Coxeter Quaternions and Reflections AMM 1946

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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


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.


4/12

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.


5/12

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,


6/12

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.


7/12

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.


8/12

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.


9/12

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.


10/12

144


(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.


11/12

19461

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.


12/12

146


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.

Coxeter Quaternions and Reflections AMM 1946

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