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7/26/2019 Coxeter Quaternions and Reflections AMM 1946
1/12
Quaternions and ReflectionsAuthor(s): H. S. M. CoxeterSource: The American Mathematical Monthly, Vol. 53, No. 3 (Mar., 1946), pp. 136-146Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2304897
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2/12
QUATERNIONS AND REFLECTIONS*
H. S. M.
COXETER,
UniversityfToronto
1.
Introduction. t is just a hundred years since Cayley began to use qua-
ternions for the
discussion of rotations. He was followed by Boole,
Donkin,
Clifford, uchheim, Klein, Hurwitz, Hathaway, Stringham, nd Study. Appar-
ently
none of
these men thought of considering
first he simpler operation of
reflection
nd
deducing a rotation as the product of two reflections.This pro-
cedure
will be
described
in
??3
and
5, and
its
consequences developed
in
the
later sections.
Every quaternion
a
=ao+a1i+aZj+a3k
determines point
P.
=
(ao,
a,,
a2,
a3)
in euclidean4-space, and a hyperplane
oxo+a1xl+a2x2+a3x3
=0. The reflection
in that hyperplane s found
to
be the
transformation ---axa/Na. This leads
easily
to the
classical
expression
x
-
axb
(Na
=
Nb
=
1)
for
the
generaldisplacementpreserving
he
origin.
Cayley
obtained
this
elegant
expressionby
brute force
s
early
as
1855. It
became
somewhat
more natural
in the
hands
of
Klein and
Hurwitz, fortyyears later.
The treatment
n
?7
will
possibly
serve
to
clarify
t still
further.
We
begin
with
a few
lgebraic lemmas,
mostly
due to
Hamilton.
2.
Elementary properties
of
quaternions. A
quaternion
s a hyper-complex
number
a
=ao+a1i+a2j+a3k, where
ao, al,
a2,
a3
are
real
numbers
scalars ),
and multiplication s definedby the rules
i2=
j2 =
k2= ijk
=-1,
which
imply
k
=
=i-kj,
ki
=j =-ik,
ij=k
=-Ji.
Thus
quaternions
form n
associative
but
non-commutative lgebra.
It is
oftenconvenient to split a quaternion
into its scalar and vector
partst
a
=
Sa
+ Va, Sa
=
ao,
Va
=
ari
+
a2j + a3k.
We define lso thecozjugatequaterniont
d
=
Sa- Va
=
ao
-
ali
-
a2j
-
a3k
and
the norm
Na
=
da
=
ad
=
a2+ a2+
a2+
a3
*
This
paper
s
an amplificationf an invited
ddress elivered t the
annual meeting f the
Mathematical ssociation f
American Chicago nNovember 4, 1945.
t
W. R.
Hamilton,
lements f
Quaternions,
ol.
1, London, 899, p.
177,
193.
By
his
pecial
convention,. 186,Sxy meansS(xy), not (Sx)y;similarly orVxyand Nxy.
t
Klein's
symbol
seems
preferable
o
Hamilton's
a.
136
7/26/2019 Coxeter Quaternions and Reflections AMM 1946
3/12
QUATERNIONS
AND
REFLECTIONS
137
In
terms
of a and d, we have Sa=2(a+d),
Va=
2(a-a).
Since I-1-
and
i=-k
=f, etc.,we
easily verify
hat
ab=
whenceNab =aab
=
baab
=
b(Na)b
=
NaNb. If Na
=
1, we call a a unit quater-
nion. To
every non-vanishing
uaterniona there
corresponds unit quaternion*
Ua
=
a/N.
A
quaternionp
is said
to be
pure
if
Sp=O. Then
P=
-p and
p2 -Np.
Thus a
pure
unit quaternion s a square
root of -1.
The
point
P.
=
(X1, X2, X3)
in
ordinary pace
may be represented y
the
pure
quaternion
x =x1i+x2j+x3k.
This representation
esembles he Argand diagram
in
the
plane,
in that the distance
PXP,
is
givenby
pxpP2
=
N(y -x). Since
xy=-(xIyI+X2y2+x3y3)+
X2 X3:
X3
XI
j+
Xl
X212
Y2 Y3
Y3
Yi
yl
Y2
we see that -Sxy
and
Vxy
are the ordinary
scalar
product
and 'vector
product
of the two vectors
PoPe,
nd
PoP11.
fx and y are pure
unit
quaternions,
then
Px
and
P,
lie on the unit
sphere around the origin
Po,
and
we
have
Z
PxPoPv
=
O, where
cos =-
Sxy= -(xy
+
yx).
Thus the conditionforP. and P, to lie in perpendiculardirectionsfromPo is
xy + yx
=
0.
LEMMA
2.1.
For any quaternion
we
can
find
a unit
quaterniony
such that
ay
=yy.
Proof.
Take any purequaternion
p
forwhich
PoP,
is
perpendicular
o
PoPva.t
Then
(Va)p
+ p(Va)
=
0.
But since the scalar Sa commuteswithp,
(Sa)p
-
p(Sa)
=
0.
By
addition,
ap
-
p=
0.
The
desired
unit
quaternion
is
y
=
Up.
* Hamilton,
p. cit.,
p. 137.
t
If
an
explicit
ormula
s
desired,
we
may
write
=
V(Va)q,
where
is
anypure
quaternion
(other hanthescalarmultiplesfVa). In fact, fp=Vaj, where =Va, then2p=ag-qa, and
2(ap + pa)
=
a(a
-
qa)
+
(aq
-
ga)
=
a2q
qa2
=
0,
a2
being
calar.
Thus
our ppeal to geometry
ouldhave
been voided.
7/26/2019 Coxeter Quaternions and Reflections AMM 1946
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138
QTJATERNIONS
AND
REFLECTIONS
[March,
LEMMA
2.2. Let a
and b be two
quaternions
aving he
ame
norm
nd
the
ame
scalar part.*Then we
can
find a unit
quaternion such that y
=yb.
Proof. If
b
=
d,
this is
covered by
Lemma 2.1;
so let us assume
b
#a
.
Since
Sa=Sb, we have
a+d=b+?i,
a-7b
=
b-a,
a(a
-
)b
=
a(b
-)b,
a(ab -
Nb) = (ab -
Na)b.
Thus
ac-cb, where
c=ab-Nb=ab-Na=a(b-a).
Since bzA,
c0;
so we
can
take
y
=
Uc.
LEMMA 2.3. Any
quaternion s
expressible s
a
power of
a
pure
quaternion.f
Proof.Since Npt= (Np) t, it will be sufficiento provethis for unitquater-
nion,a.
Since (Sa)2
-
(Va)2
=
Na =1,
such a
quaternion
may
be
expressed
as
a
=
cos
a
+
p
sin
a,
where
cos
a=
Sa and p is a
pure unit
quaternion.
Since Va
=
p
sin
a,
we
have
p
=
UVa.
Since
p2
-1,
de
Moivre's Theorem
shows
that
an
=
cos
na + p sin na
for
ny real
numbern. In
particular,
wr2a =p, so a
=p2ot/'.
Thus
a= pty
where
p
UVa
and cos 'tr
=
Sa (so
that we may
suppose
0?
t 2).
3.
Reflections
nd rotations n
three
dimensions.We
have seen that
P.
and
P,,
lie in
perpendicular
directions
from the origin
PO
if
the pure quaternions
x
and
y
satisfy he relation
xy
+
yx
=
0.
If
Ny
=
1,
so that y2
-1,
this
condition
may be expressed
as
x
=
yxy.
Since
yxy=
yy
=
-yxy, yxy
is pure
for
any position of
P,.
Thus
the linear
transformation
-->yxywhere Ny
=
1) represents
collineationwhich eaves
in-
variant
every
point
P.
in the plane
through
Po
perpendicular o
PoPV,
.e.,
the
plane
ylXl
+
Y2X2
+
y3X3
=
0.
Moreover,
t
reverses he
vector
POP,,:
y
-+y3=
y.
*
Two such quaternionssatisfy the same rank equation
x2
2mx+n
=
0, where m
=
Sa
=
Sb
and n
Na
=
Nb. For
a
general
iscussion f the
equation y
yb, ee
Arthur ayley,
On
the
quaternionquation
Q-Qq'=0,
Messenger f
Mathematics,
ol. 14,1885,
pp.
108-112.
t
Hamilton,
.
399.
7/26/2019 Coxeter Quaternions and Reflections AMM 1946
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1946]
QUATERNIONS AND REFLECTIONS 139
But the only collineationhaving these
properties s the
reflection
n the plane
of
invariant points. Hence
THEOREM
3.1. The
reflection
n the
plane
EY,yx=O
is represented y the rans-
formation
x
-yxy (Ny
=
1).
The product of two such reflections,
-*yxy and x--zxz (with Ny
=
Nz =1),
is the transformation
x
-*
zyxyz.
Accordingly,
his is a
rotation,
n
the plane
PoP11Pz,
through twice the angle
P,PoP2.
In other words, t is a rotation through
5
bout the line
PoPvv,z
where
Cos
2q
= -
Syz.
By Lemma 2.1, any unit quaternion a may be expressed as -zy, where
ay =yd=z. (Here z, like y,
is
a
pure
unit
quaternion; for,
z + -=ay-yd
=
O, and Nz
=
NaNy
=
1.)
Since =-yz, this shows that the
transformation
x
--+
xa
(Na= 1)
is
a
rotation through
$
bout
PoPva,
where cos 5=
Sa.
In
the notation of the proofof Lemma
2.3, we have a =cos ao+p sin a, where
PoPp is the unit vector along the axis of rotation, and a
=
+
24.
The sign is a
matter of convention. Taking the
plus, we find that the rotation through
w
about the x3-axis
POPk
ransforms
i
into
Pi:
1
+
k
1-k
(i + j)(1-k)
\/
/2 2
(The opposite convention would
have given i-
>-j.)
We
sum
up
in
THEOREM 3.2.* The rotation hrough
5
about the ine with direction osines
(Pl, P2,
p3)
is represented ythe ransformation->axd, where
a
cos
21
+
(pli
+
p2j
+ p3k) sin
2q.
Since the productof two rotations, -*axd and x--*bxb,
s
anotherrotation,
viz.,
x->baxba,
we can immediatelydeduce
THEOREM 3.3. All
therotations
bout
ines through
he
origin
n
ordinary
pace
form group,homomorphicothegroupof all
unit
quaternions.
Since the rotation through
q
is indistinguishable rom he rotation through
q+27r
about the same
axis,
there are
two
quaternions,
?a,
for each rotation.
*
Arthur ayley,On certain
results elating o quaterniohs, hilosophicalMagazine (3),
vol.
26, 1845, . 142.George oole,Noteson quaternions,bid., ol. 33, 1848,
. 279.W.
F.
Donkin,
On
the
geometricalheory
f
rotation,
bid.
4),
vol.
1, 1851,p.
189,
7/26/2019 Coxeter Quaternions and Reflections AMM 1946
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140
QUATERNIONS
AND REFLECTIONS
[March,
Alternatively,
we may say
that the group of
rotations
is isomorphic o the
group
of all homogeneous
quaternions, n accordance
with
the formula
x
-
ax4/Na,
or
x -+ axa-1.
(A homogeneousquaternionis the class of all scalar multiples of an ordinary
quaternion.) To
make this
an
isomorphism
rather than
an
anti-isomorphism,
we mustagree to
multiply
group elements from
ight o left.
Combining
x->axd
with
the inversion - -x,
we obtain
the transformation
x --axCa
(Na=1)
whichmay be regarded
either
as a
rotatory-inversion
f angle
4
or
as a rotatory-
reflection
f angle
+ir.
4. The general displacement.
Two
orthogonal
trihedra
at
the same
origin
can be brought nto coincidence by the successiveapplication of the following
reflections:
ne to
interchange he two
xl-axes,
another to
interchange
he two
x2-axes
withoutdisturbing
he
xi-axis),
and
a
third
if
necessary) to reversethe
x3-axis.
Thus the
generalorthogonal
transformation
n
three
dimensions
s
the
product of
at most three
reflections-an
even
or odd number
according
as the
transformation
reserves
or
reversessense.
In
particular,
any displacementor
movement )
eaving
the
origin
fixedmust
be
the
product
of
only
two
reflections
t.e.,a
rotation. This
is
a famousresult,
due
to
Euler.)
Thus the group considered
in Theorem 3.3
is the group
of all
such displacements,
and
is
a
subgroup
of
index 2 in the groupofall orthogonaltransformations.he lattercontainsalso
the rotatory-reflections
--
-axJ,
which
are
products
of three
reflections.
Similarly
in
four
dimensions,
the
general
orthogonal
transformation
i.e.,
congruent ransformation
ith
a fixed
rigin)
s a
product
of at
most
fourreflec-
tions (in
hyperplanes).
Thus
the
general
displacement leaving
the
originfixed)
is the
product
of two
or four
reflections.
ut
the
product
of
two
reflectionss a
rotation (about
the common
plane
of
the two
hyperplanes, hrough
twice the
angle
between
them).
Hence
a
displacement
is
either
a
single
rotation
or the
product
of two
rotations
about distinct
planes..
f
any point
besides
the
origin
is invariant, the displacement can only be a single rotation; for it operates
essentially
n
the
3-space perpendicular
o the line of invariant
points.
In the
general
case
of
a
double
rotation,
where
only
the
origin
s
fixed,
t is
well
known
(though
far
from
obvious)
that
the
axial
planes
of the two rotations
may
be
chosen
to be
completelyorthogonal.
This wa's first
roved by
Goursat
in
1889.
We
shall
obtain
a new
proof
n
?9.
5.
Reflections
and
rotations
n
four
dimensions.
A
point
P.
in
euclidean
4-space
has
fourCartesian coordinates
xO,
Xl,
X2,
X3)
which
may
be
interpreted*
as
the constituents
f
a
quaternion
X
=
Xo + X1i +
x2j
+
x3k.
*
A. S.
Hathaway,
uaternions
s
numbers
f
four-dimensional
pace,
Bulletin f
heAmerican
Mathematical
ociety,
ol.
4,
1897, p.
54-57.
7/26/2019 Coxeter Quaternions and Reflections AMM 1946
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1946]
QUATERNIONS
AND REFLECTIONS
141
The distance
PXPV s given
by
PZPy
(yo
-
XO) + (Y1-X1) +
(Y2
- X2)
+
(Y3-X3) =
N(Y-X).
If x and y are unit quaternions,then P. and P, lie on the
unit
hypersphere
around the
origin
Po,
and we have Z
PPOP,P
=0,
where
cos 0
=
XOyo xlyl
+ x2y2
x3y3
Sx9 =
2(xY
+ yx).
Thus the condition
for
P.
and Pv to lie
in perpendicular
directionsfrom
Po
is
xy
+ yx
=
0.
If
Ny
= 1, this condition may be expressed as
x
= -
yxy.
Thus the linear transformation-* - yxy whereNy=1) represents collinea-
tion whichleaves
invariant
every point
P.
in the
hyperplane
through
Po
per-
pendicular
to
PoPS,
i.e.,
the hyperplane
YOXO
+
ylXl
+
Y2X2
+
Y3X3
=
0.
But it reverses
the vector
PoP,:
Y-
yyy
=-y.
Hence
THEOREM
5.1.
Thereflectionn thehyperplane
y,x, =0
is representedy he
transformation
x->-yxy
(Ny=
1).
The product
of *two such
reflections,
x->-yxy
and x-?-zxz
(with
Ny
=
Nz 1), is the transformation
x
z
yxy
z
=
zyxyz.
Accordingly,
his is a rotation,
n
the plane
PoP,P5,
through
twice the
angle
PvPoPx.
In other
words,
t is a rotation
aboutthe completely
orthogonal
plane
E
Y'x=
z
x =
O
through
ngle p,
where cos
=
Sy2
=
Szy
=
Syz.
This
proves
THEOREM 5.2.
Thegeneral
otation
hroughngle
5
about
plane)
s
x
->
axb,
where
Na=Nb= 1
and Sa=Sb=cos
j4.
Conversely,the
transformation
->axb
(where
Na
=
Nb
=1) is a
rotation
wheneverSa
=
Sb; forthen,by Lemma 2.2, we can findunit quaternionsy and z
such that ay= yb=z, enabling
us to write
a=
zy,
b
=
yz.
7/26/2019 Coxeter Quaternions and Reflections AMM 1946
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142
QUATERNIONS AND
REFLECTIONS
[March,
6.
Clifford
ranslations.The productof the two
rotationsx-*axa and
x->axa
is
the so-called
left
ranslation*
x
-+
a2xI
while the product of
x-+axa and x-*jxa is the right
ranslation
x
-
xa2.
Clearly, lefttranslationsx-?ax forma
group, so do
right translationsx->xb,
and any lefttranslation ommuteswith
any right ranslation.Every left
trans-
lation has a unique expressionx->ax
(Na
=
1);
for,the equation ax
=
bx
would
imply a=b. Similarly every right
translation has a unique expression
x->xb.
Hence
THEOREM 6.1.
The group of left or
right) translations
s
isomorphic
o
the
group of unitquaternions.
(In the case ofright ranslations,we
can make this a true somorphism,
nd
not merely n
anti-isomorphism, y
lettingthe quaternion b correspond o
the
translation
x->xb,
so
that the product
ab corresponds o x-*xab =xbd.)
Comparing
this
with
Theorem
3.3,
we
see that
the
group
of
left
(or right)
translations s
homomorphic o the group of rotations
about a fixed origin n
ordinary pace, with
two Clifford ranslationsforeach
rotation.
A
Clifford
ranslation (i.e., a left or right translation)
has
the remarkable
propertyof turning
every vector
through the same angle. For,
if
Nx=1,
so
that also Nax =1, the left translationx-*ax transforms . into Pa,, and
cos
ZPLPoP.,
=
Sax?
=
Sa,
which s the same forall vectors
POP..
Similarly,x-*xb transforms
.
into
P
b,
and
cos Z
P,,POPA
Sxxb
=
Sb.
Can
a left
translation
be
also
a
right
translation?
This
would make
ax
=xb
for
every x. The case x=1 gives a=b.
Now take x to be
the y of Lemma
2.1.
Then
ya= ay=yd,
y(a-2)
=
O,
Va=
O,
and so, since Na = 1, a
=
? 1. Thus the only lefttranslations hat are also right
translationst are the
identity -*x and
the inversion -+-x.
Instead of
dceriving heorem 5.1 fromthe condition for
LZPPoPv
to be
a
right ngle, we might
have observed that x-*ax (where
Na
=
1)
must
be some
kind
of
congruent
transformation since Nax
=
Nx),
and
that
this
transforms
the
special
reflection -+-x into the
general
reflection
x
-a4x
= - a2a.
*
Felix Klein,Vorlesungeniber icht-Euklidischeeometrie, erlin, 928,p. 240.
t
WilliamThrelfall nd Herbert eifert, opologischeUntersuchung
er
DiskontinuitIts-
bereiche ndlicher ewegungsgruppenes dreidimensionalenpharischen
aumes,
Mathematische
Annalen, ol. 104, 1931,p.
10.
7/26/2019 Coxeter Quaternions and Reflections AMM 1946
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19461
QUATERNIONS AND REFLECTIONS
143
7. The group of displacements. The product of a
lefttranslation nd a right
translation s, of course, x->axb
(where Na
=
Nb =1). The
product of two dis-
displacementsof this form s another of the same form.
n
particular,
s we saw
in Theorem 5.2, a rotation s of this form
with
the special
relation Sa
=
Sb).
But t'hegeneral displacementwith a fixed rigin s the productof two rotations.
Hence
THEOREM 7.1.* The general
displacement reserving heorigin s
x -axb
(Na=Nb=1).
Of
course,
axb
is the
same
as (-
a)x(-
b). With this exception, ach displace-
ment
has a unique expression
x->axb. For, the equation axb
=a'xb'
would imply
a'Ilax
-xb'b-l for veryx,
whence a'-'a
=b'b-1
=
?
1.
In
otherwords,every
dis-
placement (with a fixedorigin)
is the product of a left translation
nd a right
translation n
just
two
ways.
Thus
the direct
product
of
the
groups
of
all
left
translations
nd of all
right
ranslations
s
homomorphic
o
the
group
of
all four-
dimensionaldisplacementspreserving
he
origin with two elementsof
the direct
product foreach displacement),
and this in turn s homomorphic
o the direct
square of
the
group of all
three-dimensional isplacementspreserving
he origin
Xwith
wo
displacements
for
each
element of the direct
square).
8. The
general orthogonal ransformation
n
four
dimensions.
The
general
opposite or sense-reversing
ransformationeaving the origin
invariant
is
the product of an odd numberof reflections.Hence, in fourdimensions, t is
either a
single
reflection
r
a product of
three. But in the latter
case
the three
reflecting yperplanes
ntersect n a line of invariant points,
and every hyper-
plane perpendicular
to
this line
is
invariant;
so
this
scarcely
differs
rom
a
ro'tatory-reflectionn ordinary
space. As such, it has an axis
and a special re-
flecting
lane.
Its
product
with
the
special
reflection
-x-
-
is
a
displacement
x->axb; so
the
rotatory-reflection
tself must
be x- >-axb,
or,
after
changing
the sign of a,
x
->
axb.
Since a a+-b b=a+b, the line of invariant points is
POPa+b.
Since a a-b b
-
(a
-
b),
the axis is
POPa_b.
We
sum
up
our
conclusion
n
THEOREM
8.1.
Every
orthogonal ransformation
n
four
dimensions
s
either
x -axb or x->
axb.
9.
The
general displacement
expressed as
a double
rotation.
By
Theorem
5.2,
the
general half-turn
s
x
->
pxq,
*
Arthur
ayley,Recherches
lterieuresur es determinants
auches,Journal
ur
die reine
und angewandteMathematik, ol.
50, 1855,p. 312.
7/26/2019 Coxeter Quaternions and Reflections AMM 1946
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QUATERNIONS AND REFLECTIONS
(March,
where p and
q
are
pure unit quaternions.
This is the half-turn
bout
the plane
containing
ll points
P.
for
which x =pxq,
or
px + xq
=
0.*
Since p
-
q and 1+pq are particularsolutions of this equation forx, we may
describe
theplane as
PoPp_sP1+p.
A rotation
hrough 7rbout
the
same plane is,
of course,
x-*ptxqt.
Replacing q by
-q (=
=q1), we deduce that x?ptxq-t
is a
rotation through thr
bout the completely
orthogonal plane
PoPp+Pl-p,.
(The fact
that these
two planes are
completely
rthogonal
s mosteasily verified
by
observing
that the product
of the half-turns
-*pxq and x-
pxq-1 is the
inversion
x->p2x=
-x.)
Thus the productof
rotationsthrough
r and uwx
bout the respective
planes
PoPprqPi?pq
s
x+pt+uxqt-u.
Setting
tr-=a+13,
ur=
a-f,
and observing
that
cosa +
P
sina
=p2a/-
(see
the proof
of Lemma 2.3),
we deduce that
the
productof rotations
through
angles a?/3
about planes
POPP,P1?ip
is
x
-*
(cos
a
+
p
sin
a))x(cos
,B
+
q
sin p).
In
other
words,
THEOREM
.1.1
Thegeneral isplacement
->axb
s the
ouble otation
hrough
angles
about
lanes
PoPpi,Pi?pq,
where
cos
a
=
Sa,
cos
P
=
Sb,
p
=
UVa,
q
=
UVb.
10. Lines in elliptic space.
The
above considerations
an
be translated
nto
termsof elliptic
geometry y
identifyingairs
of antipodal points
on
thehyper-
sphere.
Now all scalar
multiples
of a quaternion
x
represent
he
same
point
P.,
whose
oordinates
xO,
X1l
X2,
x3)
are
homogeneous.
he transformation
x
-*
-
yxy
is the
reflection
n
the polar plane
of
Pv,
and
this
is
the
same
as
the
inversion
in P, itself.We now say that the group of all displacements is preciselythe
direct
product
of the
groups
of left
and right
displacements;
accordingly,
t is
isomorphic to the
direct square of
the group
of displacements
preserving
he
origin.
Instead
of
Theorem 9.1,
we
say
that
the
general displacement
x->axb
is the
product of
rotations
through angles a?
+
about
the
respective
lines
PpT-P,?,,,
where
(1)
cosa
=
Sa,
cos,
=
Sb,
p
=
UVa,
q
=
UVb.
*
Irving tringham,n thegeometryfplanes n parabolic paceof four imensions,rans-
actions f
the American
Mathematical
ociety,
ol.
2, 1901,p.
194.
t
tdouard
Goursat,
ur
les
substitutions
rthogonales
t les divisions
6guliRres
e
1'espace,
Annales cientifiques
e l'gcole
Normale up6rieure
3), vol.
6, 1889, .
36.
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QUATERNIONS AND REFLECTIONS 145
The line
P,P,P1+,,
which s
the axis ofthe half-turn
-*pxq,
is conveniently
denotedby {p, q},* or equally well by
{
-p, -q}.
Thus any twopure
unit
quaternions etermine
line {p,
q}. The
absolute polar line is
{
-p,
q} or
{P, -q}.
Two lines
{p,
q and
{p',
q'} have, n general,wo common erpendicular
lines,
which re
thetransversals
fthefour
ines{
?p,
q and
{
?p',
q'}. These
are preserved
by either
of the
half-turns -+pxq,
x--p'xq', and so also
by their
product
(2)
x
*
p'pxqq'.
Thus they are the two
axes of this double
rotation.
Any point on
either xis
will be
reflectedirstn
{
p, q and
then n
{
p', q'};
altogether
t willbe trans-
lated
along that
axis throughtwice
the distance
between
the lines. Hence
the
two istancesetweenhe ines
{
p, q and
{
p',
q' ,
measuredlong heirommon
perpendiculars,
re
2
J ?
:
where
(3)t
cosa
-
Sp'p
--
Spp',
cosf8
=-Sqq';
and the common
perpendiculars
hemselves
re
the
lines
(4)
?
UVpp',
UVqq'
1.
It follows
rom
3)
that
the condition
or
p, q
and
{p',
q'} to intersect
s
(5)
Spp'
=
Sqq',
and then
the
angle between
them,being
half
the
angle
of the rotation
(2),
is
arc cos
(C+
Spp').
Similarly,
he
condition
or
{p, q} and {
p', q' }
to be perpen-
dicular (i.e.,
forone to
intersect he polar
of the other)
is
(6)
Spp'+ Sqq'
=
0.
Thus
the condition
for them to intersect
t right
ngles
is
(7)
SpP'
=
Sqq'
=
0.
The commonperpendicular ines (4) cease to be determinate fVpp' orVqq'
vanishes, i.e.,
if
either p'= +porq'=
+q. Lines
{p, q}
and
{p, q'}
are
saidto
be left arallel.
They
have an
infinity,
f
commonperpendiculars
(8)
{
UVpp',
UVqq'
,
where
p' may
range over all
unit pure quaternions
(except
+p).
To
verify
*
This notation
as used
by A. S. Hathaway,
uaternionpace,
Transactions
ftheAmerican
Mathematical
ociety,
ol. 3, 1902,
p. 53.
It is closely ssociatedwith the representation
f a
line n elliptic
paceby an
ordered
air ofpoints n a sphere; ee Eduard Study,Beitriige
ur
nichteuklidischeeometrie,merican ournalfMathematics,ol. 29, 1907,pp. 121-124.
t
Here we are
using
1)
with
=
-p'p,
b=
-qq'.
Plus
signs
would
have
given
he
supple-
mentary istances,
which
re equally
valid; but it
seemspreferableo use the sign
that
makes
a
and 63
mall
when
'
and
g'
are
nearly qual to
p and
q.
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QUATERNIONS AND REFLECTIONS
this we
merely have to
observe
that
the line
(8),
which intersects
{p, q
}
at
rightangles, also intersects
{ , q'}
at
right
angles.
The distance
between
these left
parallel lines,
measuredalong any of the common
perpendiculars,
s
2arc
cos
(-Sqq').
Similarly,right arallel ines
{
,
q
}
and
{
', q
}
are distant
2
arc cos (-
Spp')
along any
of an
infinity
f
commonperpendiculars
8), only now
it is
q'
that
can vary. Thus
the common
perpendicular lines
of
right parallels are
left
parallel, and
vice versa.
By the remark at
the beginning
of ?9, the condition for
the line {p, q}
to
contain
the
point
P.
is
(9)
px+xq=O.
Regarding
this as an
equation
to
be
solved
for
q
or
p, we see
that
we
can
draw
through givenpointP. just one left nd one right arallel to a given ine
{
,
q
namely{-x-1px}
and
{-
xqx-1,
q}.
Thus
the set of all
left
(or right) parallels to a
given line is
an elliptic con-
gruence:there s just
one memberofthe set
through very
point of space.
If
{
,
q
}
contains
P,,,
its
polar
line lies in the
polar plane ofP.
Replacing
q by
-
q,
we deduce
that the condition for the
line
{
,
q}
to lie in the
plane
EY,x1V
O
s
(10)*
py
=
yq.
Instead of insisting
hat the p
and q of the symbol
{
, q
}
shall be unit pure
quaternions,we could just as well
allow them to
be any two pure
quaternions
of
equal norm.
Then {p,
q
is
thesame line as
{Xp, Xq for ny non-zero calar
X;
i.e., the
two
pure
quaternions are homogeneous
coordinates
for the line. In-
stead of
(3) we must
now write
cos
a
=-Spp'/v'}Npp,
cos
3
-Sqq'/VNN;
but
instead of
4)
we
find hat the
common
perpendiculars o {p, q
}
and
{p', q'}
are
simply
{
?Vpp',
Vqq'}, or
{Vpp',
Vqq'}
and
{
Vp'p,
Vqq'}.
Formulas
(5), (6), (7),
(9), (10)
remain valid, but (8) takes
the simpler form
{
Vpp', Vqq'
} .
To
express
the line
PaPb
in the
form p, q
}
we have to find
pure
quaternions
p
and
q
satisfying
pa + aq
=
pb + bq
=
0.
We
may
take
p=ab
-b4
and
q=aZb-Da, or,
halving these,p=Vab and
q=VcVb.
Thus
the ine
PaPb
is
{
Vab, Vab
}.
Similarly, the line of intersection of planes
,x
=
0 and Eb,x,
= 0
is
{Vab,
Vba}.
*
Hathaway,
Quaternion
pace, p. 52.