eÆcien
ts.
3
e
1
e
1
e
2
e
2
e
3
Figure
1:
Hande
dness.
The
t
w
o
frames
sho
wn
are,
b
y
con
v
en
tion,
assigned
a
righ
t-
handed
orien
tation.
Both
e
1
e
2
and
e
1
e
2
e
3
giv
e
rise
to
righ
t-handed
pseudoscalars
for
their
resp ectiv
e
algebras.
The
pseudoscalar
is,
b
y
con
v
en
tion,
c
hosen
to
b e
right-hande
d.
This
is
equiv
alen
t
to
sa
ying
that
the
generating
frame
fe
1
;
e
2
;
e
3
g
is
righ
t-handed.
If
a
left-handed
set
of
orthonormal
v
ectors
is
m
ultipli
ed
together
the
result
is
I
.
There
is
no
in
trinsic
de nition
of
handedness
|
it
is
a
con
v
en
tion
adopted
to
mak
e
our
life
easier.
In
3-d
a
righ
t-handed
frame
is
constructed
as
follo
ws.
Align
y
our
th
um
b
along
the
e
3
direction.
Then
the
grip
of
y
our
righ
t
hand
sp eci es
the
direction
in
whic
h
e
1
rotates
on
to
e
2
(Fig.
1).
The
handedness
of
a
frame
c
hanges
sign
if
the
p ositions
of
an
y
t
w
o
v
ectors
are
sw
app ed.
2
Pro
ducts
in
G
3
An
y
t
w
o
v
ectors
in
the
algebra,
a
and
b
sa
y
,
can
b e
m
ultipli
ed
with
the
geometric
pro duct,
and
w
e
ha
v
e
ab
=
a
b
+
a
^
b:
(2.1)
No
w
the
biv
ector
a
^
b
b elongs
to
a
3-d
space,
spanned
b
y
the
fe
i
^
e
j
g.
If
w
e
expand
out
in
a
basis,
a
=
3
X
i=1
a
i
e
i
;
b
=
3
X
i=1
b
i
e
i
;
(2.2)
w
e
nd
that
the
comp onen
ts
of
the
outer
pro duct
are
giv
en
b
y
a
^
b
=
(a
2
b
3
b
3
a
2
)e
2
^
e
3
+
(a
3
b
1
a
1
b
3
)e
3
^
e
1
+
(a
1
b
2
a
2
b
1
)e
1
^
e
2
:
(2.3)
The
comp onen
ts
are
the
same
as
those
of
the
cross
pro duct,
but
the
result
is
a
biv
ector
rather
than
a
v
ector.
T
o
understand
the
relationship
b et
w
een
these
w
e
rst
need
to
establish
the
prop erties
of
some
of
the
new
pro ducts
pro
vided
b
y
our
3-d
algebra.
4
2.1
V
ectors
and
Biv
ectors
The
three
basis
biv
ectors
satisfy
(e
1
e
2
)
2
=
(e
2
e
3
)
2
=
(e
3
e
1
)
2
=
1
(2.4)
and
eac
h
biv
ector
generates
90
o
rotations
in
its
o
wn
plane.
So,
for
example,
w
e
see
that
e
1
(e
1
^
e
2
)
=
e
1
(e
1
e
2
)
=
e
2
;
(2.5)
whic
h
returns
a
v
ector.
The
geometric
pro duct
for
v
ectors
extends
to
all
ob
jects
in
the
algebra,
so
w
e
can
form
expressions
suc
h
as
aB
,
where
B
is
a
general
biv
ector.
But
w
e
ha
v
e
no
w
seen
that
e
1
(e
2
^
e
3
)
is
a
triv
ector,
so
the
result
of
the
pro duct
a
B
can
clearly
con
tain
b oth
v
ector
and
triv
ector
terms.
T
o
help
understand
the
prop erties
of
the
pro duct
a
B
w
e
rst
decomp ose
a
in
to
terms
in
and
out
of
the
plane,
a
=
a
k
+
a
?
(2.6)
(see
Fig.
2).
W
e
can
no
w
write
aB
=
(a
k
+
a
?
)B
.
Supp ose
that
w
e
also
write
B
=
a
k
^
b
(2.7)
where
b
is
orthogonal
to
a
k
in
the
B
plane
(Fig.
2).
W
e
see
that
a
k
B
=
a
k
(a
k
^
b)
=
a
k
(a
k
b)
=
(a
k
)
2
b
(2.8)
whic
h
is
a
v
ector
in
the
b
direction.
On
the
other
hand
a
?
B
=
a
?
(a
k
^
b)
=
a
?
a
k
b
(2.9)
is
the
geometric
pro duct
of
3
orthogonal
v
ectors,
and
so
is
a
triv
ector.
As
exp ected,
the
geometric
pro duct
of
the
v
ector
a
and
the
biv
ector
B
has
resulted
in
t
w
o
terms,
a
v
ector
and
a
triv
ector.
W
e
therefore
write
aB
=
a
B
+
a
^
B
(2.10)
where
the
dot
is
generalised
to
mean
the
lowest
grade
part
of
the
result,
while
the
w
edge
means
the
highest
grade
part
of
the
result.
2.2
Inner
Pro
duct
a
B
F
rom
Eq.
(2.8)
w
e
see
that
the
a B
=
a
k
B
term
pro
jects
on
to
the
comp onen
t
of
a
in
the
plane,
and
then
rotates
this
through
90
o
and
dilates
b
y
the
magnitude
of
B
.
W
e
also
see
that
a
B
=
a
k
2
b
=
(a
k
b)a
k
=
B
a
;
(2.11)
5
a
k
a
?
a
B
b
a
=
a
k
+
a
?
Figure
2:
A
ve
ctor
and
a
plane.
The
v
ector
a
is
decomp osed
in
to
a
sum
of
t
w
o
v
ectors,
one
lying
in
the
plane
and
the
other
p erp endicular
to
it.
so
the
dot
pro duct
b et
w
een
a
v
ector
and
a
biv
ector
is
an
tisymmet
ric.
W
e
use
this
to
de ne
the
inner
pro duct
of
a
v
ector
and
a
biv
ector
as
a
B
=
1
2
(a
B
B
a):
(2.12)
T
o
see
that
this
alw
a
ys
returns
a
v
ector,
consider
the
inner
pro duct
a
(b
^
c).
F
ollo
wing
the
rules
for
the
geometric
pro duct
w
e
form:
a
(b
^
c
)
=
1
2
a
(bc
c
b)
=(a
b)c
(a
c
)b
1
2
(bac
ca
b)
=2(a
b
)c
2(a
c
)b
+
1
2
(bc
cb)a
=2(a
b
)c
2(a
c
)b
+
(b
^
c)a
;
(2.13)
where
w
e
ha
v
e
made
rep eated
use
of
the
rearrangemen
t
ba
=
2a
b
ab:
(2.14)
It
follo
ws
imm
ediatel
y
that
a
(b
^
c
)
=
1
2
a(b
^
c
)
(b
^
c
)a
=
(a
b)c
(a
c
)b;
(2.15)
whic
h
is
indeed
a
pure
v
ector.
This
is
one
of
the
most
useful
results
in
geometric
algebra
and
is
w
orth
memorisi
ng.
2.3
Outer
Pro
duct
a
^
B
F
rom
Eq.
(2.9),
the
a
^
B
term
pro
jects
on
to
the
comp onen
t
p erp endicular
to
the
plane,
and
returns
a
triv
ector.
This
term
is
symmetric
a
^
B
=
a
?
a
k
b
=
a
k
ba
?
=
B
^
a:
(2.16)
6
W
e
therefore
de ne
the
outer
pro duct
of
a
v
ector
and
a
biv
ector
as
a
^
B
=
1
2
(a
B
+
B
a
):
(2.17)
V
arious
argumen
ts
can
b e
used
to
sho
w
that
this
is
a
pure
triv
ector
(see
later).
W
e
no
w
ha
v
e
a
de nition
of
the
outer
pro duct
of
three
v
ectors,
a
^
(b
^
c).
This
is
the
grade-3
part
of
the
geometric
pro duct.
W
e
denote
the
op eration
of
pro
jecting
on
to
the
terms
of
a
giv
en
grade
with
the
h
i
r
sym
b ol,
where
r
is
the
required
grade.
Using
this
w
e
can
write
a
^
(b
^
c
)
=
ha(b
^
c)i
3
=
ha
(bc
b
c
)i
3
:
(2.18)
But
in
the
nal
term
a
(b
c)
is
a
v
ector
(grade-1)
so
do es
not
con
tribute.
It
follo
ws
that
a
^
(b
^
c
)
=
ha(bc
)i
3
=
habc
i
3
;
(2.19)
where
w
e
ha
v
e
used
the
fact
that
the
geometric
pro duct
is
asso ciativ
e
to
remo
v
e
the
brac
k
ets.
It
follo
ws
from
this
simple
deriv
ation
that
the
outer
pro duct
is
also
asso ciat-
iv
e,
(a
^
b)
^
c
=
a
^
(b
^
c
)
=
a
^
b
^
c:
(2.20)
This
is
true
in
general.
The
triv
ector
a
^
b
^
c
can
b e
pictured
as
the
parallelepip ed
formed
b
y
sw
eeping
a
^
b
along
c
(see
Fig.
3).
The
same
result
is
obtained
b
y
sw
eeping
b
^
c
along
a
,
whic
h
is
the
geometric
w
a
y
of
picturing
the
asso ciativit
y
of
the
outer
pro duct.
The
other
main
prop ert
y
of
the
outer
pro duct
is
that
it
is
an
tisymmetri
c
on
ev
ery
pair
of
v
ectors,
a
^
b
^
c
=
b
^
a
^
c
=
c
^
a
^
b
;
etc.
(2.21)
This
expresses
the
geometric
result
that
sw
apping
an
y
t
w
o
v
ectors
rev
erses
the
orien
t-
ation
(handedness)
of
the
pro duct.
2.4
The
Biv
ector
Algebra
Our
three
indep enden
t
biv
ectors
also
giv
e
us
a
further
new
pro duct
to
consider.
When
m
ultiply
ing
t
w
o
biv
ectors
w
e
nd,
for
example,
that
(e
1
^
e
2
)(e
2
^
e
3
)
=
e
1
e
2
e
2
e
3
=
e
1
e
3
;
(2.22)
resulting
in
a
third
biv
ector.
W
e
also
nd
that
(e
2
^
e
3
)(e
1
^
e
2
)
=
e
3
e
2
e
2
e
1
=
e
3
e
1
=
e
1
e
3
;
(2.23)
7
a
b
c
a
b
c
a
^
b
b
^
c
Figure
3:
The
T
rive
ctor.
The
result
of
sw
eeping
a
^
b
along
c
is
a
directed
v
olume,
or
triv
ector.
The
same
triv
ector
is
obtained
b
y
sw
eeping
b
^
c
along
a.
so
the
pro duct
is
an
tisymmetri
c.
The
symmetric
con
tribution
v
anishes
b ecause
the
t
w
o
planes
are
p erp endicular.
If
w
e
in
tro duce
the
follo
wing
lab elling
for
the
basis
biv
ectors:
B
1
=
e
2
e
3
;
B
2
=
e
3
e
1
;
B
3
=
e
1
e
2
(2.24)
w
e
nd
that
the
comm
utator
satis es
B
i
B
j
B
j
B
i
=
2
ij
k
B
k
:
(2.25)
This
algebra
is
closely
link
ed
to
3-d
rotations,
and
will
b e
familiar
from
the
quan
tum
theory
of
angular
momen
tum
.
It
is
useful
to
in
tro duce
a
sym
b ol
for
one-half
the
comm
utator
of
2
biv
ectors.
W
e
call
this
the
c
ommutator
pr
o
duct
and
denote
it
with
a
cross,
so
A
B
=
1
2
(AB
B
A):
(2.26)
The
comm
utator
pro duct
of
t
w
o
biv
ectors
alw
a
ys
results
in
a
third
biv
ector
(or
zero).
The
basis
biv
ectors
all
square
to
1,
and
all
an
ticomm
ute.
These
are
the
prop erties
of
the
generators
of
the
quaternion
algebra.
This
observ
ation
helps
to
sort
out
some
of
the
problems
encoun
tered
with
the
quaternions.
Hamilton
attempted
to
iden
tify
pure
quaternions
(n
ull
scalar
part)
with
v
ectors,
but
w
e
no
w
see
that
they
are
actually
bive
ctors.
This
has
an
imp ortan
t
consequence
when
w
e
lo ok
at
their
b eha
viour
under
re ections.
Hamilton
also
imp osed
the
condition
ij
k
=
1
on
his
unit
quaternions,
whereas
w
e
ha
v
e
B
1
B
2
B
3
=
e
2
e
3
e
3
e
1
e
1
e
2
=
+1:
(2.27)
T
o
set
up
a
direct
map
w
e
m
ust
ip
a
sign
somewhere,
for
example
in
the
y
comp onen
t:
i
$
B
1
;
j
$
B
2
;
k
$
B
3
:
(2.28)
This
sho
ws
us
that
the
quaternions
w
ere
left-hande
d,
ev
en
though
the
i;
j
;
k
w
ere
in
terpreted
as
a
righ
t-handed
set
of
v
ectors.
Not
surprisingly
,
this
w
as
a
source
of
some
confusion!
8
e
1
e
3
e
2
^
e
3
I
Figure
4:
The
pr
o
duct
of
a
ve
ctor
and
a
trive
ctor.
The
diagram
sho
ws
the
result
of
the
pro duct
e
1
I
=
e
1
(e
1
e
2
e
3
)
=
e
2
e
3
2.5
Pro
ducts
In
v
olving
the
Pseudoscalar
The
pseudoscalar
I
=
e
1
e
2
e
3
is
the
unique
righ
t-handed
unit
triv
ector
in
the
algebra.
This
giv
es
us
a
n
um
b er
of
new
pro ducts
to
consider.
W
e
start
b
y
forming
the
pro duct
of
I
with
the
v
ector
e
1
,
I
e
1
=
e
1
e
2
e
3
e
1
=
e
1
e
2
e
1
e
3
=
e
2
e
3
:
(2.29)
The
result
is
a
biv
ector
|
the
plane
p erp endicular
to
the
original
v
ector
(see
Fig.
4).
The
pro duct
of
a
grade-1
v
ector
with
the
grade-3
pseudoscalar
is
therefore
a
grade-2
biv
ector.
Rev
ersing
the
order
w
e
nd
that
e
1
I
=
e
1
e
1
e
2
e
3
=
e
2
e
3
:
(2.30)
The
result
is
therefore
indep enden
t
of
order
|
the
pseudoscalar
comm
utes
with
all
v
ectors
in
3-d,
I
a
=
a
I
;
for
all
a
:
(2.31)
It
follo
ws
that
I
comm
utes
with
all
elemen
ts
in
the
algebra.
This
is
alw
a
ys
the
case
for
the
pseudoscalar
in
spaces
of
o dd
dimension.
In
ev
en
dimensions,
the
pseudoscalar
an
ticomm
utes
with
all
v
ectors,
as
can
b e
easily
c
hec
k
ed
in
2-d.
W
e
can
no
w
express
eac
h
of
our
basis
biv
ectors
as
the
pro duct
of
the
pseudoscalar
and
a
dual
v
ector,
e
1
e
2
=
I
e
3
;
e
2
e
3
=
I
e
1
;
e
3
e
1
=
I
e
2
:
(2.32)
This
op eration
of
m
ultiplyi
ng
b
y
the
pseudoscalar
is
called
a
duality
transformation.
W
e
next
form
the
square
of
the
pseudoscalar
I
2
=
e
1
e
2
e
3
e
1
e
2
e
3
=
e
1
e
2
e
1
e
2
=
1:
(2.33)
So
the
pseudoscalar
comm
utes
with
all
elemen
ts
and
squares
to
1.
It
is
therefore
a
further
candidate
for
a
unit
imaginary
.
In
some
ph
ysical
applications
this
is
the
correct
9
one
to
use,
whereas
for
others
it
is
one
of
the
biv
ectors.
These
di eren
t
p ossibilities
pro
vide
us
with
a
v
ery
ric
h
geometric
language.
Finally
,
w
e
consider
the
pro duct
of
a
biv
ector
and
the
pseudoscalar:
I
(e
1
^
e
2
)
=
I
e
1
e
2
e
3
e
3
=
I
I
e
3
=
e
3
:
(2.34)
So
the
result
of
the
pro duct
of
I
with
the
biv
ector
formed
from
e
1
and
e
2
is
e
3
,
that
is,
min
us
the
v
ector
p erp endicular
to
the
e
1
^
e
2
plane.
This
a ords
a
de nition
of
the
v
ector
cross
pro duct
in
3-d
as
a
b
=
I
(a
^
b
):
(2.35)
The
b old
sym
b ol
should
not
b e
confused
with
the
sym
b ol
for
the
comm
utator
pro duct.
The
latter
is
extremely
useful,
whereas
the
v
ector
cross
pro duct
is
largely
redundan
t
no
w
that
w
e
ha
v
e
the
outer
pro duct
a
v
ailable.
Equation
(2.35)
sho
ws
ho
w
the
cross
pro duct
is
a
biv
ector
in
disguise,
the
biv
ector
b eing
mapp ed
to
a
v
ector
b
y
a
dualit
y
op eration.
It
is
also
no
w
clear
wh
y
the
pro duct
only
exists
in
3-d
|
this
is
the
only
space
for
whic
h
the
dual
of
a
biv
ector
is
a
v
ector.
W
e
will
ha
v
e
little
further
use
for
the
cross
pro duct
and
will
rarely
emplo
y
it
from
no
w
on.
This
means
w
e
can
also
do
a
w
a
y
with
the
a
wkw
ard
distinction
b et
w
een
axial
and
p olar
v
ectors.
Instead
w
e
just
talk
of
v
ectors
and
biv
ectors.
The
dualit
y
op eration
in
3-d
pro
vides
an
alternativ
e
w
a
y
to
understand
the
geometric
pro duct
a
B
of
a
v
ector
and
a
biv
ector.
W
e
write
B
=
I
b
in
terms
of
its
dual
v
ector
b,
so
that
w
e
no
w
ha
v
e
aB
=
I
a
b
=
I
(a
b
+
a
^
b):
(2.36)
This
demonstrates
that
the
symmetric
part
of
the
pro duct
generates
the
triv
ector
a
^
B
=
I
(a
b
)
=
1
2
(aB
+
B
a
);
(2.37)
whereas
the
an
tisymmetri
c
part
returns
a
v
ector
a
B
=
I
(a
^
b)
=
1
2
(aB
B
a
):
(2.38)
This
justi es
the
de nition
of
the
inner
and
outer
pro ducts
b et
w
een
a
v
ector
and
biv
ector.
As
with
pairs
of
v
ectors,
these
com
bine
to
return
the
geometric
pro duct,
a
B
=
a
B
+
a
^
B
:
(2.39)
3
F
urther
De nitions
An
imp ortan
t
op eration
in
GA
is
that
of
rev
ersing
the
order
of
v
ectors
in
an
y
pro duct.
This
is
denoted
with
a
dagger,
A
y
.
Scalars
and
v
ectors
are
in
v
arian
t
under
rev
ersion,
10
but
biv
ectors
c
hange
sign,
(e
1
e
2
)
y
=
e
2
e
1
=
e
1
e
2
:
(3.1)
Similarly
,
w
e
see
that
I
y
=
e
3
e
2
e
1
=
e
1
e
3
e
2
=
e
1
e
2
e
3
=
I
:
(3.2)
A
general
m
ultiv
ector
in
3-d
can
b e
written
M
=
+
a
+
B
+
I
:
(3.3)
F
rom
the
ab o
v
e
w
e
see
that
M
y
=
+
a
B
I
:
(3.4)
The
c
hoice
of
the
dagger
sym
b ol
re ects
the
fact
that,
if
one
c
ho oses
to
adopt
a
Her-
mitian
matrix
represen
tation
for
the
v
ector
generators,
the
rev
erse
op eration
corres-
p onds
to
the
Hermitian
adjoin
t
for
matrices.
It
is
also
useful
to
adopt
the
op
er
ator
or
dering
c
onvention
that,
in
the
absence
of
brac
k
ets,
inner
and
outer
pr
o
ducts
ar
e
p
erforme
d
b
efor
e
ge
ometric
pr
o
ducts.
This
cleans
up
expressions
b
y
enabling
us
to
remo
v
e
unnecessary
brac
k
ets.
F
or
example,
on
the
righ
t-hand
side
of
Eq.
(2.35)
w
e
can
no
w
write
a
b
=
I
a
^
b:
(3.5)
W
e
ha
v
e
already
in
tro duced
the
h
i
r
notation
for
pro
jecting
on
to
the
terms
of
grade-r .
F
or
the
op eration
of
pro
jecting
on
to
the
scalar
comp onen
t
w
e
usually
drop
the
subscript
0
and
write
hAB
i
=
hAB
i
0
(3.6)
for
the
scalar
part
of
the
pro duct
of
t
w
o
arbitrary
m
ultiv
ec
tors.
The
scalar
pro duct
is
alw
a
ys
symmetri
c
hAB
i
=
hB
Ai:
(3.7)
It
follo
ws
that
hA
B
C
i
=
hC
A
B
i:
(3.8)
This
cyclic
reordering
prop ert
y
is
v
ery
useful
in
practice.
11
a
a
0
m
a
k
a
?
Hyp erplane
a
=
a
?
+
a
k
a
0
=
a
?
a
k
Figure
5:
A
r
e e
ction
in
the
plane
p
erp
endicular
to
m.
4
Re ections
Supp ose
that
w
e
re ect
the
v
ector
a
in
the
(h
yp er)plane
orthogonal
to
some
unit
v
ector
m
(m
2
=
1).
The
comp onen
t
of
a
parallel
to
m
c
hanges
sign,
whereas
the
p erp endicular
comp onen
t
is
unc
hanged.
The
parallel
comp onen
t
is
the
pro
jection
on
to
m:
a
k
=
a
m
m :
(4.1)
(NB
op erator
ordering
con
v
en
tion
in
force
here.)
The
p erp endicular
comp onen
t
is
the
remainder
a
?
=
a
a
m
m
=
(am
a
m)m
=
a
^
m
m :
(4.2)
This
sho
ws
ho
w
the
w
edge
pro duct
pro
jects
on
to
the
comp onen
ts
p erp endicular
to
a
v
ector.
The
result
of
the
re ection
is
therefore
a
0
=
a
?
a
k
=
a
m
m
+
a
^
m
m
=
(m
a
+
m
^
a)m
=
ma
m :
(4.3)
This
remark
ably
compact
form
ula
only
arises
in
geometric
algebra.
W
e
can
start
to
see
no
w
that
geometric
pro ducts
arise
naturally
when
op
er
ating
on
v
ectors.
It
is
simple
to
c
hec
k
that
our
form
ula
has
the
required
prop erties.
F
or
an
y
v
ector
m
in
the
m
direction
w
e
ha
v
e
m(m )m
=
mm m
=
m
(4.4)
12
and
so
m
is
re ected.
Similarly
,
for
an
y
v
ector
n
p erp endicular
to
m
w
e
ha
v
e
m(n
)m
=
mnm
=
n
m m
=
n
(4.5)
and
so
n
is
una ected.
W
e
can
also
giv
e
a
simple
pro of
that
inner
pro ducts
are
unc
hanged
b
y
re ections,
a
0
b
0
=
(
ma
m )
(
m
bm )
=
hm ammbm i
=
hm abmi
=
hm mabi
=
a
b
:
(4.6)
W
e
next
construct
the
transformation
la
w
for
the
biv
ector
a
^
b
under
re ection
of
b oth
a
and
b.
W
e
obtain
a
0
^
b
0
=
(
ma
m )
^
(
mbm)
=
1
2
(m a
m mbm
m bmmam)
=
1
2
m(ab
ba)m
=
m
a
^
b
m :
(4.7)
W
e
reco
v
er
essen
tially
the
same
la
w,
but
with
a
crucial
sign
di erence.
Biv
ectors
do
not
transform
as
v
ectors
under
re ections.
This
is
the
reason
for
the
confusing
distinction
b et
w
een
p olar
and
axial
v
ectors
in
3-d.
Axial
v
ectors
in
v
ariably
arise
as
the
result
of
the
cross
pro duct.
They
are
really
biv
ectors
and
should
b e
treated
as
suc
h.
This
also
explains
wh
y
19th
cen
tury
mathematicians
w
ere
confused
b
y
the
transformation
prop erties
of
the
quaternions.
They
w
ere
exp ected
to
transform
as
v
ectors
under
re ections,
but
actually
transform
as
biv
ectors
(i.e.
with
the
opp osite
sign).
5
Rotations
F
or
man
y
y
ears,
Hamilton
struggled
with
the
problem
of
nding
a
compact
represen
t-
ation
for
rotations
in
3-d.
His
goal
w
as
to
generalise
to
represen
tation
of
2-d
rotations
as
a
complex
phase
c
hange.
The
k
ey
to
nding
the
correct
form
ula
is
to
use
that
result
that
a
r
otation
in
the
plane
gener
ate
d
by
two
unit
ve
ctors
m
and
n
is
achieve
d
by
suc-
c
essive
r
e e
ctions
in
the
(hyp
er)planes
p
erp
endicular
to
m
and
n.
This
is
illustrated
in
Fig.
6.
It
is
clear
that
an
y
comp onen
t
of
a
outside
the
m ^ n
plane
is
un
touc
hed.
It
is
also
a
simple
exercise
in
trigonometry
to
con rm
that
the
angle
b et
w
een
the
initial
v
ector
a
and
the
nal
v
ector
a
0
0
is
t
wice
the
angle
b et
w
een
m
and
n
.
The
result
of
the
successiv
e
re ections
is
therefore
to
rotate
through
2
in
the
m
^
n
plane,
where
m
n
=
cos( ).
So
ho
w
do es
this
lo ok
in
GA?
a
0
=
mam
(5.1)
a
00
=
n
a
0
n
=
n
(
ma
m )n
=
n
m a
m n
(5.2)
13
a
a
0
a
00
m
n
m
^
n
Figure
6:
A
R
otation
fr
om
2
R
e e
ctions.
a
0
is
the
result
of
re ecting
a
in
the
plane
p erp endicular
to
m .
a
00
is
the
result
of
re ecting
a
0
in
the
plane
p erp endicular
to
n.
This
is
b eginning
to
lo ok
v
ery
simple!
W
e
de ne
the
r
otor
R
b
y
R
=
nm:
(5.3)
Note
the
ge
ometric
pro duct
here!
W
e
can
no
w
write
a
rotation
as
a
7!
RaR
y
(5.4)
Incredibly
,
this
form
ula
w
orks
for
an
y
grade
of
m
ultiv
e
ctor,
in
an
y
dimension,
of
an
y
signature!
T
o
mak
e
con
tact
with
the
2-d
result
w
e
rst
expand
R
as
R
=
nm
=
n
m
+
n
^
m
=
cos ( )
+
n
^
m:
(5.5)
So
what
is
the
magnitude
of
the
biv
ector
n
^
m ?
(n
^
m)
(n
^
m)
=
hn
^
m
n
^
mi
=
hnm
n
^
mi
=
n
[m
(n ^
m )]
=
n
(m
cos( )
n
)
=
cos
2
( )
1
=
sin
2
( ):
(5.6)
W
e
therefore
de ne
a
unit
biv
ector
in
the
m
^
n
plane
b
y
^
B
=
m
^
n=
sin( );
^
B
2
=
1:
(5.7)
This
c
hoice
of
orien
tation
(m
^ n
rather
than
n ^ m )
ensures
that
the
biv
ector
has
the
same
orien
tation
as
the
rotation,
as
can
b e
seen
in
Fig.
6.
14
In
terms
of
the
biv
ector
^
B
w
e
no
w
ha
v
e
R
=
cos( )
^
B
sin( ):
(5.8)
Lo ok
familiar?
This
is
nothing
else
than
the
p olar
decomp osition
of
a
complex
n
um
b er,
with
the
unit
imaginary
replaced
b
y
the
unit
biv
ector
^
B
.
W
e
can
therefore
write
R
=
exp(
^
B
):
(5.9)
The
exp onen
tial
here
is
de ned
in
terms
of
its
p o
w
er
series
in
the
normal
w
a
y
.
It
is
p ossible
to
sho
w
that
this
series
is
absolutely
con
v
ergen
t
for
an
y
m
ultiv
ector
argumen
t.
(Exp onen
tiating
a
m
ultiv
e
ctor
is
essen
tially
the
same
as
exp onen
tiating
a
matrix).
No
w
recall
that
our
form
ula
w
as
for
a
rotation
through
2 .
If
w
e
w
an
t
to
rotate
through
,
the
appropriate
rotor
is
R
=
exp
f
^
B
=2g
(5.10)
whic
h
giv
es
us
the
nal
form
ula
a
7!
e
^
B
=2
a
e
^
B
=2
:
(5.11)
This
describ es
a
rotation
through
in
the
^
B
plane,
with
orien
tation
sp eci ed
b
y
^
B
.
The
GA
description
forces
us
to
think
of
rotations
taking
place
in
a
plane
as
opp osed
to
ab out
an
axis.
The
latter
is
an
en
tirely
3-d
concept,
whereas
the
concept
of
a
plane
is
quite
general.
Rotors
are
one
of
the
fundamen
tal
concepts
in
geometric
algebra.
Since
the
rotor
R
is
a
geometric
pro duct
of
t
w
o
unit
v
ectors,
w
e
see
imme
diately
that
RR
y
=
n
m (n
m )
y
=
n
m mn
=
1
=
R
y
R:
(5.12)
This
pro
vides
a
quic
k
pro of
that
our
form
ula
has
the
correct
prop ert
y
of
preserving
lengths
and
angles,
a
0
b
0
=
(RaR
y
)
(RbR
y
)
=
hRaR
y
Rb R
y
i
=
hRabR
y
i
=
a
b:
(5.13)
No
w
supp ose
that
the
t
w
o
v
ectors
forming
the
biv
ector
B
=
a
^
b
are
b oth
rotated.
What
is
the
expression
for
the
resulting
biv
ector?
T
o
nd
this
w
e
form
B
0
=
a
0
^
b
0
=
1
2
(a
0
b
0
b
0
a
0
)
=
1
2
(RaR
y
Rb R
y
Rb R
y
RaR
y
)
=
1
2
(RabR
y
RbaR
y
)
=
1
2
R(ab
ba
)R
y
=
Ra
^
bR
y
=
RB
R
y
:
(5.14)
Biv
ectors
are
rotated
using
precisely
the
same
form
ula
as
v
ectors!
The
same
turns
out
to
b e
true
for
all
geometric
ob
jects
represen
ted
b
y
m
ultiv
ec
tors.
This
is
one
of
the
most
attractiv
e
features
of
geometric
algebra.
15
6
Prop
erties
of
Rotors
Let
us
consider
the
problem
of
rotating
a
unit
v
ector
n
1
in
to
another
unit
v
ector
n
2
in
3-d
space,
where
the
angle
b et
w
een
these
t
w
o
v
ectors
in
.
What
is
the
rotor
R
whic
h
p erforms
suc
h
a
rotation?
If
R
is
the
rotor
w
e
require
then
it
m
ust
satisfy
n
2
=
Rn
1
R
y
whic
h,
under
m
ultiplic
ation
on
the
righ
t
b
y
R
giv
es,
n
2
R
=
Rn
1
:
(6.1)
No
w
consider
the
quan
tit
y
(1
+
n
2
n
1
).
Since
n
1
2
=
n
2
2
=
1,
w
e
see
that
n
2
(1
+
n
2
n
1
)
=
n
2
+
n
1
(6.2)
(1
+
n
2
n
1
)n
1
=
n
1
+
n
2
(6.3)
so
equation
(6.1)
is
satis ed
if
R
/
(1
+
n
2
n
1
).
It
remains
simply
to
normalize
R
so
that
it
satis es
RR
y
=
1.
If
R
=
(1
+
n
2
n
1
)
w
e
obtain
RR
y
=
2
(1
+
n
2
n
1
)(1
+
n
1
n
2
)
=
2
2
(1
+
n
2
n
1
);
(6.4)
whic
h
giv
es
us
the
follo
wing
form
ula
for
R:
R
=
1
+
n
2
n
1
p
2(1
+
n
2
n
1
)
:
(6.5)
W
e
can
reco
v
er
our
earlier
expression
b
y
rst
noting
that
p
2(1
+
n
2
n
1
)
=
2
cos ( =2):
(6.6)
The
rotor
R
can
b e
no
w
written
as
R
=
cos( =2)
+
n
2
^
n
1
jn
2
^
n
1
j
sin( =2)
=
exp
2
n
1
^
n
2
jn
2
^
n
1
j
;
(6.7)
where
jn
2
^
n
1
j
is
the
magnitude
of
the
biv
ector
n
2
^
n
1
,
de ned
b
y
jn
2
^
n
1
j
=
(n
2
^
n
1
)
(n
1
^
n
2
)
1=2
:
(6.8)
In
this
w
a
y
the
rotor
is
again
written
as
the
exp onen
tial
of
a
biv
ector,
reco
v
ering
Eq.
(5.9).
An
alternativ
e
represen
tation,
a
v
ailable
only
in
3-d,
is
to
in
tro duce
the
dual
v
ector
n
and
write
the
rotor
as
R
=
exp
I
2
n
=
cos ( =2)
I
n
sin( =2):
(6.9)
This
generates
a
rotation
of
radians
ab out
an
axis
parallel
to
the
unit
v
ector
n
in
a
righ
t-handed
screw
sense.
(This
is
precisely
ho
w
3-D
rotations
are
represen
ted
in
the
quaternion
algebra.)
16
6.1
Comp
osition
La
w
A
feature
of
the
rotor
treatmen
t
of
rotations
is
the
ease
with
whic
h
rotations
can
no
w
b e
com
bined.
Supp ose
that
the
rotor
R
1
tak
es
the
v
ector
a
to
the
v
ector
b,
b
=
R
1
aR
y
1
:
(6.10)
If
the
v
ector
b
is
no
w
rotated
b
y
a
second
rotor
R
2
to
the
v
ector
c,
w
e
ha
v
e
c
=
R
2
bR
y
2
;
(6.11)
and
therefore
c
=
(R
2
R
1
)a(R
2
R
1
)
y
:
(6.12)
The
com
bined
rotation
is
therefore
generated
b
y
the
comp osite
rotor
R
=
R
2
R
1
:
(6.13)
This
is
the
gr
oup
c
omp
osition
la
w
for
rotors.
It
is
straigh
tforw
ard
to
c
hec
k
that
this
results
in
a
new
rotor.
This
comp osition
rule
has
t
w
o
imp ortan
t
features.
The
rst
is
that
in
nding
the
comp osite
rotor
a
maximum
of
16
m
ultipli
cations
is
required.
This
compares
fa
v
ourably
with
the
27
required
when
m
ultiply
ing
together
2
rotation
matrices.
The
second
is
that
w
e
ha
v
e
far
b etter
con
trol
o
v
er
n
umerical
errors
when
com
bining
rotors.
If
n
umerical
errors
do
arise,
the
w
orst
that
can
happ en
is
that
the
rotor
is
no
longer
normalised
exactly
to
1.
This
is
easily
recti ed
b
y
rescaling.
No
suc
h
simple
metho d
is
a
v
ailable
with
rotation
matrices.
If
n
umerical
errors
mean
that
the
matrix
is
no
longer
orthogonal
there
is
no
simple
metho d
to
reco
v
er
the
\nearest"
orthogonal
matrix.
6.2
F
rames
and
Recipro
cals
A
frequen
tly-encoun
tered
problem
is
ho
w
to
nd
the
rotor
giv
en
t
w
o
arbitrary
sets
of
v
ectors,
kno
wn
to
b e
related
m
y
a
rotation.
T
o
solv
e
this
problem
w
e
m
ust
rst
in
tro duce
the
notion
of
a
r
e
cipr
o
c
al
frame.
Giv
en
a
set
of
linearly
indep enden
t
v
ectors
fe
i
g
(where
no
w
no
assumption
of
orthonormalit
y
is
made),
the
recipro cal
frame,
fe
i
g,
is
de ned
suc
h
that
e
i
e
j
=
Æ
i
j
:
(6.14)
W
e
construct
suc
h
a
recipro cal
frame
in
n-dimensions
as
follo
ws:
e
j
=
(
1)
j
1
e
1
^
e
2
^
^
e
j
^
^
e
n
I
1
e
(6.15)
17
where
I
e
=
e
1
^
e
2
^
^
e
n
and
e
j
indicates
that
e
j
is
missing
from
the
pro duct.
In
three
dimensions
this
is
a
v
ery
simple
op eration
and
the
recipro cal
frame
v
ectors
for
a
linearly
indep enden
t
set
of
v
ectors
fe
i
g
are
as
follo
ws:
e
1
=
1
I
e
2
^
e
3
e
2
=
1
I
e
3
^
e
1
(6.16)
e
3
=
1
I
e
1
^
e
2
;
where
I
=
e
3
^
e
2
^
e
1
.
A
v
ector
a
can
b e
expanded
in
either
frame
as
follo
ws
(summation
con
v
en
tion
in
force)
a
=
a
j
e
j
(a
e
j
)e
j
(6.17)
a
=
a
j
e
j
(a
e
j
)e
j
:
(6.18)
The
iden
ti cation
of
a
j
with
a
e
j
is
obtained
b
y
dotting
the
equation
a
=
a
j
e
j
with
e
i
.
Similarly
,
the
iden
ti cation
of
a
j
with
a
e
j
is
obtained
b
y
dotting
a
=
a
j
e
j
with
e
i
.
So,
giv
en
a
general
frame
and
a
v
ector,
the
recipro cal
frame
is
needed
to
construct
the
comp onen
ts
of
the
v
ector
in
the
c
hosen
frame.
Of
course,
for
orthonormal
frames
there
is
no
distinction
b et
w
een
the
frame
and
its
recipro cal.
Supp ose
no
w
that
w
e
ha
v
e
t
w
o
sets
of
v
ectors
in
3-d
(not
necessarily
orthonormal)
fe
k
g
and
ff
k
g
whic
h
w
e
kno
w
are
related
b
y
a
rotation.
W
e
hence
kno
w
that
f
k
=
Re
k
R
y
(6.19)
and
w
e
seek
a
simple
expression
for
the
rotor
R.
As
w
e
are
in
3-d,
w
e
can
write
R
=
e
B
=2
and
R
y
=
e
B
=2
=
cos(jB
j=2)
+
sin
(jB
j=2)B
=jB
j:
(6.20)
W
e
therefore
nd
that
e
k
R
y
e
k
=
e
k
[cos
(jB
j=2)
+
sin(jB
j=2)B
=jB
j]e
k
=
3
cos (jB
j=2)
sin
(jB
j=2)B
=jB
j
=
4
cos (jB
j=2)
R
y
:
(6.21)
W
e
no
w
form
f
k
e
k
=
Re
k
R
y
e
k
=
4
cos (jB
j=2)R
1:
(6.22)
It
follo
ws
that
R
is
a
scalar
m
ultiple
of
1
+
f
k
e
k
.
W
e
therefore
establish
the
simple
form
ula
R
=
1
+
f
k
e
k
j1
+
f
k
e
k
j
=
p
(
y
)
(6.23)
where
=
1
+
f
k
e
k
.
This
neat
form
ula
reco
v
ers
the
rotor
directly
from
the
frame
v
ectors.
It
w
orks
in
all
cases
except
when
the
rotation
is
through
180
o
,
in
whic
h
case
=
0.
This
is
easily
handled
as
a
sp ecial
case.
18
6.3
Rotation
Matrices
The
con
v
en
tional
w
a
y
to
treat
rotations
is
through
the
application
of
3
3
orthogonal
matrices,
whic
h
are
applied
to
the
co ordinates
of
a
v
ector
in
a
giv
en
xed
orthonormal
frame.
If
w
e
denote
this
frame
b
y
fe
k
g
w
e
ha
v
e
a
=
a
k
e
k
and
a
0
=
RaR
y
=
a
0
k
e
k
:
(6.24)
The
comp onen
ts
of
the
rotated
v
ector
a
0
are
related
to
the
original
comp onen
ts
b
y
a
0
i
=
R
ij
a
j
(6.25)
where
R
is
an
orthogonal
matrix.
F
rom
the
preceding
w
e
ha
v
e
a
0
i
=
e
i
a
0
=
e
i
(RaR
y
)
=
e
i
(Re
j
R
y
)a
j
:
(6.26)
It
follo
ws
that
the
matrix
comp onen
ts
are
giv
en
b
y
R
ij
=
e
i
(Re
j
R
y
):
(6.27)
As
exp ected,
the
comp onen
ts
dep end
quadratically
on
the
rotor
R.
It
follo
ws
that
R
and
R
enco de
the
same
rotation.
Ev
en
for
the
simplest
rotations,
one
can
see
that
the
rotor
enco ding
is
signi can
tly
more
compact
than
the
matrix
expression.
Giv
en
a
rotation
matrix
R
ij
one
can
reco
v
er
the
rotor
eÆcien
tly
b
y
adapting
Eq.
(6.23).
W
e
de ne
=
1
+
R
ij
e
i
e
j
=
1
+
Re
j
R
y
e
j
(6.28)
so
that
the
rotor
is
giv
en
b
y
R
=
p
(
y
)
:
(6.29)
This
result
mak
es
it
easy
to
con
v
ert
from
the
standard
form
ulation
to
the
geometric
algebra
framew
ork.
6.4
Euler
Angles
The
Standard
Euler
angle
form
ulation
of
rotations
is
to
express
an
y
rotation
as
a
com
bination
of
3
rotations:
1st:
rotate
ab out
the
e
3
axis
2nd:
rotate
ab out
the
rotated
e
1
axis
3rd:
rotate
ab out
the
rotated
e
3
axis
19
In
traditional
accoun
ts
this
is
in
v
olv
es
de ning
a
set
of
3
rotation
matrices
A
1
=
0
@
cos
sin
0
sin
cos
0
0
0
1
1
A
;
A
2
=
0
@
1
0
0
0
cos
sin
0
sin
cos
1
A
A
3
=
0
@
cos
sin
0
sin
cos
0
0
0
1
1
A
(6.30)
w
e
are
then
told
to
apply
these
matrices
in
r
everse
order
to
form
A
=
A
1
A
2
A
3
(6.31)
so
that
the
co ordinates
transform
as
x
0
=
Ax.
This
matrix
ordering
is
often
con-
fusing
and
is
justi ed
using
argumen
ts
based
on
mixtures
of
\activ
e"
and
\passiv
e"
transformations.
It
is
therefore
instructiv
e
to
see
ho
w
this
lo oks
in
geometric
algebra.
W
e
start
b
y
de ning
the
rotor
R
1
=
exp
I
2
e
3
;
(6.32)
whic
h
generates
a
rotation
ab out
the
e
3
axis.
Next
w
e
need
a
rotation
ab out
the
rotated
e
1
axis,
whic
h
is
generated
b
y
R
2
=
exp
I
2
e
0
1
(6.33)
where
e
0
1
=
R
1
e
1
R
y
1
.
One
can
see,
then,
that
R
2
=
R
1
exp
I
2
e
1
R
y
1
:
(6.34)
Finally
,
w
e
rotate
ab out
the
new
3-axis,
whic
h
requires
the
rotor
R
3
=
exp
I
2
e
00
3
(6.35)
where
e
00
3
=
R
2
R
1
e
3
R
y
1
R
y
2
.
In
this
case
the
rotor
can
b e
written
as
R
3
=
R
2
R
1
exp
I
2
e
3
R
y
1
R
y
2
:
(6.36)
No
w
forming
the
com
bined
rotor
R
w
e
nd
that
R
=
R
3
R
2
R
1
=
R
2
R
1
exp
I
2
e
3
R
y
1
R
y
2
R
2
R
1
=
R
1
exp
I
2
e
1
R
y
1
R
1
exp
I
2
e
3
=
exp
I
2
e
3
exp
I
2
e
1
exp
I
2
e
3
:
(6.37)
20
This
fully
explains
the
order
in
whic
h
the
rotations
are
applied,
and
a
v
oids
all
complic-
ations
connected
with
c
hanging
frames
midw
a
y
through
the
calculation,
or
attempting
to
distinguish
rotations
of
co ordinates,
rotations
of
co ordinate
axes,
and
(\activ
e")
rotations
of
v
ectors.
Despite
the
clean
form
of
the
Euler
angle
formalism
in
geometric
algebra,
this
is
rarely
an
optimal
enco ding
for
rotations.
Giv
en
an
arbitrary
rotor,
its
decomp osition
in
to
Euler
angles
is
not
straigh
tforw
ard,
and
the
pro duct
form
ula
is
equally
messy
.
In
practice
it
is
b est
to
either
w
ork
directly
with
the
rotor
R,
or
with
its
biv
ector
generator
B
,
R
=
exp(
B
=2).
6.5
In
terp
olating
Rotors
Rotors
are
elemen
t
s
of
a
four-dimensional
space,
normalised
to
1.
They
can
b e
rep-
resen
ted
as
p oin
ts
on
a
3-spher
e
|
the
set
of
unit
v
ectors
in
four
dimensions.
This
is
the
rotor
gr
oup
manifold.
A
t
an
y
p oin
t
on
the
manifold,
the
tangent
sp
ac
e
is
three-
dimensional.
This
is
the
analog
of
the
tangen
t
plane
to
a
sphere
in
three
dimensions.
Rotors
therefore
require
three
parameters
to
sp ecify
them
uniquely
.
The
simplest
c
hoice
of
parameters
is
directly
in
terms
of
the
biv
ector
generators,
with
jB
2
j
:
(6.38)
The
rotors
R
and
R
generate
the
same
rotation,
b ecause
of
their
double-sided
action.
It
follo
ws
that
the
r
otation
group
manifold
is
more
complicated
than
the
rotor
group
manifold
|
it
is
a
pro
jectiv
e
3-sphere
with
p oin
ts
R
and
R
iden
ti ed.
This
is
one
reason
wh
y
it
is
usually
easier
to
w
ork
with
rotors.
Supp ose
w
e
are
giv
en
t
w
o
estimates
of
a
rotation,
R
0
and
R
1
,
ho
w
do
w
e
nd
the
mid-p oin
t?
With
rotors
this
is
remark
ably
easy!
Supp ose
that
the
rotors
are
R
0
and
R
1
.
W
e
rst
mak
e
sure
they
ha
v
e
the
smallest
angle
b et
w
een
them
in
four
dimensions.
This
is
done
b
y
ensuring
that
hR
0
R
y
1
i
=
cos
>
0:
(6.39)
If
this
inequalit
y
is
not
satis ed,
then
the
sign
of
one
of
the
rotors
should
b e
ipp ed.
The
`shortest'
path
b et
w
een
the
rotors
on
the
group
manifold
is
de ned
b
y
R()
=
R
0
exp(B
);
(6.40)
where
R(0)
=
R
0
;
R(1)
=
R
1
:
(6.41)
It
follo
ws
that
w
e
can
nd
B
from
exp
(B
)
=
R
y
0
R
1
:
(6.42)
21
The
path
de ned
b
y
exp(B
)
is
an
in
v
arian
t
construct.
If
b oth
endp oin
ts
are
trans-
formed,
the
path
transforms
in
the
same
w
a
y
.
The
midp oin
t
is
R
1=2
=
R
0
exp(B
=2);
(6.43)
whic
h
therefore
generates
the
midp oin
t
rotation.
This
is
quite
general
|
it
w
orks
for
an
y
rotor
group
(or
an
y
Lie
gr
oup
).
F
or
rotations
in
three
dimensions
w
e
can
do
ev
en
b etter.
R
0
and
R
1
can
b e
view
ed
as
t
w
o
unit
v
ectors
in
a
four-dimensional
space.
The
path
exp
(B
)
lies
in
the
plane
sp eci ed
b
y
these
v
ectors:
the
rotors
can
therefore
b e
treated
as
unit
v
ectors
in
four
dimensions.
The
path
b et
w
een
them
lies
en
tirely
in
the
plane
of
the
t
w
o
rotors,
and
therefore
de nes
a
segmen
t
of
a
circle.
The
rotor
path
b et
w
een
R
0
and
R
1
can
b e
written
as
R()
=
R
0
cos
+
sin
^
B
;
(6.44)
where
w
e
ha
v
e
used
B
=
^
B
.
But
w
e
kno
w
that
exp
(B
)
=
cos
+
sin
^
B
=
R
y
0
R
1
:
(6.45)
It
follo
ws
that
R()
=
R
0
sin
sin
cos
+
sin
(R
y
0
R
1
cos
)
(6.46)
=
1
sin
sin(1
)
R
0
+
sin
R
1
;
(6.47)
whic
h
satis es
R()R
y
()
=
1
for
all
.
The
midp oin
t
rotor
is
therefore
simply
R
1=2
=
sin( =2)
sin
(R
0
+
R
1
):
(6.48)
This
giv
es
us
a
remark
ably
simple
prescription
for
nding
the
midp oin
t:
add
the
r
otors
and
normalise
the
r
esult.
By
comparison,
the
equiv
alen
t
matrix
is
quadratic
in
R,
and
so
is
m
uc
h
more
diÆcult
to
express
in
terms
of
the
t
w
o
endp oin
t
rotation
matrices.
Supp ose
no
w
that
w
e
ha
v
e
a
n
um
b er
of
estimates
for
a
rotation
and
w
an
ted
to
nd
the
a
v
erage.
Again
the
answ
er
is
simple.
First
one
c
ho oses
the
sign
of
the
rotors
so
that
they
are
all
in
the
`closest'
con guration.
This
will
normally
b e
easy
if
the
rotations
are
all
roughly
equal.
If
some
of
the
rotations
are
quite
di eren
t
then
one
migh
t
ha
v
e
to
searc
h
around
for
the
closest
con guration,
though
in
these
cases
the
a
v
erage
of
suc
h
rotations
is
not
a
useful
concept.
Once
one
has
all
of
the
rotors
c
hosen,
one
simply
adds
them
up
and
normalises
the
result
to
obtain
the
a
v
erage.
This
sort
of
calculation
can
b e
useful
in
computer
vision
problems
where
one
has
a
n
um
b er
of
estimates
of
the
relativ
e
rotations
b et
w
een
cameras,
and
their
a
v
erage
is
required.
The
lesson
here
is
that
problems
in
v
olving
rotations
can
b e
simpli ed
b
y
w
orking
with
rotors
and
relaxing
the
normalisation
criteria.
This
enables
us
to
w
ork
in
a
four-
dimensional
linear
space
and
is
the
basis
for
a
simpli ed
calculus
for
rotations.
22
7
Di eren
tiation
for
m
ultiv
ector
quan
tities
Here
w
e
giv
e
a
brief
discussion
of
the
pro cess
of
di eren
tiating
with
resp ect
to
an
y
m
ultiv
ector.
Ha
ving
a
v
ailable
suc
h
a
calculus
means
that,
in
practice,
it
is
easy
to
tak
e
deriv
ativ
es
with
resp ect
to
rotors
and
v
ectors
and
this
mak
es
man
y
least-squares
minim
iz
ation
problems
m
uc
h
simpler
to
deal
with.
In
computer
vision
and
motion
analysis
one
tends
to
dra
w
frequen
tly
on
an
approac
h
whic
h
minim
iz
es
some
expression
in
order
to
nd
the
relev
an
t
rotations
and
translations
{
this
is
a
standard
tec
hnique
for
an
y
estimation
problem
in
the
presence
of
uncertain
t
y
.
If
X
is
a
mixed-grade
m
ultiv
ector,
X
=
P
r
X
r
,
and
F
(X
)
is
a
general
m
ultiv
ector-
v
alued
function
of
X
,
then
the
deriv
ativ
e
of
F
in
the
A
`direction'
is
written
as
A
@
X
F
(X
)
(here
w
e
use
to
denote
the
scalar
part
of
the
pro duct
of
t
w
o
m
ultiv
ec
tors,
i.e.
A
B
hAB
i),
and
is
de ned
as
A
@
X
F
(X
)
lim
!0
F
(X
+
A)
F
(X
)
:
(7.1)
F
or
the
limit
on
the
righ
t
hand
side
to
mak
e
sense
A
m
ust
con
tain
only
grades
whic
h
are
con
tained
in
X
and
no
others.
If
X
con
tains
no
terms
of
grade-r
and
A
r
is
a
homogeneous
m
ultiv
ec
tor,
then
w
e
de ne
A
r
@
X
=
0.
This
de nition
of
the
deriv
ativ
e
also
ensures
that
the
op erator
A
@
X
is
a
scalar
op erator
and
satis es
all
of
the
usual
partial
deriv
ativ
e
prop erties.
W
e
can
no
w
use
the
ab o
v
e
de nition
of
the
directional
deriv
ativ
e
to
form
ulate
a
general
expression
for
the
m
ultiv
ec
tor
deriv
ativ
e
@
X
without
reference
to
one
particular
direction.
This
is
accomplished
b
y
in
tro ducing
an
arbitrary
frame
fe
j
g
and
extending
this
to
a
basis
(v
ectors,
biv
ectors,
etc..)
for
the
en
tire
algebra,
fe
J
g.
Then
@
X
is
de ned
as
@
X
X
J
e
J
(e
J
@
X
);
(7.2)
where
fe
J
g
is
an
extended
basis
built
out
of
the
recipro cal
frame.
The
directional
deriv
ativ
e,
e
J
@
X
,
is
only
non-zero
when
e
J
is
one
of
the
grades
con
tained
in
X
(as
previously
discussed)
so
that
@
X
inherits
the
m
ultiv
ec
tor
prop erties
of
its
argumen
t
X
.
Although
w
e
ha
v
e
here
de ned
the
m
ultiv
e
ctor
deriv
ativ
e
using
an
extended
basis,
it
should
b e
noted
that
the
sum
o
v
er
all
the
basis
ensures
that
@
X
is
indep enden
t
of
the
c
hoice
of
fe
j
g
and
so
all
of
the
prop erties
of
@
X
can
b e
form
ulated
in
a
frame-free
manner.
One
of
the
most
useful
results
concerning
m
ultiv
ector
deriv
ativ
es
is
@
X
hX
B
i
=
B
;
(7.3)
where
w
e
assume
that
B
and
X
con
tain
the
same
grades.
(If
the
grades
are
di eren
t
then
only
the
terms
in
B
whic
h
share
grades
with
X
are
pro duced
on
the
righ
t.)
F
rom
this
basic
result
one
can
also
see
that
@
X
hX
y
B
i
=
@
X
hX
B
y
i
=
B
y
:
(7.4)
23
7.1
Rotor
Calculus
An
y
function
of
a
rotation
can
b e
view
ed
as
taking
its
v
alues
o
v
er
the
group
manifold.
In
most
of
what
follo
ws
w
e
are
in
terested
in
scalar
functions,
though
there
is
no
reason
to
restrict
to
this
case.
The
deriv
ativ
e
of
the
function
with
resp ect
to
a
rotor
de nes
a
v
ector
in
the
tangen
t
space
at
eac
h
p oin
t
on
the
group
manifold.
The
v
ector
p oin
ts
in
the
direction
of
steep est
increase
of
the
function.
This
can
all
b e
made
mathematically
rigorous
and
is
the
sub
ject
of
di er
ential
ge
ometry.
The
problem
is
that
m
uc
h
o
this
is
o
v
er-complicated
for
the
relativ
ely
simple
minim
isation
problems
encoun
tered
in
computer
vision.
W
orking
in
trinsically
on
the
group
manifold
in
v
olv
es
in
tro ducing
lo cal
co ordinates
(suc
h
as
the
Euler
angles)
and
di eren
tiating
with
resp ect
to
eac
h
of
these
in
turn.
The
resulting
calculations
can
b e
long
and
messy
and
often
hide
the
simplici
t
y
of
the
answ
er.
Geometric
algebra
pro
vides
us
with
a
more
elegan
t
and
simpler
alternativ
e.
W
e
relax
the
rotor
normalisation
constrain
t
and
replace
R
b
y
|
a
general
elemen
t
of
the
ev
en
subalgebra.
It
is
straigh
tforw
ard
to
sho
w
that
the
deriv
ativ
e
op erator
de ned
ab o
v
e
reduces
to
a
simple
form
if
w
e
rst
decomp ose
in
terms
of
the
fe
i
g
basis
as
=
0
+
3
X
k =1
k
I
e
k
(7.5)
where
the
f
0
;
:
:
:
;
3
g
are
a
set
of
scalar
comp onen
ts.
The
m
ultiv
e
ctor
deriv
ativ
e
then
b ecomes
@
=
@
@
0
3
X
k =1
I
e
k
@
@
k
:
(7.6)
This
deriv
ativ
e
is
indep enden
t
of
the
c
hosen
frame.
It
satis es
the
basic
result
@
h
Ai
=
A
(7.7)
where
A
is
an
ev
en-grade
m
ultiv
ector
(indep enden
t
of
).
All
further
results
for
@
are
built
up
from
this
basic
result
and
Leibniz'
rule
for
the
deriv
ativ
e
of
a
pro duct.
The
basic
tric
k
no
w
is
to
re-write
a
rotation
as
RaR
y
=
a
1
:
(7.8)
This
w
orks
b ecause
an
y
ev
en
m
ultiv
ec
tor
can
b e
written
as
=
1=2
R
(7.9)
where
R
is
a
rotor,
=
y
and
=
0
if
and
only
if
=
0.
The
in
v
erse
of
is
then
1
=
1=2
R
y
(7.10)
24
so
that
1
=
RR
y
=
1:
(7.11)
The
equalit
y
of
equation
(7.8)
follo
ws
immediatel
y
.
If
one
imagines
a
function
o
v
er
a
sphere
in
three
dimensions,
one
can
extend
this
to
a
function
o
v
er
all
space
b
y
attac
hing
the
same
v
alue
to
all
p oin
ts
on
eac
h
line
from
the
origin.
The
extension
R
7!
do es
precisely
this,
but
in
a
four
dimensional
space.
W
e
are
no
w
able
to
di eren
tiate
functions
of
the
rotation
quite
simply
.
The
t
ypical
application
is
to
a
scalar
of
the
t
yp e
(RaR
y
)
b
=
hRaR
y
bi
=
h
a
1
bi:
(7.12)
T
o
di eren
tiate
this
w
e
need
a
result
for
the
deriv
ativ
e
of
the
in
v
erse
of
a
m
ultiv
ec
tor.
W
e
start
b
y
letting
M
b e
a
constan
t
m
ultiv
ec
tor,
and
deriv
e
0
=
@
h
1
M
i
=
1
M
+
_
@
h
_
1
M
i;
(7.13)
where
the
o
v
erdots
denote
the
scop e
of
the
deriv
ativ
e.
It
follo
ws
that
_
@
h
_
1
M
i
=
1
M
:
(7.14)
But
in
this
form
ula
w
e
can
no
w
let
M
b ecome
a
function
of
,
as
only
the
rst
term,
1
,
is
acted
on
b
y
the
di eren
tial
op erator.
W
e
can
therefore
replace
M
b
y
M
1
to
obtain
the
useful
form
ula
_
@
h
_
1
M
i
=
1
M
1
:
(7.15)
So,
let
us
no
w
consider
the
problem
of
nding
the
rotor
R
whic
h
`most
closely'
rotates
the
v
ectors
fu
i
g
on
to
the
v
ectors
fv
i
g,
i
=
1;
:::;
n.
More
precisely
,
w
e
wish
to
nd
the
rotor
R
whic
h
minim
ize
s
=
n
X
i=1
(v
i
Ru
i
R
y
)
2
:
(7.16)
Expanding
giv
es
=
n
X
i=1
(v
i
2
v
i
Ru
i
R
y
Ru
i
R
y
v
i
+
R(u
i
2
)R
y
)
=
n
X
i=1
(v
i
2
+
u
i
2
)
2hv
i
Ru
i
R
y
i
:
(7.17)
25
W
e
no
w
replace
Ru
i
R
y
with
u
i
1
and
di eren
tiate,
forming
@
(
)
=
2
n
X
i=1
@
hv
i
u
i
1
i
=
2
n
X
i=1
_
@
h
_
A
i
i
+
_
@
hB
i
_
1
i
;
where
A
i
=
u
i
1
v
i
and
B
i
=
v
i
u
i
(using
the
cyclic
reordering
prop ert
y).
The
rst
term
is
easily
ev
aluated
to
giv
e
A
i
.
T
o
ev
aluate
the
second
term
w
e
can
use
equation
(7.15).
One
can
then
substitute
=
R
and
note
that
R
1
=
R
y
as
RR
y
=
1.
W
e
then
ha
v
e
@
(
)
=
2
n
X
i=1
u
i
1
v
i
1
(v
i
u
i
)
1
=
2
1
n
X
i=1
(
u
i
1
)v
i
v
i
(
u
i
1
)
=
4R
y
n
X
i=1
v
i
^
(Ru
i
R
y
):
(7.18)
Th
us
the
rotor
whic
h
minim
iz
es
the
least-squares
expression
(R)
=
P
n
i=1
(v
i
Ru
i
R
y
)
2
m
ust
satisfy
n
X
i=1
v
i
^
(Ru
i
R
y
)
=
0:
(7.19)
This
is
in
tuitiv
ely
ob
vious
{
w
e
w
an
t
the
R
whic
h
mak
es
u
i
`most
parallel'
to
v
i
in
the
a
v
erage
sense.
The
big
adv
an
tage
of
the
approac
h
used
here
is
that
one
nev
er
lea
v
es
the
geometric
algebra
of
space,
and
the
resultan
t
biv
ector
is
ev
aluated
in
the
same
space,
rather
than
in
some
abstract
tangen
t
space
on
the
group
manifold.
The
solution
of
equation
(7.19)
for
R
will
utilize
the
linear
algebra
framew
ork
of
geometric
algebra
and
is
describ ed
in
the
follo
wing
section.
8
Linear
algebra
Geometric
algebra
is
a
v
ery
natural
framew
ork
for
the
study
of
linear
functions
and
non-
orthonormal
frames.
Here
w
e
will
giv
e
a
brief
accoun
t
of
ho
w
geometric
algebra
deals
with
linear
algebra;
w
e
do
this
since
man
y
computer
vision
and
engineering
problems
can
b e
form
ulated
as
problems
in
linear
algebra.
26
If
w
e
tak
e
a
linear
function
F(a
)
whic
h
maps
v
ectors
to
v
ectors
in
the
same
space
then
it
is
p ossible
to
extend
F
to
act
linearly
on
m
ultiv
ectors.
This
extension
of
F
is
giv
en
b
y
F(a
1
^
a
2
^
:
:
:
^
a
r
)
=
F(a
1
)
^
F(a
2
)
^
:
:
:
^
F(a
r
):
(8.1)
The
extended
function
preserv
es
grade
since
F
maps
an
r -grade
m
ultiv
ec
tor
to
another
r -grade
m
ultiv
ec
tor.
The
adjoint
to
F
is
written
as
F
and
de ned
b
y
F(a
)
=
e
i
hF(e
i
)ai;
(8.2)
where,
as
b efore,
fe
i
g
is
an
arbitrary
frame
and
fe
i
g
is
its
recipro cal
frame.
This
de nition
ensures
that
a
F(b
)
=
b
F
(a
);
(8.3)
so
the
adjoin
t
represen
ts
the
function
corresp onding
to
the
transp ose
of
the
matrix
whic
h
is
represen
ted
b
y
F.
If
F
=
F
the
function
is
said
to
b e
self-adjoint,
or
symmetric.
Symme
tric
functions
satisfy
e
i
^
F(e
i
)
=
0;
(8.4)
and
this
ensures
that
the
an
y
matrix
represen
ting
F
is
symmetri
c.
Similarly
,
if
F
=
F
then
the
function
is
an
tisymme
tric.
As
an
illustration
of
the
use
of
linear
algebra
tec
hniques,
w
e
will
discuss
the
solution
of
equation
(7.19).
W
e
rst
re-write
the
equation
as
e
i
^
"
n
X
i=1
R
(e
i
v
i
)u
i
R
y
#
=
0:
(8.5)
W
e
no
w
in
tro duce
a
function
F
de ned
b
y
F(a)
=
n
X
i=1
(a
v
i
)u
i
:
(8.6)
Equation
(8.5)
can
then
b e
written
as
e
i
^
RF(e
i
)R
y
=
0:
(8.7)
Let
us
no
w
de ne
another
function
R
mapping
v
ectors
on
to
v
ectors
suc
h
that
R(a)
=
RaR
y
.
With
these
de nitions
equation
(8.7)
tak
es
the
form
e
i
^
RF(e
i
)
=
0;
(8.8)
27
whic
h
tells
us
that
RF
is
symmet
ric.
W
e
no
w
p erform
a
singular-v
alue
decomp osition
(SVD)
on
F,
whic
h
enables
us
to
write
F
=
SD
(8.9)
where
S
is
an
orthogonal
transformation
and
D
is
symmetri
c.
Comparing
with
(8.8)
w
e
see
that
a
solution
is
pro
vided
b
y
R
=
S
1
=
S:
(8.10)
The
rotation
R
(and
hence
the
rotor
R)
is
therefore
found
directly
from
the
SVD
of
the
function
F.
9
Elasticit
y
The
sub
ject
of
the
b eha
viour
of
solids
under
applied
stress
is
one
of
the
oldest
in
ph
ysics.
Despite
its
great
history
,
the
sub
ject
is
still
rapidly
ev
olving,
driv
en
b
y
adv
ances
in
engineering,
and
the
adv
en
t
of
new
materials
with
un
usual
prop erties.
Here
w
e
review
ho
w
the
com
bination
of
linear
algebra
and
geometric
calculus
is
applied
to
the
sub
ject
of
elasticit
y
.
W
e
sho
w
ho
w
arbitrary
,
nonlinear
strains
can
b e
handled,
b efore
reducing
to
the
simpler
linearised
theory
.
W
e
also
lo ok
at
the
b eha
viour
of
an
elastic
lamen
t,
whic
h
is
the
simplest
system
to
extend
to
the
nonlinear
regime.
9.1
The
Displacemen
t
Field
The
cen
tral
idea
in
treating
elastic
deformations
is
essen
tially
the
same
as
that
used
in
rigid
b o dy
dynamics.
W
e
imagine
an
undeformed,
reference
con guration
and
denote
a
p osition
in
this
with
the
v
ector
x.
Eac
h
p oin
t
in
the
reference
con guration
maps
to
a
p oin
t
y
in
the
ph
ysical
con guration.
The
map
b et
w
een
these
is
a
function
of
p osition
and
time,
whic
h
w
e
write
as
y
=
f
(x;
t):
(9.1)
(See
Figure
7.)
No
w
consider
t
w
o
p oin
ts
x
and
x
+
a
,
close
together
in
the
reference
con guration.
The
distance
b et
w
een
these
is
jx
+
a
xj
=
ja
j:
(9.2)
The
images
of
these
t
w
o
p oin
ts
in
space
are,
dropping
the
time
dep endence,
f
(x)
and
f
(x
+
a
).
The
v
ector
b et
w
een
these
is,
to
rst
order
in
,
f
(x
+
a
)
f
(x)
=
a
r
f
(x):
(9.3)
28
x
x
+
a
f
(x)
f
(x
+
a
)
Figure
7:
A
n
Elastic
Deformation.
The
nonlinear
function
f
(x)
maps
a
p oin
t
in
the
reference
con guration
to
a
p oin
t
in
space.
The
directional
deriv
ativ
es
of
f
(x)
tell
us
ab out
the
strains
in
the
material.
The
directional
deriv
ativ
es
of
f
(x;
t)
therefore
con
tain
information
ab out
the
lo cal
distortion
of
the
material.
This
information
is
summarised
in
the
linear
function
f
(a
)
=
f
(a;
x
;
t)
=
a
r
f
(x;
t):
(9.4)
This
is
a
time-dep enden
t
linear
function
of
a
,
de ned
for
eac
h
p oin
t
x
in
the
reference
con guration.
One
w
a
y
to
think
of
the
function
f
(a)
is
as
follo
ws:
supp ose
that
the
material
is
lled
with
a
series
of
curv
es
(these
could
b e
realised
ph
ysically
using
dy
es
in
the
formation
pro cess,
lik
e
in
a
m
ulticoloured
eraser).
If
the
tangen
t
v
ector
to
one
of
these
curv
es
in
the
undistorted
medium
is
giv
en
b
y
the
v
ector
a
then,
after
the
distortion,
this
v
ector
transforms
to
f
(a).
The
distance
b et
w
een
the
images
of
the
p oin
ts
x
and
x
+
a
is
no
w
jf
(a
)j
=
p
(f
(a)
2
):
(9.5)
The
f
(a)
2
term
can
b e
written
as
f
(a)
2
=
hf
(a)f
(a)i
=
ha
f
f
(a)i
=
a
f
f
(a);
(9.6)
where
f
is
the
adjoin
t
(transp ose)
of
the
linear
function
f
.
The
function
g
=
f
f
is
therefore
resp onsible
for
the
c
hange
in
distance
b et
w
een
p oin
ts
in
the
undistorted
and
distorted
medium
.
This
m
ust
therefore
b e
directly
related
to
the
str
ain
tensor
for
the
solid.
The
strain
in
the
undistorted
medium
should
b e
zero,
whic
h
corresp onds
to
f
b eing
the
iden
tit
y
.
A
suitable
de nition
for
the
strain
tensor
E
(a)
is
therefore
E
(a)
=
1
2
(g (a)
a
):
(9.7)
29
The
factor
of
1=2
is
included
so
that
the
strain
tensor
has
the
correct
linearisation
prop erties.
An
alternativ
e
de nition
of
the
strain
tensor,
whic
h
has
a
n
um
b er
of
features
to
re-
commend
it,
is
E
(a)
=
1
2
ln
g (a):
(9.8)
T
o
date,
this
alternativ
e
de nition
has
not
b een
seriously
considered.
One
adv
an
tage
of
this
c
hoice
is
that
T
r(E
)
=
ln
(det
f
)
(9.9)
so
the
trace
of
the
strain
tensor
is
directly
related
to
the
v
olume
scale
factor.
Whic
h
of
the
p ossible
de nitions
of
the
strain
tensor
should
b e
used
can
ultimately
only
b e
settled
b
y
the
accuracy
of
the
predictions
of
mo dels
based
on
the
di eren
t
c
hoices.
The
strain
tensor
is
symmetric
and
tells
us
ab out
the
strains
in
the
distorted
b o dy
.
The
function
E
(a)
tak
es
as
its
argumen
t
a
v
ector
in
the
xed,
undistorted
medium,
and
it
returns
a
v
ector
in
the
same
medium.
As
with
rigid
b o dy
dynamics,
this
turns
out
to
b e
the
easiest
w
a
y
to
w
ork.
The
function
g (a
)
is
symmetri
c,
whic
h
ensures
that
an
y
o
v
erall
rotational
comp onen
t
of
the
strain
has
b een
factored
out.
g (a
)
is
also
p ositiv
e
de nite,
so
its
prop erties
are
most
naturally
discussed
in
terms
of
its
eigen
v
ectors.
These
are
the
directions
in
the
reference
b o dy
whic
h
are
stretc
hed,
but
not
rotated,
for
a
giv
en
strain.
9.2
Stress
and
the
Balance
Equations
The
con
tact
force
b et
w
een
t
w
o
surfaces
in
the
medium
is
a
function
of
the
normal
to
the
surface
(and
p osition
and
time).
Cauc
h
y
sho
w
ed
that,
giv
en
suitable
con
tin
uit
y
conditions,
the
force
m
ust
b e
a
linear
function
of
the
normal.
W
e
write
this
as
T
(n
).
Since
T
(
n
)
=
T
(n
)
it
follo
ws
that
Newton's
third
la
w
is
immedi
ately
satis ed.
The
function
T
(n
)
tak
es
as
its
argumen
t
a
v
ector
in
the
reference
con guration,
and
returns
a
v
ector
in
the
material
b o dy
(see
Fig.
8).
W
e
need
a
means
to
pull
this
bac
k
to
the
reference
con guration,
so
that
the
stress
can
b e
related
to
the
strain
there.
T
o
see
ho
w
to
do
this
w
e
need
to
consider
the
balance
equations
(i.e.
force
la
ws)
in
the
material
b o dy
.
The
total
force
on
an
elemen
t
of
v
olume
V
is
found
b
y
in
tegrating
T
(n
)
o
v
er
the
surface
of
the
elemen
t,
so
w
e
ha
v
e
Z
@
2
y
@
t
2
dV
=
I
T
(ds);
(9.10)
30
n
T
(n
)
Figure
8:
The
Str
ess.
The
stress
tensor
T
(n
)
returns
the
force
in
the
material
b o dy
o
v
er
the
plane
with
normal
n,
in
the
reference
b o dy
.
where
=
(x)
is
the
densit
y
in
the
undistorted
medium.
A
simple
application
of
the
div
ergence
theorem
con
v
erts
the
surface
in
tegral
to
a
v
olume
in
tegral,
I
T
(ds)
=
Z
T
(
r
)
dV
(9.11)
from
whic
h
w
e
can
read
o
the
dynamical
equation
@
2
y
@
t
2
=
T
(
r
):
(9.12)
The
second
balance
equation
is
found
b
y
considering
the
total
couple
on
a
v
olume
elemen
t,
and
relating
this
to
the
c
hange
in
angular
momen
tum
.
The
total
couple
ab out
the
p oin
t
y
0
is
M
=
I
(y
y
0
)
^
T
(ds):
(9.13)
This
m
ust
b e
equated
with
the
c
hange
in
angular
momen
tum
,
@
@
t
Z
(y
y
0
)
^
_
y
dV
=
Z
(y
y
0
)
^
T
(
r
)
dV
:
(9.14)
Applying
the
div
ergence
theorem
again,
w
e
nd
that
angular
momen
tum
balance
is
satis ed
pro
vided
@
i
y
^
T
(e
i
)
=
f
(e
i
)
^
T
(e
i
)
=
0:
(9.15)
It
follo
ws
that
the
tensor
T
(n
)
is
symmetric,
where
T
(n)
=
f
1
T
(n
):
(9.16)
This
is
the
( rst)
Piola-Kirc
ho
stress
tensor.
T
(n)
is
a
symmetri
c
tensor
de ned
en
tirely
in
the
reference
con guration,
since
the
f
1
term
maps
the
v
ector
T
(n
)
bac
k
to
the
reference
cop
y
.
The
Piola-Kirc
ho
tensor
is
the
one
that
w
e
m
ust
relate
to
the
strain
tensor,
via
the
constitutiv
e
relations
of
the
material.
31
9.3
Constitutiv
e
Equations
The
strain
tensor
can,
in
principle,
b e
a
quite
general
function
of
the
applied
stresses.
Complications
can
include
a
lac
k
of
homogeneit
y
and
isotrop
y
,
viscosit
y
,
thermal
and
c
hemical
prop erties,
and
a
dep endence
on
the
history
of
the
b o dy
.
F
or
a
wide
range
of
applications,
ho
w
ev
er,
w
e
can
restrict
to
the
simplest
case
of
a
line
ar,
isotr
opic,
homo
gene
ous
(LIH)
b o dy
.
In
these
the
stresses
and
strains
are
related
linearly
b
y
just
t
w
o
parameters,
the
bulk
mo dulus
B
and
the
shear
mo dulus
G.
F
or
LIH
media
the
relation
b et
w
een
the
applied
stress
T
(a
)
and
the
strain
E
(a)
is
T
(a
)
=
2GE
(a )
+
(B
2
3
G)T
r(E
)a :
(9.17)
The
bulk
mo dulus
B
describ es
ho
w
the
b o dy
resp onds
to
isotropic
pressure,
as
is
the
case
when
the
b o dy
is
imm
ersed
in
a
liquid.
The
applied
stress
is
then
a
uniform
pressure
P
in
all
directions,
so
w
e
ha
v
e
T
(a)
=
P
a:
(9.18)
The
sign
is
negativ
e,
b ecause
the
force
is
a
compression,
rather
than
a
stretc
h.
T
aking
the
trace
of
b oth
sides
of
equation
(9.17)
giv
es
3P
=
3B
T
r(E
):
(9.19)
The
distortion
in
the
medium
will
b e
giv
en
b
y
f
(x)
=
x
+
x
0
;
(9.20)
where
x
0
is
the
v
ector
from
the
origin
to
its
image
in
the
ph
ysical
con guration,
and
is
the
scale
factor.
It
follo
ws
that
f
(a)
=
a;
(9.21)
and
hence
T
r(E
)
=
3(
2
1)=2.
The
bulk
mo dulus
is
then
giv
en
b
y
B
=
2P
3(
2
1)
:
(9.22)
If
w
e
no
w
linearise
b
y
setting
=
1
+
,
w
e
reco
v
er
the
familiar
result
that
B
=
P
3
=
P
V
V
(9.23)
where
V
is
the
v
olume.
Since
the
force
is
a
compression,
the
c
hange
in
v
olume
V
will
b e
negativ
e.
When
a
stress
is
applied
along
a
single
direction,
the
b o dy
will
resp ond
b
y
stretc
hing
along
the
direction
of
the
applied
force,
and
con
tracting
in
the
other
t
w
o
directions.
The
relativ
e
sizes
of
these
e ects
is
con
trolled
b
y
the
shear
mo dulus
G
|
the
second
of
the
t
w
o
main
elastic
parameters.
Giv
en
a
set
of
constitutiv
e
relations
and
the
balance
equations,
one
has
enough
information
to
compute
the
ev
olution
of
the
system.
The
resulting
equations
are,
in
general,
highly
complicated
and
nonlinear,
ev
en
if
the
material
itself
is
linear.
F
or
this
reason
it
is
usual
to
w
ork
in
the
linear
regime
of
small
deformations
whenev
er
p ossible.
32
9.4
Linearised
Elasticit
y
Supp ose
that
the
elastic
deformation
can
b e
written
as
x
0
=
f
(x )
=
x
+
x
0
+
(9.24)
where
is
a
v
ector
eld.
The
directional
deriv
ativ
es
of
this
are
denoted
with
an
underbar,
so
(a)
=
a
r
:
(9.25)
W
orking
to
rst
order,
the
strain
tensor
b ecomes
E
(a)
=
1
2
(a)
+
(a)
:
(9.26)
The
stress
tensor
T
(a
)
giv
es
the
applied
force
o
v
er
the
surface
p erp endicular
to
a
in
the
reference
con guration.
In
the
linearised
theory
,
this
is
the
same
as
the
force
in
the
material
b o dy
(to
rst
order).
Assuming
an
LIH
material,
w
e
then
reco
v
er
the
dynamical
equations
Gr
2
+
(B
+
1
3
G)r
r
=
@
2
@
t
2
:
(9.27)
F
or
man
y
applications
w
e
assume
a
harmonic
time
v
ariation
cos(!
t),
for
whic
h
w
e
reco
v
er
the
ve
ctor
Helmholtz
e
quation,
v
2
l
r
r
+
v
2
t
r
(r
^
)
+
!
2
=
0:
(9.28)
Here
the
equation
has
b een
expressed
in
terms
of
longitudinal
and
transv
erse
sound
sp eeds
v
l
and
v
t
,
giv
en
b
y
v
2
l
=
B
+
4
3
G
;
v
2
t
=
G
:
(9.29)
The
v
ector
Helmholtz
equation
is
used
to
study
man
y
phenomena,
ranging
from
oscil-
lations
of
an
elastic
sphere
to
the
propagation
of
w
a
v
es
created
b
y
an
earthquak
e.
9.5
The
Elastic
Filamen
t
W
e
no
w
treat
the
b ending
and
t
wisting
of
an
elastic
lamen
t
under
static
loads.
Sup-
p ose
that
the
lamen
t
is
describ ed
b
y
the
curv
e
x().
W
e
will
c
ho ose
to
b e
the
aÆne
parameter
along
the
curv
e,
so
that
x
0
=
@
x
is
a
unit
v
ector.
This
v
ector
can
b e
iden
ti ed
with
the
third
v
ector
of
an
orthonormal
frame,
x
0
=
f
3
=
R()e
3
R
y
():
(9.30)
33
The
remaining
t
w
o
v
ectors
then
determine
t
w
o
directions
p erp endicular
to
the
lamen
t,
and
can
b e
used
to
describ e
an
y
in
ternal
t
wisting
in
the
lamen
t.
With
this
approac
h,
b oth
the
b ending
and
t
wisting
of
the
lamen
t
are
describ ed
in
the
single
equation
for
the
rotor
R.
A
thin
b eam
or
lamen
t
has
sti ness
to
b ending.
When
it
is
b en
t,
a
b ending
momen
t
(couple)
is
set
up
whic
h
is
linearly
related
to
the
curv
ature.
In
terms
of
the
t
w
o
principal
directions
in
the
lamen
t,
the
appropriate
form
ula
for
the
b ending
momen
t
is
M
=
Y
I
R
;
(9.31)
where
Y
is
Y
oung's
mo dulus,
I
is
the
relev
an
t
principal
momen
t
of
area,
and
R
is
the
radius
of
curv
ature
in
the
plane
of
the
b ending.
The
radius
of
curv
ature
is
determined
b
y
the
magnitude
of
the
pro
jection
of
the
v
ector
f
0
3
in
to
the
relev
an
t
plane.
So
the
radius
of
curv
ature
in
the
f
1
f
3
plane,
for
example,
is
giv
en
b
y
1
R
1
=
jf
0
3
(f
1
f
3
)f
3
f
1
j
=
jf
1
f
0
3
j:
(9.32)
T
o
compute
f
0
3
w
e
rst
need
to
establish
an
imp ortan
t
result
for
the
deriv
ativ
e
of
a
rotor.
Rotors
are
normalised
to
unit
y
,
so
RR
y
=
1.
Di eren
tiating
this,
w
e
obtain
R
0
R
y
+
RR
y
0
=
0:
(9.33)
It
follo
ws
that
R
0
R
y
=
RR
y
0
=
(R
0
R
y
)
y
:
(9.34)
The
quan
tit
y
R
0
R
y
is
therefore
equal
to
min
us
its
rev
erse
(and
has
ev
en
grade)
so
m
ust
b e
a
pure
biv
ector.
W
e
set
this
equal
to
=2,
so
that
R
0
=
1
2
R
=
1
2
R
B
;
(9.35)
where
B
=
R
y
R.
It
follo
ws
that
f
0
3
=
R
0
e
3
R
y
+
Re
3
R
y
0
=
1
2
(
f
3
+
f
3
)
=
f
3
;
(9.36)
so
the
radius
of
curv
ature
just
pic
ks
out
one
co eÆcien
t
of
.
Equation
(9.31)
can
corresp ondingly
b e
used
to
nd
the
curv
ature
induced
b
y
an
ap-
plied
couple
C
.
With
C
and
giv
en
in
terms
of
comp onen
ts
b
y
C
=
X
k
c
k
I
f
k
;
=
X
k
!
k
I
f
k
;
(9.37)
34
w
e
nd
that
the
curv
ature
and
the
couple
are
related
b
y
c
1
=
Y
i
1
!
1
;
c
2
=
Y
i
2
!
2
;
(9.38)
where
i
1
is
the
momen
t
of
area
measured
p erp endicular
to
the
f
1
direction.
In
addition
to
its
sti ness
to
b ending,
the
lamen
t
has
a
sti ness
to
torsion.
F
or
the
case
of
elastic
b eha
viour,
the
t
wist
in
the
f
1
f
2
plane
is
prop ortional
to
the
applied
torque,
and
w
e
ha
v
e
c
3
=
Gi
3
f
0
1
f
2
=
Gi
3
!
3
:
(9.39)
The
applied
couple
C
and
the
`curv
ature
biv
ectors'
and
B
are
therefore
related
b
y
C
=
Y
(i
1
!
1
I
f
1
+
i
2
!
2
I
f
2
)
+
Gi
3
!
3
I
f
3
=
RI
(
B
)R
y
;
(9.40)
whic
h
de nes
the
linear
function
I
(whic
h
maps
biv
ectors
to
biv
ectors).
W
e
can
in
v
ert
this
relation
to
giv
e
B
=
I
1
(R
y
C
R);
(9.41)
whic
h
expresses
the
curv
ature
biv
ector
B
in
terms
of
the
applied
couple
C
and
the
elastic
constan
ts.
The
full
set
of
equations
are
no
w
(9.30)
and
(9.41),
together
with
the
rotor
equation
dR
d
=
1
2
R(
B
+
0
);
(9.42)
where
the
biv
ector
0
expresses
the
natural
shap e
of
the
lamen
t.
An
adv
an
tage
of
this
set
of
equations
is
that
lo cally
small
distortions
of
the
lamen
t
can
b e
allo
w
ed
to
build
up
in
to
large,
global
deviations.
An
in
teresting
simple
case
is
that
of
a
wr
ench,
where
C
(x
)
=
C
0
+
f
^
x;
(9.43)
where
C
0
and
f
are
resp ectiv
ely
the
couple
and
force
applied
at
the
ends.
A
wrenc
h
suc
h
as
this
describ es
the
general
case
of
a
ligh
t
lamen
t
loaded
at
its
ends.
Figure
9
sho
ws
the
t
yp e
of
distortion
that
can
result.
35
Figure
9:
A
lament
lo
ade
d
at
its
two
ends.
Tw
o
directions
are
sho
wn,
though
there
is
also
considerable
structure
in
the
third.
The
material
has
i
1
=
i
2
and
a
zero
P
oisson
ratio.
10
Relev
an
t
P
ap
ers
The
follo
wing
list
of
pap ers,
courses,
and
notes
discuss
in
detail
some
of
the
applications
outlined
in
the
talks.
C.J.L.
Doran
and
A.N.
Lasen
b
y
,
Lecture
Notes
to
accompan
y
4th
y
ear
undergraduate
course
on
Physic
al
Applic
ations
of
Ge
ometric
A
lgebr
a,
2000.
Av
ailable
at
http://www.
mra
o.
cam
.ac
.u
k/
cli
ffo
rd/
pt
III
cou
rse
/.
Da
vid
Hestenes,
New
F
oundations
for
Classic
al
Me
chanics
(Second
Edition).
Published
b
y
Klu
w
er
Academic.
J.
Lasen
b
y
and
A.
Stev
enson.
Using
geometric
algebra
in
optical
motion
capture.
In
E.
Ba
yro
and
G.
Sob czyk,
editors,
Ge
ometric
algebr
a:
A
ge
ometric
appr
o
ach
to
c
omputer
vision,
neur
al
and
quantum
c
omputing,
r
ob
otics
and
engine
ering.
L.
Dorst,
S.
Mann
and
T.
Bouma.
GABLE:
A
Matlab
T
utorial
for
Ge
ometric
A
lgebr
a.
Av
ailable
at
www.carol.w
ins
.u
va.
nl/
le
o/
cli
ffo
rd/
ga
ble
bet
a.h
tm
l
36
C.J.L.
Doran.
Ba
y
esian
inference
and
geometric
algebra:
an
application
to
camera
lo calization.
In
E.
Ba
yro
and
G.
Sob czyk,
editors,
Ge
ometric
algebr
a:
A
ge
ometric
appr
o
ach
to
c
omputer
vision,
neur
al
and
quantum
c
omputing,
r
ob
otics
and
engine
ering.
J.
Lasen
b
y
,
W.J.
Fitzgerald,
A.N.
Lasen
b
y
and
C.J.L.
Doran.
New
geometric
metho ds
for
computer
vision
{
an
application
to
structure
and
motion
estimation.
International
Journal
of
Computer
Vision,
26(3),
191-213.
1998.
J.
Clemen
ts.
1999
Be
am
buckling
using
ge
ometric
algebr
a.
M.Eng.
pro
ject
rep ort,
Cam
bridge
Univ
ersit
y
Engineering
Departmen
t.
L.
Dorst.
Honing
geometric
algebra
for
its
use
in
the
computer
sciences.
In
G.
Sommer,
editor,
Ge
ometric
Computing
with
Cli or
d
A
lgebr
as.
Springer.
M.
Ringer
and
J.
Lasen
b
y
,
2000
Mo
del
ling
and
tr
acking
articulate
d
motion
fr
om
multiple
c
amer
a
views.
Cam
bridge
Univ
ersit
y
Engineering
Departmen
t
Rep ort
CUED/F-INFENG/-TR.378.
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