Geometry and mappings of screws with applications to the hybrid control of robotic manipulators


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Geometry and mappings of screws with applications to the hybrid control of robotic manipulators
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vi, 297 leaves : ill. ; 28 cm.
Lipkin, Harvey
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Subjects / Keywords:
Screws, Theory of   ( lcsh )
Orthographic projection   ( lcsh )
Manipulators (Mechanism)   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1985.
Includes bibliographical references (leaves 290-296).
Statement of Responsibility:
by Harvey Lipkin.
General Note:
General Note:

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University of Florida
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Copyright 1985


Harvey Lipkin


The author wishes to express his sincere appreciation

to his committee members, Professors Gary Matthew, George

N. Sandor, Ralph Selfridge, John Staudhammer and Lou

Torfason as well as to Professor Delbert Tesar, who have

all helped make this work possible through their continued

encouragement and support. Special thanks are given to the

committee chairman, Professor Joseph Duffy, for his

guidance and enthusiasm throughout all phases of this

work. The author is indebted to Professor Kenneth H. Hunt

for his liberal applications of geometric stimulation, both

directly and through antipodal correspondence. Professor

Eric J.F. Primrose is gratefully thanked for sharing his

comments and geometrical insight on many topics.

The author greatly appreciates the efforts of Ms.

Denise Gater in typing this dissertation under such arduous

chronological constraints. Further, the Roger Gater

Courier Service has been most invaluable.

Finally, the author has depended upon the moral

support and motivation extended to him by his wife, Leslie,

and son, Matthew.

The financial support of the National Science

Foundation (Grant No. MEA83-24725) and of the Westinghouse


Research and Development Center, Pittsburgh, PA, is

gratefully acknowledged in making the completion of this

work possible.



ACKNOWLEDGMENTS ...................................... iii

ABSTRACT ............................................. vi


1 INTRODUCTION.................................... 1

2 PROJECTIVE GEOMETRY .......................... 4

2.1 A Projective Formulation of Coordinates. 6
2.2 Homogeneous Coordinates ................. 19
2.3 Projective Transformations ............... 39

3 METRICAL GEOMETRY ............................ 57

3.1 Projective Metrics ........................ 62
3.2 Elliptic Geometry......................... 77
3.3 Euclidean Geometry............. .......... 88
3.4 Vector Formulations ..................... 110

4 INVERSIVE GEOMETRY......................... .. 132

4.1 The Elliptic Polarity of Screws.......... 135
4.2 Quaternion Mappings ..................... 146
4.3 Ball's Planar Representation of the
Two-System.............................. 166
4.4 Self-Polar Screw Systems ................ 187

5 APPLICATIONS.................................. 205

5.1 Hybrid Control........................... 208
5.2 Noninvariant Methods .................... 225
5.3 Noninvariant Properties .................. 245
5.4 Invariant Formulations .................. 265


BIBLIOGRAPHY ......................................... 290

BIOGRAPHICAL SKETCH .................................. 297

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Harvey Lipkin

August 1985

Chairman: Joseph Duffy
Major Department: Mechanical Engineering

The nature and invariant properties of the elliptic

polarity of screws in Euclidean space is established using

a new series of mappings, central to which is a quaternion

representation. The mappings are used to generalize Ball's

planar representation of the two-system of screws which is

shown to be a mapping on a complex plane where the elliptic

polarity induces a conformal mapping. Screw theory is

applied to the hybrid control of robotic manipulators where

it is demonstrated that a current theory based on

orthogonall" projection yields noninvariant results. New

invariant methods of hybrid control are detailed by

introducing invariant kinestatic filters.



Research may start from definite problems whose
importance it recognizes and whose solution is
sought more or less directly by all forces. But
equally legitimate is the other method of re-
search which only selects the field of its acti-
vity and, contrary to the first method, freely
reconnoitres in the search for problems which are
capable of solution. Different individuals will
hold different views as to the relative value of
these two methods. If the first method leads to
greater penetration it is also easily exposed to
the danger of unproductivity. To the second
method we owe the acquisition of large and new
fields, in which the details of many things re-
main to be determined and explored by the first
method. A Clebsch [1871, p. 6]

The theory of screws was first established by Ball

[1900] after a quarter century of development. Although

most mathematicians of the late nineteenth century were

acquainted with screw theory, very few actually pursued

its development. Consequently, in the early twentieth

century it virtually faded into obscurity. However, with

a revived interest in spatial mechanisms in the 1960s and

robotics in the 1970s, screw theory is being slowly res-

urrected in either the guise of dual vectors or Plucker

coordinates. An excellent treatment of Plucker coordinates

in application to mechanisms has been given by Woo and

Freudenstein [1970]. Recent texts by Hunt [1978] and

Bottema and Roth [1979] have investigated and applied

screws in various kinematic areas.


However, the problem that was initially investigated

for this dissertation, the orthogonalityy" of screws, was

not treatable using the current literature in the field of

kinematics. It was necessary to reconsider some of the

geometrical work of the nineteenth century and try to put

it in a modern perspective with improvements of notation,

methods, etc. The beginning of this dissertation deals with

a systematic development of the geometry of points, planes

and lines to provide a broad basis for understanding the

subtleties of screw theory in a unified framework. As such,

the author has tried to integrate some of the pertaining

accomplishments of the great late nineteenth century geometers:

Ball, Cayley, Clifford, Grassmann, Klein and Plucker. When-

ever possible, original papers were referred to in order to

understand the spirit and implications between the lines

which are forever lost in most modern presentations. Ori-

ginal quotes have been included throughout to compensate

somewhat for this lamentable circumstance.

The geometrical development rightly begins with projec-

tive geometry, which is the most general, in Chapter 2. In

order to commence on an analytical basis at the outset,

coordinates are introduced in a metric-free manner for

points. They are then generalized using determinant princi-

ples which elegantly display dualistic properties. Projec-

tive transformations leave incident relations invariant and

are developed based on a tetrahedron principle.

Metrical geometry is developed in Chapter 3 from pro-

jective geometry based on Cayley's conception of the

Absolute. Elliptic geometry is detailed with emphasis on

determining elliptic relations for screws that appear simi-

lar to projective ones. Here the elliptic polarity is first

introduced. Elements of Euclidean geometry are then derived

with what appears to be a new contribution in deducing

Euclidean collineations based directly on the invariance

of the Absolute. Vectors are introduced and are used to

characterize screws in particular.

Chapter 4 deals with the role of the elliptic polarity

in Euclidean space. It uses the previous developments to

explain the motivation for and the importance of this work.

Unless specifically noted all developments in this chapter

are claimed to be original. A series of new mappings of

screws is introduced, central to which is a quaternion

representation. Ball's planar representation of the two-

system of screws is generalized as a mapping on an inversive

plane where the elliptic polarity induces a conformal mapping.

In Chapter 5 screw theory is applied to the hybrid control

of robotic manipulators. It is demonstrated that a current

theory based on orthogonall" projection yields noninvariant

results under Euclidean translations and changes in the unit

of length. By way of introducing invariant kinestatic fil-

ters, new invariant methods of hybrid control are presented.



Throughout the ages, from the ancient Egyptians and
Euclid to Poncelet and Steiner, geometry has been
based on the concept of measurement, which is defined
in terms of the relation of congruence. It was von
Staudt (1798-1867) who first saw the possibility of
constructing a logical geometry without this concept.
Since his time there has been an increasing tendency
to focus attention on the much simpler relation of
incidence, which is expressed by such phrases as "The
point A lies on the line p" or "The line p passes
through the point A." H.S.M. Coxeter [1942, p. 18]

During the nineteenth century a great deal of

investigation centered about geometrical relations which

are independent of metrical concepts such as distance,

angle, area, etc. Metric-free geometric relations form the

subject of projective geometry of which the property of

incidence is paramount, such as a point is incident to a

plane or a line cuts a conic. For many years projective

geometry was viewed as a relatively insignificant area

within the domain of Euclidean geometry. The situation

took a radical turn in 1859 when Cayley demonstrated that

projective geometry was actually the most general and that

Euclidean geometry was merely a specialization where a

certain configuration was deemed fixed and used as a

reference for measurement. Later, Klein demonstrated how

non-Euclidean geometries could be included under Cayley's

principle which is explained in Chapter 3.

Thus, there arose a contradiction. Since coordinates

were typically based upon metrical considerations, how

could such coordinates be logically applied to projective

relations since metrical geometry is subordinate? Klein

supplied an answer to this by suggesting the use of von

Staudt's projective constructions which are employed to

define the algebra of points and is presented in a modified

form in Section 2.1. First nonhomogeneous coordinates are

introduced which are then used to define the homogeneous

coordinates of points.

In Section 2.2 determinant formulations are utilized

to develop homogeneous coordinates in the plane for points

and lines and in space for points, planes, lines and screws

following extensionsal principles attributable to

Grassmann. This enables a rather elegant and complete

symmetry to be displayed amongst dual elements. Plucker's

ray and axis line coordinates are detailed and their

important idential relation is formulated which is

subsequently used throughout the entire work.

Projective transformations of space preserve incidence

relations and are detailed in Section 2.3. An important

tetrahedron relationship is introduced which unifies

projective transformations for points, planes and lines or

screws. Polarities are developed as correlations whose

properties are symmetrical with their inverses and are then

employed in Chapter 3.

Section 2.1 A Projective Formulation of Coordinates

In the first place, it is important to realize
that when coordinates are used, in projective
Geometry, they are not coordinates in the
ordinary metrical sense, i.e. the numerical
measures of certain spatial magnitudes. On
the contrary, they are a set of numbers,
arbitrarily but systematically assigned to
different points, like the numbers of houses
in a street, and serving only, from a philo-
sophical standpoint, as convenient designations
for points which the investigation wishes to
distinguish. But for the brevity of the
alphabet, in fact, they might, as in Euclid,
be replaced by letters. Bertrand Russell [1897,
pp. 118-119]

Projective geometry deals with the incidence properties

of elements and figures without employing any form of measure-

ment or metric. However, various metrical subgeometries may

be derived from the more general projective geometry by re-

quiring that certain figures be designated as an absolute

reference. Thus to employ metrical concepts, such as Cartesian

coordinates, in the analytical development of projective geom-

etry is to incur a logical contradiction since metrical rela-

tions are subordinate.

In this section, it is demonstrated that coordinates may

be introduced into projective geometry in a metric-free manner.

Coordinates are important in the analytical development of

projective geometry since they are the tools for the derivation

of compact expressions for complex relations and further, they

provide insight which may not be evident from a synthetic de-

velopment. Coordinates also provide an essential vehicle for

numerical calculations which may be used to verify and illus-

trate relations unambiguously. Since coordinates are routinely

used in metrical geometries it is highly desirable to demon-

strate how they may be first utilized projectively and then

specialized for metrical application.

The introduction of coordinates in projective geometry

presented here is not intended to be an exhaustive treatment

but rather an illustrative one. The intent is to draw a

sharp distinction between projective and metrical concepts

at the outset and to provide a logical sequence of development

from projective to metrical geometry. The analysis in this

section is based primarily on the complete and systematic

methods of Veblen and Young [1910, 1917] to whom frequent

reference is made.

It was however von Staudt [1857] who first demonstrated

that analytic methods may be introduced into geometry on a

strictly projective basis. For this purpose he invented the

algebra of throws. The method was later simplified by

Hessenberg [1905] and is presented here. First, the addition

and multiplication of points on a line is defined using two

special projective constructions. Then this algebra of points

is demonstrated to be isomorphic to the field of real numbers

and is extended to include the concept of infinity in a con-

sistent manner. A unique real number is associated with each

point on the line with the exception of a single point which

assumes a correspondence with infinity. The unique real

number associated with each point is called the nonhomogeneous

coordinate of the point on the line. The apparently exceptional

role of the point associated with infinity is removed upon the

introduction of homogeneous coordinates.

In order to assign coordinates to points on a line U, it

is first required to select three distinct points x0, x1 and

x. which together are referred to as a scale. By the special

nature of the constructions which define addition and multipli-

cation, the points of the scale are endowed with the properties

of 0, 1 and -. Figures 2.1.1 and 2.1.2 illustrate the construc-

tions for addition and multiplication which can also be used

respectively to define subtraction and division.

Using the operation of addition it is possible to label

all the points corresponding to integers, e.g. x1 + x1 = x2,

x1 + x2 = x3, etc. Next, using the integers with the operation

of division it is possible to label all points corresponding

to rational numbers. Thus it can be shown that the points

corresponding to rational numbers together with the two con-

structions are isomorphic to the field of rational numbers

which includes the properties of associativity, communitivity,

distributivity and the existence of inverses.

Next, it is necessary to show that for all points on the

line there is a corresponding real number and that together

with the constructions they are isomorphic to the field of real

numbers. This transition is made from the rational points

by invoking a Dedekind cut which is detailed in Coxeter [1942],

Veblen and Young [1910].

For the notation used here, the nonhomogeneous coordinates

of the points x0, x1, xa, are given by the subscripts

0, 1, a, where the point x is excluded. It is important

to note that the constructions yield



X0 Xa Xb Xa+b Xoo

Figure 2.1.1

Addition of two points. A fixed line U0
through x0 meets the two distinct fixed lines
U and U in points r and s respectively.
The lines x r and x s meet U and U respec-
a b a
tively at r' and s. The line r s meets U at
x a+b which yields the sum of the two points,
x + xb = x a+b. By reversing the latter
steps, subtraction is analogously constructed,
eg. xa = xa+b xb.


Xo X1 Xa Xb Xab Xoo

Figure 2.1.2

Multiplication of two points. Through the points
x0, x1, x. are drawn respectively the fixed
lines U0, U U with U0, U1 meeting at r and
U U. meeting at s The lines x r and Xb s
meet U and U respectively at r and s. Line
r s meets U at xab which yields the product of
the two points, x xb = x ab. By reversing
the latter steps, division is analogously
defined, eg. xa = xab xb.
a ab

x + x = x (1)
a 0

x x = x (x x ) (2)

x x = x (xa x0 ) (3)

which are consistent with the usual properties associated

with infinity.

Although the three distinct points comprising the scale

are selected arbitrarily, the addition and multiplication

constructions impart them with the special properties asso-

ciated with 0, 1 and w. However, from a projective standpoint,

all points have identical properties. Most generally, the

fundamental theoremof projective geometry, Schreier and

Sperner [1935], states that in a space of dimension n, there

is a unique projective transformation between a pair of n+2

elements providing that for each part of the pair no n+l ele-

ments belong to a space of dimension n-l (.see Section 2.3 for

n=3). For points on a line n=l and thus three distinct points

are related by a projective transformation. Three distinct

new points may be chosen as another scale and all other points

relabelled in terms of it. By way of projective transforma-

tions, all scales and subsequently all coordinates are pro-

jectively equivalent. Nevertheless, when a particular scale

is employed, the scale points will still have the special

properites of 0, 1 and due to the definitions of the con-

structions. The approach taken here is to use the addition

and multiplication constructions solely for the purpose of

labelling the points. These constructions are not utilized

further, such as to analytically express projective trans-

formations in terms of nonhomogeneous coordinates. Thus,

the special properties of the scale points do not enter into

any further analytical development.

Projective homogeneous coordinates are introduced by

associating a pair of numbers a0, al, which can be written
as the 2xl array [a0 a 1] with the nonhomogenous coordinate

a of the point x such that

a (4)

The homogeneous coordinates of xa are not unique since [Aa0 Aa1]T
X0O also satisfies (4). The coordinates [0 X]T, X#0 are asso-

ciated with the point x Thus to every pair of homogeneous

coordinates, with the exception of [0 0] there corresponds a

unique point of the line and to every point of the line there

corresponds a pair of coordinates, which to a scalar multiple,

is unique.

Projective homogeneous coordinates for points in a plane

are developed from the homogeneous coordinates of points in

three distinct lines. As illustrated in Fig. 2.1.3, three

noncollinear points are selected as the vertices of a triangle

of reference and are used to establish two scale points on each

side, w0, w x0, x and y0, y.. A fourth point which does

not lie on one of the edges is selected as the unit point.

Lines through the unit point and each vertex establish on

the sides opposite the unit points wI, x1 and yl, thus complet-

ing the three scales. Together, the three vertices and the unit




/;* \

/ \ s S
"; .-";'*..


Xl Xa
[F iol

Wi 10111T



Figure 2.1.3 Projective homogeneous coordinate for the plane.


point are said to form a reference frame and respectively

their homogeneous coordinates are designated by

[1 0 O]T

[0 1 0]T

[0 0 I]T


[1 1 I]T (5)

Every point on a side of the triangle may now be assigned

a triple of homogeneous coordinates [a a a a2]T. For example,

on line U each of the scale points is characterized by a2=0.

In terms of the homogeneous coordinates of U every point has

the form [a0 al]T which is now denoted in the planar system

by [a0 a1 0] Analogously, points on U and Ux have coordi-
w x
nates of the form [0 a1 a2]T and [a0 0 a2]T.

For any point not on a side of the triangle, all of its

coordinates [a0 a1 a2] will be nonzero. Let its projections

on U and U from the opposite vertices be the points x and
x y a
Ya with coordinates determined by their respective scales,
[xa0 0 xa2 ]T and [ya0 al 0]T It is possible to select the

coordinates of the point such that

al Yal a2 x a2(6)

0 a0 a0 Xa0

that is in the ratios,

a0:al:a2 = xaOY a0:xa0 al:Xa2 Ya0

The coordinates of xa and ya can thus be expressed respectively

as [a0 0 a2T] and [a0 a 0]T For the system of coordinates

to be consistent, it is necessary to demonstrate that the

coordinates of wa the projection of point a on U ,are propor-

tional to [0 a a2]T This can be proved using an argument

involving cross ratios which is detailed in Veblen and Young

[1910, pp. 175-176]. Therefore, to a nonzero scalar multiple,

the projective homogeneous coordinates of a point on a plane

are given by the triple [a0 a1 a2]T. It should be noted that

the four points chosen for the reference frame, no three of

which are collinear, are otherwise arbitrary. Thus any four

such points may be chosen since by the fundamental theorem of

projective geometry they are related by a unique projective


The method of using coordinates on three lines to establish

coordinates on the plane leads to a recursive algorithm that

may be used for a space of any dimension and is briefly described

here for three dimensions.

Four noncoplanar points are chosen as the vertices of a

tetrahedron of reference and a fifth point not incident with

a face is selected as the unit point. As illustrated in Fig.

2.1.4, the four vertices and unit point are respectively

assigned the coordinates,

[1 0 0 O]T

[0 1 0 0]T

[0 0 1 O]T

[0 0 0 1]T

[o 0 0 ]T

[ 0 0 0 T [0 1 0 01

Figure 2.1.4

Tetrahedron of reference for projective homogeneous
coordinates in space.


[1 1 11T (8)

Lines through the unit point and each vertex establish

unit points on the opposite sides respectively,

[0 1 1 1]

[1 0 1 1]T

[1 1 0 1]T

[1 1 1 0] (9)

and together with the vertices, they are used to establish a

planar reference system on each face.

Every point which is incident to a face of the tetrahedron

may now be assigned a quadruple of homogeneous coordinates

[a0 a a2 aa3] For example, on the face opposite the vertex

[0 0 0 1]T, each of the points is characterized by a3=0. This

face has precisely the same planar reference system (5) described

previously and thus has the form [a0 a1 a2] which is now de-

noted in the spatial system by [a0 a1 a2 0] Analogously,

points on the remaining faces are characterized respectively by

a0=0, al=0, a2=0.

For any point not incident to a face of the tetrahedron,

all of its coordinates [a0 a1 a2 a3]T are nonzero. It is neces-

sary to show that the projections of the point onto the four

faces can be assigned coordinates in the planar systems given

respectively by

[0 a1 a2 a3]T

[a0 0 a2 a3 T

[a0 a1 0 a3]T

[a0 a1 a2 0]T (10)

This has been proved by Veblen and Young [1910, pp. 194-195].

It should be noted that the five points chosen for the

reference frame, no four of which are coplanar, are otherwise

arbitrary. Thus any such five points may be selected since

by the fundamental theoremof projective geometry they are

related by a unique projective transformation.

In the preceding developments, only the projective homo-

geneous coordinates of points in spaces of one, two and three

dimensions were considered. One-dimensional spaces are also

described by lines in a plane through a point or planes through

a line and an analogous procedure may be developed to assign

coordinates. Further, for two-dimensional spaces such as a

lines in a plane or planes through a point and for three-

dimensional spaces where for example the assemblage of all

planes forms a three-space, coordinates may be assigned.

In the two-dimensional space of a plane, it is possible

to simultaneously assign point and line coordinates and in

three-space it is possible to assign point and plane coordi-

nates without incurring any logical inconsistencies. In the

next section such a methodology is presented and is then

extended by determinant principles to assign coordinates to

lines and screws in projective three-space.

Section 2.2 Homogeneous Coordinates

A point given by its coordinates and a point
determined by its equation, or geometrically
speaking by an infinite number of planes inter-
secting each other in that point, are quite
different ideas, not to be confounded with one
another. That is the case also with regard to a
plane given by its coordinates and a plane repre-
sented by its equation, or considered as contain-
ing an infinite number of points. Hence is derived
a double signification of a right line. It may
be considered as the geometrical locus of points,
or described by a point moving along it, and
accordingly represented by two equations in x, y,
z each representing a plane containing that line.
But it may likewise be considered as the inter-
section of an infinite number of planes, or as
enveloped by one of these planes, turning round
it like an axis; accordingly it is represented
by two equations in t, u, v, each representing
an arbitrary point of the line. The passage from
one of the two conceptions to the other is a dis-
continuous one.

The geometrical constitution of space, hitherto
referred either to points or to planes, may as
well be referred to right lines. According to
the double definition of such lines, there occurs
to us a double construction of space.

In the first construction we imagine infinite
space to be transversed by lines themselves
consisting of points. An infinite number of such
lines pass in all directions through any given
point; each of these lines may be regarded as
described by a moving point. This constitution
of space is admitted when, in optics, we consider
luminous points as sending out in all directions
luminous rays, or, in mechanics, forces acting on
points in every direction.

In the second construction infinite space is like-
wise regarded as transversed by right lines, but
these lines are determined by means of planes
passing through them. Every plane contains an
infinite number of right lines having within it
every position and direction, around each of which
the plane may turn. We refer to this second con-
ception when, in optics, we regard, instead of rays,
the corresponding fronts of waves and their conse-
cutive intersections, or when, in mechanics, accord-
ing to POINSOT'S ingenious philosophical views, we

introduce into its fundamental principles
"couples," as well entitled to occupy their
place as ordinary forces. The instantaneous
axes of rotation are right lines of the
second description. J. Plucker [1865, pp.

The dualistic properties of projective geometry may

be elegantly expressed in an analytic manner by employing

homogeneous coordinates and determinant principles. The

main objective of this section is the development of projec-

tive homogeneous coordinates in space, particularly line

coordinates and their extensions to screws. However, it is

appropriate to commence with point and line coordinates in

the plane since this provides a foundation for the applica-

tion of extensional principles to three-dimensional space,

Forder [1940].

A line passing through two points may be described as

the join of the two points and dually, the intersection point

of two lines may be described as the meet of the two lines.

Essentially, the principle of duality in the plane is that

incidence relations remain valid when the roles of points

and lines are interchanged (along with an appropriate altera-

tion in terminology such as replacing join with meet). For

example, the previous relations may be expressed as, a line

(point) is the join (meet) of two points (lines).

In the plane, line coordinates are developed by first

demonstrating, as in Coxeter [1942, pp. 78-80], that the

condition for a point and a line to be incident may be

expressed as the linear relation

S0w0 + Slwl + S2w2 = 0 (1)

This relation can also be expressed more compactly by

Sw = 0 (2)

for which

S = [SO Sl S21T, w = [w0 w1 w2] (3)

The 3x1 arrays S and w are respectively projective homogeneous

line and point coordinates in the plane and are unique to a

nonzero scalar factor, i.e. only the ratios of the coordinates

are significant. Assuming in (2) that the coordinates S have

constant values while the coordinates w are free to vary, then

(2) represents the locus of points which are incident to the

line S, or in other words, the equation of line S. Dually, if

the coordinates w are assumed constant while the coordinates S

are free to vary, then (2) represents the pencil of lines which

are incident to the point w, or in other words, the equation of

point w.

Equation (2) may be applied to express line coordinates

in terms of point coordinates and reciprocally. Let x and y

be two given distinct points on S and let w be any variable

point on the line. Then using (2) yields three equations

[w x y] S = 0 (4)

for which the vanishing of the determinant

|w x yj = 0

yields an equation in w

w01x1 y21 + wllx2 Y01 + w2xo0 Yll = 0 (6)


lxi y.j = x. y. x. y. (7)

is the ij minor in (5). Based on the expansion (6), it is

useful to define the determinants of the nonsquare arrays

[w] (or w) and [xy] as the 3x1 column arrays

wI = [w0 w w2]T (8)

xyl = [jx1 y21 Ix2 yol Ixo y ]T. (9)

Thus equations (5) or (6) may be expressed simply as

w |Ixy| = 0 or w Ixyl = 0 (10)

Comparing (10) with (2) yields the coordinates of line S to

a scalar multiple in terms of points x and y,

S = jxyl. (11)

Since x and y are assumed distinct, then from (5) it is

readily deduced that any point on the line S may be expressed

in the form

w = ax + By (12)

where scalars a and $ are both not simultaneously zero.

Equation (12) is referred to as the freedom equation of the

line and any point on it is given by the ratio a:3. Using

(.12) it may be shown that the coordinates of S are, to a

scalar factor, independent of which two distinct points are

chosen. For example, replacing x and y with (a0x + 80y) and

(alx + B1y) in (11) yields

S = aB Ixyl (13)

where the scalar |aSj 0 since the points are assumed distinct.

Because of the duality principle, the development of

line coordinates in terms of point coordinates is completely

analogous to the preceding analysis. Let T and U be two given

distinct lines incident to w, then if the variable line S is

also concurrent,

[S T U]Tw = 0 (14)

for which

|S T U| = 0 (15)

yields an equation in S,

SOIT1 U21 + SlT 2 U0 + S2 ITO Ull = 0 (16)

or equivalently,

,slI IT UI = 0 (17)

Comparing (17) with (\2) yields the coordinates of point w

to a scalar multiple in terms of lines T and U,

w = IT U.


Analogous to (12), the freedom equation for any line

incident to w is deduced from (15)

S = aT + BU (19)

where a and 6 are both not simultaneously zero.

In Section 2.1 the vertices of the reference triangle

were assigned point coordinates corresponding to the columns

of 13 (where In is the nxn identity matrix) and the first

column represented what is usually called the origin. By

substituting pairs of vertices in (11) it is shown that the

line coordinates of any side and the point coordinates of

the opposite vertex are given by the same column of I3. The

side opposite the origin is often called the line at infinity

though no reference to distance is implied here.

For the plane, point and line coordinates can be summarized


xi IT (20)

x y IT U|. (21)

In (20), point and line coordinates are given respectively by

the three lxl determinants of [x] and [T] selected in the

order 0, 1, 2. In (21) line and point coordinates are given

respectively by the three 2x2 determinants of [xy] and [TU]

selected in the order 12, 20, 01.

In higher order spaces many of the developments for the

plane may be further generalized by application of extensional

determinant principles which were first introduced by

Grassmann, see Forder [1940], Klein [1908]. For three-dimensional

projective space, the point may be chosen as the fundamental

element and then a line is the join of two points and a plane

is the join of three points. Dually, the plane may be selected

as the fundamental element and then a line is the meet of

two planes and the point is the meet of three planes. In

space, the point and plane are dual elements whereas the

line is self-dual. The duality principle in space asserts

that incidence relations that are valid for points, lines and

planes remain valid when their roles are interchanged with

planes, lines and points along with an appropriate alteration

in terminology.

The condition for a point and plane to be incident may be

expressed as the linear relation

SOw0 + S1w1 + S2w2 + S3w3 = 0 (22)

or equivalently

Sw = 0 (23)

for which now

S = [S0SIS2S3 T w = [w0wlW2w3] (24)

The 4x4 arrays S and w are respectively projective homogeneous

plane and point coordinates and are unique to a nonzero scalar

factor since only the ratios of the coordinates are significant.

Equation (23) may represent either the locus of points incident

to plane S or dually, the bundle of planes incident to point w,

the difference being respectively whether S or w is assumed

fixed while the other is free to vary.

Plane coordinates may be developed from point coordinates

by requiring that one variable and three given noncollinear

points be incident to a plane

[w x y z]Ts = 0 (25)

for which

|w x y zI = 0 (26)

yields an equation in the variable point w

w0 xly2z3l + wllx2yOz3 + w2x0oylz3l w3jx1l0Z21 = 0 (27)

and Ixiyjzkl is the ijk minor in (26). Alternately, (27) may

be expressed more compactly using nonsquare determinants

wIT Ixyz| = 0 or wT xyzl = 0 (28)


IwI = [w0 w1 w w3]T (29)

Ixyz| = [Ixl 2z31x2yOz3l1xoylz3 xlyOz2|]T (30)

Comparing (30) with (23) yields the coordinates of plane S to
a scalar factor in terms of the points x, y and z,

S = Ix y zI. (31)

Since x, y and z are noncollinear, then from (26) it is
deduced that any point on the plane S may be expressed in the


w = ax + By + yz (32)

where the scalars a, 6 and y are all not simultaneously zero.

Equation (32) is the freedom equation of the plane and any point

on it is specified by the ratios a: ( :y. Substituting for x,

y and z in (31), any other three noncollinear points on the

plane S, a.x + $.y + y.z, i=l,2,3 yields

S = |ae y Ixyz| (33)

where the scalar jacSyj 0 since the points are noncollinear.

Briefly, by the principle of duality, the development of

plane coordinates from point coordinates is entirely analogous

to the preceding development. Let one variable plane and

three given planes, which themselves do not meet in a line, all

be concurrent at a point

[S T U V]w = 0 (34)

for which

IS T U V = 0 (35)

yields an equation in the variable plane S,

S01T1U2V31 + SI1T2UoV31 + S21TOU1V31 + S3IT1U0V21 = 0 (36)

or equivalently

S I T ITU V = 0.


Comparing (37) with (26) yields the coordinates of point w to

a scalar factor in terms of planes T, U and V,

w = IT U V1. (38)

Further, the freedom equation of the planes through w is

deduced from (35),

S = aT + BU + yV (39)

for a, 8 and y not all simultaneously zero.

In Section 2.1 the vertices of the reference tetrahedron

were assigned coordinates corresponding to the columns of 14*

By substituting triples of vertices in (31) it may be shown

that the plane coordinates of any face and the point coordinates

of the opposite vertex are given by the same column of 14. For

instance, the first column corresponds to the vertex at the

origin and to the face that is often referred to as the plane

at infinity.

In space, point and plane coordinates may be summarized

and line coordinates introduced using the nonsquare determinants,

x| IT| (40)

Ix y IT U (41)
Ix y z T U V (42)

In (40) point and plane coordinates are given respectively

by the four lxl determinants of [x] and [T] selected in the

order 0, 1, 2, 3. For (42), plane and point coordinates are

given respectively by the four 3x3 determinants of [x y z] and

[T U V] selected in the order 123, 203, 013, 102.

Plucker's ray and axis line coordinates, Plucker [1865,

1866], are defined by (41) respectively as the six 2x2 determi-

nants of [x y] and [T U] selected in the order 01, 02, 03, 23,

31, 12. Ray line coordinates p represent the join of two

points x, y and axis coordinates P represent the meet of two

planes and

P= P01P02P03P23P31P12 'T j = Ixijl (43)

= 3T T.U. (44)
P = [P01P02P03P23P31P12 Pij = (44)

or more briefly,

p = x y P = IT U (45)

Referring to Fig. 2.2.1, the relationship between ray and

axis coordinates is derived from the incidence relations of

two points with two planes,

Tx = 0 (46)

TTy = 0 (47)

Ux = 0 (48)

U y = 0. (49)

Forming in turn, T -(48) U *C(46), T -(49) U -(47),

x -(47) y *(46) and x *(49) y -(48) yields respectively,


Figure 2.2.1 A line is formed as either the join of two points
or the meet of two planes.

[P ]x = 0 (50)

[P*]y = 0 (51)

[p*]T = 0 (52)

[p*]U = 0 (53)

where the rank two skew-symmetric arrays are given by

01 02 03


P30 P31 P32 -

P01 P02 P03

[p*]= 10 12 13 (55)

20 P21 P23

P30 P31 P32 _

From (50), (51) and (54), the rows (or columns) of [P ] are

four planes through the line, each incident with a vertex

of the reference tetrahedron. From (52), (53) and (55), the

rows (or columns) of [p ] are four points on the line, each

incident with a face of the reference tetrahedron.

Forming either (50). y (51)* xT or (52). UT (53). TT

yields the matrix equation

[P*][p*] = 0 .


04 04 04 04
o r-1 04 r

r-4 0 C) .
+ + + +

Q4 a4 04 04
--i 0 0 H

(i CD DI fri




CD "m ( Y
04 04 04 04

rq H
o H 04 CN
04 04 04 04
+ + + +

H 0 H 0
H4 0 H-I 0
o C- 0 C( )



OY m mm m N
04 04 0 0

a4 04 04 04




o | r -i 0 o (Y)


0 ~N 0N O0


r N 0 C n
o i-i o i o 1-1
a4 04104 0 4 a I 04


I4 r-L4 a4 1 a4 C 4


In Table 2.2.1, (56) has been expanded into components to

yield (57). Then the off-diagonal terms are used to form the

twelve equations in (58) written as a 4 x 4 array. Comparing

in turn five equations in (58) given by the positions 30, 10,

13, 03, 01 yields

23 P 31 P 12 P 01 P 02 = P03
.... p (5 9 )
01 02 P03 23 P31 12

where i is a nonzero scalar.

It should be noted that it is not necessary to explicitly

include p since it is included implicitly by only ascribing

significance to the ratios of homogeneous coordinates. Thus

setting i=l and arranging (59) in matrix form yields

P = Ap (60)

p = AP (61)


A = (62)

A = A, AA = 16. (63)

Since in (60) and (61), p and P are derived from dual

elements, the induced linear transformation A is a correlation

which is signified by tilda. The existence of this correla-

tion is due to the fact that in three-dimensional space lines

are self-dual elements and thus A represents the identical

correlation of lines. The matrix A represents a very simple

method of transforming between ray and axis coordinates, the

first and last three components are merely exchanged. Since

a double application of the exchange yields back the initial

values, then as expressed by (63), A is clearly an involution


In ray coordinates, the condition for two lines to be

incident may be obtained by letting p be the join of w, x and

q be the join of yz. When the lines intersect then all four

points are coplanar and

|w x y zI = 0 repeated, (26)

which is expanded by the first two columns to yield

p A q = 0 or lw xl A ly z = 0. (64)

Alternatively, in axis coordinates let P and Q be the same

two lines where P is the meet of X, T and Q is the meet of

U, V. When the lines intersect then all four planes are

concurrent at a point and the expansion of

|S T U VI = 0 repeated, (34)


PT A Q = 0 or IS TIT AJU VI = 0. (65)

Using (60), (61) with (64), (65) gives alternative expressions

for intersection,

p TQ = 0 or Lw xlT LU V, = 0 (66)

PTq = 0 or Is TIT x y| = 0. (67)

Because dual coordinates are used in (66), (67) their form is

analogous to (23) which expresses the incidence of a point

and a plane.

Since only the ratios of homogeneous coordinates are

significant, the six coordinates of a line represent five

parameters. However, only four independent parameters are

required to specify a line in space and therefore six coordi-

nates must be related by a single equation. In terms of ray

and axis coordinates, this relation is obtained by expanding

the singular determinants

Ixy xy = 0 TU TU = 0 (68)

to yield

p A p = 0 PA P = 0 (69)

or equivalently

p P = P Tp = 0. (70)

It is interesting to examine the line coordinates cor-

responding to the tetrahedron of reference. Referring to

Fig. 2.2.2, the vertices along with their opposing faces

are both labelled 0, 1, 2, 3 and their coordinates correspond

in order to the columns of 1I. Forming the ray coordinates

of the six joins of vertex pairs and the six meets of face



03 31 2

a) 12

001 1

\ 2 02 -

b)3 03


Figure 2.2.2 The tetrahedron of reference labelled with
a) point coordinates and ray coordinates,
b) plane coordinates and axis coordinates.

pairs both in the order 01, 02, 03, 23, 31, 12, then the result-

ing coordinates of both sets correspond in order to the columns

of 16. Opposite edges of the tetrahedron (i.e. nonintersecting),

one expressed in ray coordinates and the other in axis coordi-

nates, correspond to the same column of 16'

Unlike point and plane coordinates, there are in general

no freedom equations for lines using line coordinates correspond-

ing to (32) and (39). (There are however freedom equations for

lines in terms of either two points or two planes.) For

example, let q,r be two lines and 11,A be two scalars and


p = pq + Xr (71)

For p to be a line it must satisfy the quadratic form of (69).


p A p = 2 p q A r (72)

which only vanishes when q and r intersect.

In general, a linear combination of lines in either ray

or axis coordinates is defined here as a screw, which includes

lines as special cases. Since the homogeneous coordinates of

a screw need not satisfy (69), it follows that there are 5

screws in space. When two screws satisfy any of the equivalent


T = 0 PT Q (73)
p Aq=0 ,P AQ= (73)

P Q = 0

T q = 0
, Qq = 0


the screws are said to be reciprocal, a property which is

analogous to incidence.

For a general linear combination of n screws (or lines),

1 < n < 6, the freedom equations

p = q + r (75)

P = XIQ + R

where X1 n are all not simultaneously zero, are said

to describe an n-system of screws, Ball [1900], Hunt [1978].

Screws may also be viewed abstractly as points in a five-

dimensional space as suggested by F. Klein (see Jessop [1903] ).

Lines in this space lie on a surface which is given by (69).

Dually, hyperplanes in this space correspond to five-systems

of screws, which also may be described by six homogeneous

coordinates. By generalizing the development of coordinates

presented here, systems of coordinates may be systematically

derived for n-systems, 1 < n < 5. Detailed references on the

generation of extensional systems of coordinates based on

determinant properties are given by Forder [1940], Sommerville

[1929], Hodge and Pedoe [1947, 1952], but will not be considered

here further.

Section 2.3 Projective Transformations

I shall enunciate two general principles
which I have habitually emphasized and have put
into the foreground in these fundamental geometric
discussions. Although in this generality they
sound at first somewhat obscure, they will, with
concrete illustrations, soon become clear. One
of them is that the geometric properties of any
figures must be expressible in formulas which
are not changed when one changes the coordinate
system, i.e. when one subjects all the points of
the figure simultaneously to one of our
transformations; and, conversely, any formula
which, in this sense, is invariant under the
group of these coordinate transformations must
represent a geometric property. As simplest
examples, which all of you know, let me remind you
of the expression for the distance or for the
angle, in the figure of two points or of two
lines. We shall have to do repeatedly with
these and with many other similar formulas in
the following pages. For the sake of clearness,
I shall give a trivial example of non-invariant
formulas: The equation y = 0, for the figure
consisting of the point (x,y) of the plane, says
that this point lies on the x axis, which is,
after all, a thoroughly unessential fact, foreign
to the nature of the figure, useful only in serving
to describe it. Likewise, every non-invariant
equation represents some relation of the figure
to external, arbitrarily added, things, in particu-
lar to the coordinate system, but it does not re-
present any geometric property of the figure.

The second principle has to do with a system of
analytic magnitudes which are formed from the
coordinates of points 1, 2, such as X, Y,
and N, for example. If this system has the property
of transforming into itself, in a definite way,
under a transformation of coordinates, i.e., if
the system of magnitudes formed from the new
coordinates of the points 1, 2, expresses
itself in terms exclusively of these magnitudes
formed in the same way from the old coordinates
(the coordinates themselves not appearing explicitly),
then we say that the system defines a new geometric
configuration, i.e., one which is independent of the
coordinate system. In fact, we shall classify all
analytic expressions according to their behavior
under coordinate transformation, and we shall define
as geometrically equivalent two series of expressions
which transform in the same way. Felix Klein [1908,
pp. 25-26]

This section examines linear projective transformations

of homogeneous coordinates. Klein [1872] has enunciated a

definition of geometry which, except for minor extensions,

is still very applicable today. Essentially, Klein stated

that a geometry is defined as the properties of a space which

remain invariant under all transformations of space (or the

coordinate system) by a group of transformations.

For projective geometry, the group of transformations is

characterized by those which preserve relations of incidence.

Commencing with the group of projective point collineations,

the corresponding induced collineations for planes, lines and

screws are developed with respect to an elegant tetrahedronal

principle employing determinant relations. Using a simple

device, many of the results for collineations are extended to

the nongroup of correlations. An analysis of projective trans-

formations not only identifies important invariant relations

but also forms a foundation for developing metrical geometries

in Chapter 3.

A collineation is a one-to-one linear transformation in

which each element of space is mapped into a corresponding

element of the same type (e.g. point to point) whereas a cor-

relation differs in that each element is mapped into a corres-

ponding dual element (e.g. point to plane). A projective

transformation is uniquely determined by five pairs of

corresponding points in space provided that no four of the

five points in either pair are coplanar. For the collineation,


it = Kx (1)

where p is included explicitly as a factor of proportionality

and where the 4x4 matrix K is given by

K = [A B C D]T (2)

Since only the ratios of homogeneous coordinates are signifi-

cant, the four equations in (1) can be reduced to three ratios

of equations by, for example, dividing the last three equations

by the first equation and thus the explicit factor p is

eliminated. Multiplying out the ratios and expressing them

in matrix form yields

-tx tx A

-t2x t0x B = [0] (3)

-tx T t0xT C
3 0


where t0 t3 are the coordinates of t and where the 3x16

matrix multiplies the 16x1 column array containing the unknown

coefficients. Substitution for t and x by five pairs of

corresponding points yields 15 homogeneous equations which

are sufficient to solve for 15 ratios involving the elements

of K. Thus the projective collineation is uniquely determined

to a scalar factor and K is nonsingular since the mapping is


In (1) the factor p was explicitly included to facilitate

in the solution for K. However, it is convenient to absorb

the factor by substituting u=i which is permissible provided

it is understood that only the ratios are significant,

t = Kx.

A projective collineation of points also induces a

projective collineation of planes which may be determined

using incidence properties. Let x be incident to plane T,

Tx = 0 (5

The induced transformation k maps T into another plane X

X = kT (6

such that incidence is preserved,

XTt = 0. (7

Substituting (4) C6) in (7),

TT(kTK)x = 0 (8

and comparing with [5) yields

k = KT (9

to an arbitrary nonzero scalar multiple. Matrix k can be

calculated by replacing each element of K with its cofactor

(signed minor), and dividing by the scalar IKI = IA B C DI

(although this last step is not essential),



or more simply as

k = [a b c d] (11)

The four 4x3 determinants in (10) have been formed from ABCD

in the order 123, 203, 013, 102 which is also the same order

used in expanding each of these nonsquare determinants into

components, see (2.2.42).

There is a useful geometric interpretation for K and k.

Let ABCD represent the coordinates of four planes whose equa-

tions can be written as

[A B C D]Tw = 0. (12)

Since K is nonsingular then (12) has no solution other than

w=0 which does not represent a point in homogeneous coordinates.

Thus the four planes do not have a common point and they there-

fore form a tetrahedron. In (10), each row is the meet of

three planes and is thus a vertex of the same tetrahedron.

The vertices abcd are respectively opposite the faces

ABCD since (2), (9) and (11) yield the incidence relations

[A B C D]T [a b c d] = 14 (13)

Additionally, K can be expressed in terms of k and by analogy

with (10),

K = [lb c dl Ic a di ja b dl lb a cl]T/ a b c d (14)

The collineation of points not only induces a collineation

of planes, but also induces a collineation of lines. Let x,y

and t,u be a pair of corresponding points

t=Kx u=Ky


Forming the join of each pair, then the lines are expressed

in ray coordinates by

p = Ixyl q = Itul


and the line p is transformed into the line q. Substituting

(15) into (16) yields

q = jKx Kyl

and then substituting (2) in (17) gives the nonsquare



BT x





C y

D y



The first coordinate of q is given by

q01 = ATxB y A yB x

= AT[xyT yxT]B

= A [p*]B


where [p ] is the skew-symmetric metrix given in (2.2.55)

where elements are p.ij = |x.y.|. Expanding the bilinear
ij 1 J

expression in (19) yields after some manipulations,

q01 = AB Tp. (20)

The remaining components of q are determined by analogy with

(20) which yields


and which is more concisely expressed by

q = Kp. (22)

The six 4x2 determinants in (21) have been formed from the

planes ABCD in the order 01, 02, 03, 23, 31, 12 which is

also the same order used in expanding each of the nonsquare

determinants into components, see (2.2.41).

Analogously, the induced collineation for axis coordinates

can be developed from a pair of corresponding planes,

X = kT Y = kU (23)

Forming the meet of each pair in axis coordinates,

L = ITUI M = IXYl (24)

then the collineation transforms line L into line M and using

(23) in (24) yields

M = |kT kUl. (25)

Expanding the terms in (.25) and using (11) gives a result

which is analogous to (21),

M = [jab| lacl lad Icdl Idb l Ibc ]T L (25)

or more concisely,

M = k L. (26)

Since a collineation preserves incidence properties, then if

p and L are incident then so are q and M,

LTp = 0 (27)

MTq = 0 (28)

Substitution of (22), (26) in (.28) yields

T ^T
L (k K)p = 0 (29)

which leads to a result analogous to (9),

^ ^-T
k = K (30)

upon comparing (27) and (29) for general intersecting lines

L,p. Although (30) is correct to a scalar multiple, it can

be shown that it is actually an algebraic identity when the

plane collineation is given by (2) and (14) and the point

collineation is given by (10) and (11).

Figure 2.3.1 is used to illustrate the geometric inter-

pretations of the induced ray and axis line collineations.

From (21), the collineation of ray coordinates is given by K

whose rows are the axis coordinates for the six edges of the

illustrated tetrahedron. From (25), the collineation of


Figure 2.3.1

IauI C

C I Ibcl

lab I b


\ I AD I

The tetrahedron associated with a collineation
labelled with a) point coordinates and ray
coordinates, b) plane coordinates and axis

axis coordinates is given by k whose rows are the ray coordi-

nates for the six edges of the tetrahedron. Thus the same

tetrahedron is intimately related to the four collineations

K, k, K, k.

This tetrahedron relationship leads to two important

algebraic identities for the induced line collineations,

^T ~ ^

^T ~ ^
k A k =


ja b c dIA.

. I3

Briefly, the identities (31) (32) are the conditions for the

rows of K and k to be edges of the tetrahedron in Fig. 2.3.1.

These identities are demonstrated by first substituting (30) in-

to (31) (32) to yield

|a b c dIA = k A kT

|A B C DjA = K A K

and where additionally

a b c dl A B C DI = 1.




Relations (33), (34) are easily proved by substituting in the

values of k and K in (21) and (25) and by noting that,





IabI A Icd| = a b c d




since the left sides are merely the Laplacian expansions of

the right sides by the first two columns. Any other product

such as

T a
ab| A ad| = Ja b a d| = 0


clearly vanishes since two columns are identical.

Using incidence properties, (31) and (32) are now veri-

fied to a scalar multiple. Since p, q, L, M are lines then

they satisfy the identical relations given by (2.2.69),

T ~
p A p = 0

T q
q A q = 0

T ~
L A L = 0

MT AM =0.

Substituting (22), (26) into(.40) yields

T T = 0
p (K A K)p = 0

T ^T -~
L (k A k)L = 0.

Comparing (39) and (41) for independent p and L yields (31),

(32) to a nonzero scalar multiple,

^T A ^= A
K A K = pA

^T A ^ ~
k A k = p]A.

Equation (42) (or (31), (32)) is important since it

yields the conditions which are necessary for a 6x6 matrix

to represent an induced projective collineation of lines.





A 6x6 matrix contains 36 elements and since only the ratios

are significant there are 35 independent parameters. In

(42), either of the equations represents only 21 different

scalar equations since the matrix equations are symmetric.

Because these equations are not all homogeneous, the scalar

multiplier p is eliminated by considering the 20 ratios of

equations (in a manner analogous to (3)). Thus the 35

parameters are related by 20 constraint equations to yield

35-20=15 independent parameters to describe an induced

projective collineation of lines. This is in agreement with

the projective point and plane collineations whose associated

4x4 matrices yield 15 independent ratios.

A linear combination of lines, as previously defined by

p = X q + Ar (2.2.75)

P = XQ +. AR (2.2.76)

is in general a screw. The induced projective collineation

of screws in space is identical to that for lines by way of


Kp = XAKq + A Kr kP = 1 kQ + A kR. (43)

If a 6x6 matrix does not satisfy (42) then it is not

an induced projective collineation of lines. It is possible

for this to occur in two ways. Firstly, if the rows of the

matrix all represent lines then they cannot form a tetrahedron.

Secondly, if any row is a screw then the matrix is not an

induced collineation. General transformations of this type

have been investigated by Ball [1900], which he called

homographic transformations. It should be noted that in

the literature the terms homographic transformations and

projective transformations are often used synonymously but

are used distinctively here. Effectively, Ball treated

screws as points in a projective five-space and therefore

a homographic collineation is the most general one-to-one

linear transformation of these points.

It is not difficult to verify that nonsingular pro-

jective point and plane collineations each form a group

of transformations under the operation of composition or

matrix multiplication. Nonsingular induced projective line

collineations also form a group of transformations and it is

useful to demonstrate the property of closure. Using ray

coordinate transformations let K, J be two nonsingular
collineations and it is necessary to show that KJ is also

such a collineation, i.e. it satisfies (42),

^^ ~ ^^ ^T ^T ~ ^ ^
(KJ) A (KJ) = J (K A K)J

^T ^
= pj A J

= iXA (44)

where yX is a nonzero scalar multiplier.

Since induced collineations constitute a group, then

it follows from (42) that the bilinear forms,

T ( T
p A q ,P A Q (45)

and in particular the quadratic forms

T ( T
p A p AP (46)

are invariant expressions with respect to induced collinea-

tions. From linear principles, this is true whether p,q

(P,Q) are lines or screws. The forms (45) are often

referred to as the mutual moment of two lines or screws

although no metrical connotation is implied here. When

screws are considered as points in a five-space then the

lines of three-space are represented as points on the

quadric surface

T pT (
p A p = 0 P AP= 0 (47)

which is sometimes called a Grassmannian, Hodge and Pedoe

[1952]. In the group of homographic transformations of

screws (i.e. nonsingular 6x6 matrices), induced projective

line transformations constitute a subgroup which leaves the

quadric (47) invariant (transforms into itself).

Projective transformations are classified as either

collineations or correlations. Correlations are linear

one-to-one transformations which map each element into a

dual element. Since the product of two correlations is a

collineation, it follows that correlations do not possess

the property of closure and thus do not form a group.

However, collectively collineations and correlations form

the group of projective transformations.

In the development of correlations it is useful to

first define two distinct spaces where elements in one

are denoted by a prime. Consider the incidence of a

point x on a plane T,

T x = 0

and the correlation,

X = Ax t = XT.



Since projective transformations must preserve incidence

relations then,

t X = T (X A)x = 0


and by comparing (47) and (49) then to a scalar multiple,

S= A-T
X = A A


The inverse transformations between the two spaces are given

by inverting (48),

T = X t'

, x =A X


or equivalently using (50),

T = A t
T = A t

x = T
1 x = X.


Figure 2.3.2 illustrates the mapping between the two spaces

described by (48) and (52). It is noted that the two point

Figure 2.3.2 The transformation of two distinct spaces under
a correlation.

to plane transformations (solid lines) are transposes of

each other as are the two plane to point transformations

(dotted lines).

Correlations of a single space onto itself may be

deduced by allowing the two distinct spaces to coincide.

For the correlation to be well-defined it is necessary that

TA = A = XT (53)

for which either i = -1 and A,X and skew-symmetric or p = 1

and A,X are symmetric. Skew-symmetric correlations are

called null polarities and since the matrix is of an even

order it is generally nonsingular. Null polarities have

many interesting properties, especially in relation to the

linear complex, Jessop [19031, Busemann and Kelly [1953].

Symmetric correlations are referred to as polarities and are

used to establish metrics in Section 3.1. Generally, the

only correlations that are employed here subsequently are


The development of polarities (or more generally correla-

tions) is facilitated by introducing the polarity I ,

I I = I (54)
n n n

where I is the nxn identity collineation. A polarity may
be expressed as a product of In and a symmetrical collinea-

tion k,K,

A = I4k = kI4 (54)

X = I4K = KI4. (55)

In this manner the results obtained for collineations may be

applied directly to polarities. The induced polarity of

lines corresponding to k and K is respectively

I6k = kI6 (56)

bA A
I6K = KI (57)

Analogous to (42), induced polarities have the tetrahedron

property. For example, using (56) and (42)

~ ^ T ^ ^T T
(I6k) A (I6k) = k (I6 A I6)k

^ T ^
= k Ak

= pA. (58)

In Chapter 3 it is shown that polarities may be

employed in the development of Euclidean and non-Euclidean

geometries using Cayley's conception of the Absolute.



Space is another framework which we impose on the
world. Whence are the first principles of geometry
derived? Are they imposed on us by logic?
Lobatschewsky, by inventing non-Euclidean geometries,
has shown that this is not the case. Is space
revealed to us by our senses? No; for the space
revealed to us by our senses is absolutely different
from the space of geometry. Is geometry derived from
experience? Careful discussion will give the
answer--no! We therefore conclude that the principles
of geometry are only conventions; but these
conventions are not arbitrary, and if transported into
another world (which I shall call the non-Euclidean
world, and which I shall endeavor to describe), we
shall find ourselves compelled to adopt more of them.

In mechanics we shall be led to analogous conclusions,
and we shall see that the principles of this science,
although more directly based on experience, still
share the conventional character of the geometrical
postulates. So far, nominalism triumphs; but we now
come to the physical sciences, properly so called, and
here the scene changes. We meet with hypotheses of
another kind, and we full grasp how fruitful they
are. No doubt at the outset theories seem unsound,
and the history of science shows us how ephemeral they
are; but they do not entirely perish, and of each of
them some traces still remain. It is these traces
which we must try to discover, because in them and in
them alone is the true reality. H. Poincare [1905,
pp. xxv-xxvi]

Cayley has shown how metrical concepts may be

introduced into geometry on a purely projective basis.

That is, a figure such as a quadric surface is designated

as a fixed reference, the Absolute, and metrical

properties, are those properties of figures which take on

significance in relation to the Absolute. This is the

starting point for the systematic development of metrical

geometries in Section 3.1. Metrical relations are developed

using projective coordinates and hence the seemingly

self-contradictory name of projective metrics. This

section simultaneously treats the common properties of

hyperbolic, Euclidean and elliptic geometries in a general

manner using the notion of an Absolute polarity as an

invariant connection of dual elements in space. Using a

definition by Clifford, an analytical generalization for

determining the pitches and axes of screws is given which

apparently may be also found in Buchheim [1884b]. Metrical

collineations are defined as those which leave the Absolute

invariant and form a subgroup of projective collineations.

Norms are then introduced as functions of the Absolute and

enables the development of metrical coordinates where the

components themselves are significant not just their

ratios. Elements of projective space are then assigned a

norm of unity although, in the general case, this leads to

two sets of metrical coordinates for an element which

differ in sign. Elements with a nonunity norm are defined

as new types of space elements which have an associated

weight or magnitude.

Since the properties of the various metrical

geometries vary considerably, Section 3.2 deals exclusively

with elliptic geometry. First, the elliptic polarity is

introduced which has a close connection with interpreting

the coordinates of a space element in terms of dual

coordinates and forms a basis for some of the developments

in Chapter 4. It is shown that properties which are often

erroneously associated with n-dimensional "Euclidean"

spaces, such as orthogonalityy," are actually properties of

elliptic space when homogeneous coordinates are utilized.

This is particularly important with respect to some later

developments dealing with the orthogonalityy" of screws

which actually signifies that two screws are elliptic

conjugates. Specializing a previous formulation, it is

shown that screws in elliptic space have two axes which are

elliptic polars and two respective pitches that are

reciprocal. These results agree with ones given by

Clifford [1873] along with the notion that screws of pitch

1 have properties of free vectors. Other investigations

dealing with screws in elliptic space are given by Buchheim

[1884a, 1884b], Cox [1882], Heath [1885] and Ball [1900].

In collineation form, the elliptic polarity of lines

and screws is similar in appearance to the important

identical relation between ray coordinates and axis

coordinates which is a symmetrical correlation. Table

3.2.1 summarizes a number of relations in elliptic geometry

which appear very similar to expressions in projective

geometry. This exemplifies why it is necessary to have an

unambiguous notation to delineate collineations from

correlations and ray coordinates from axis coordinates. A

systematic development commencing with projective geometry

makes it possible to delineate the distinct but similar


Euclidean geometry is distinguished by the fact that

the Absolute polarity is singular and consequently many

relations must be approached as limiting cases. As shown

in Section 3.3, the singularity introduces an asymmetric

character to dual expressions which does not exist for

projective or elliptic geometry. The general formulation

in Section 3.1 is specialized for an interesting

development of the pitch and axis of a screw which is

expressible as a linear combination of a unique line and

its polar. It is also shown that what is often referred to

as the "dual" operator w, where w2 = 0, is merely the

Euclidean polarity expressed in form amenable to

biquaternions. Based on the Euclidean Absolute, norms are

defined which are then used to introduce Euclidean metrical

coordinates. Elements of projective space are assigned

norms of unity and points are given a unique set of

coordinates, unlike planes, lines and screws which have two

sets of coordinates that differ in sign. New space

elements are defined which have the property of weight or

magnitude since their norms are nonunity.

It is typical to study a geometry in terms of

examining its group of transformations. However, the group

of transformations in Euclidean space always appears to be

an entity given a priori from which subsequent geometric

properties are then derived. Here, beginning with the

general group of projective collineations and Cayley's

Absolute, it is shown how the corresponding group of

Euclidean transformations may be deduced as those which

leave the Euclidean Absolute invariant. Although the

procedure is not complex, it appears to have been

previously overlooked.

In Section 3.4 polar and axial vectors in Euclidean

space are introduced by way of Klein's second principle

given at the beginning of Section 2.3. Then using

Hamilton's vectors, a polar vector is defined as the

difference between two points. By introducing vectors and

making the point a more fundamental element in Euclidean

space than the plane, the ambiguity of signs for the

metrical coordinates of planes, lines and screws is

examined. The ambiguity is only resolvable for new space

elements that are then introduced namely, plane-sects,

geometric couples, line vectors and screw vectors which are

all distinguished by a magnitude and an unambiguous

associated direction. The ray coordinates and axis

coordinates of line vectors and screw vectors are then

expressed in a formulation which is typical of modern

presentations, especially the ones using dual vectors such

as in Brand [1947]. Finally, in application to the area of

mechanics, twists and wrenches are introduced along with

the formulation of virtual work. When a body is in static

equilibrium under impressed wrenches its virtual work

vanishes, a property which is shown to be analogous to the

projective property of incidence.

Section 3.1 Projective Metrics

I remark in conclusion, that, in my own point of
view, the more systematic course in the present
introductory memoir on the geometrical part of
the subject of quantics, would have been to ignore
altogether the notions of distance and metrical
geometry; for the theory in effect is, that the
metrical properties of a figure are not the pro-
perties of the figure considered per se apart
from everything else, but its properties when
considered in connexion with another figure,
viz the conic termed the Absolute. The original
figure might comprise a conic; for instance, we
might consider the properties of the figure
formed by two or more conics, and we are then
in the region of pure descriptive geometry: we
pass out of it into metrical geometry by fixing
upon a conic of the figure as a standard of
reference and calling it the Absolute. Metrical
geometry is thus a part of descriptive geometry,
and descriptive geometry is all geometry, and
reciprocally; and if this be admitted, there is
no ground for the consideration, in an intro-
ductory memoir, of the special subject of
metrical geometry; but as the notions of dis-
tance and of metrical geometry could not,
without explanation, be thus ignored, it was
necessary to refer to them in order to show that
they are thus included in descriptive geometry.
Arthur Cayley [1859, pp. 592]

In the preceding chapter, it has been demonstrated that

homogeneous coordinates may be introduced into geometry

without recourse to a form of measure, or in other words, a

metric. Homogeneous coordinates are well-suited for examin-

ing incidence relations which comprises the domain of projec-

tive geometry, or "descriptive geometry" as Cayley referred

to it. Metrical geometries such as elliptic, Euclidean and

hyperbolic may be developed from projective geometry by

establishing one or more figures as a fixed reference, which

Cayley called the Absolute. In three-dimensional space,

various metrical or so-called Cayley-Klein geometries may

be developed by defining the Absolute to be a point locus

together with a plane envelope of a quadric surface.

Projective homogeneous coordinates may be adapted for

metrical geometries and in doing so they may also be en-

dowed with the additional property of magnitude, which is

a function of the Absolute.

It is most interesting to note, that prior to the

landmark paper "A Sixth Memoir on Quantics," Cayley [1859],

projective geometry was considered merely to be a somewhat

poorer subject in what was then the all-pervasive geometry

of Euclid. Initially, not even Cayley recognized the scope

of his dictum, "descriptive geometry is all geometry," since

he had only considered the geometries known to him at that

time, Euclidean and spherical, the former of which he pre-

sented for only one and two dimensions. It was left to

F. Klein [1871, 1873], some twelve years hence, to demon-

strate that the elliptic geometry of Riemann and the hyper-

bolic geometry of Lobatchewsky and Boylai, the so-called

non-Euclidean geometries, may be developed by selecting the

Absolute to be respectively an imaginary or real figure.

In this section, metrical geometries are developed in

a general format which may then be specialized to yield

hyperbolic, Euclidean and elliptic geometries, the latter

two of which are investigated in the succeeding sections.

Instead of commencing the development here with an Absolute

quadric, it is preferred to first establish an Absolute

polarity and its adjoint, Coxeter [1965].


A correlation is a linear transformation which maps

each element of space into a dual element. A polarity is a

symmetric correlation which can be represented by a symmetri-

cal matrix and may be utilized to establish an invariant

connection of space between dual elements. This is essen-

tially equivalent to the Cayley-Klein development since

there is generally a one-to-one relationship between polari-

ties and quadric surfaces. It should be particularly noted

that since metrical geometries are specializations within

projective geometry, that metrical geometries must preserve

projective properties, in particular, relations of incidence.

The diagonal polarity T and its adjoint H where

1 .
TT = (1)
S 1



n = (2)

are used to establish elliptic, Euclidean and hyperbolic

geometries for = 1,0,-l respectively. Since the polarities

become singular in the Euclidean case, the appropriate de-

velopment considers the limiting case 0.

For the polar relations,

X = 7x (3)

x = rX (4)

the plane X is said to be the polar of point x and the

point x is said to be the pole of plane X. Two points x,y

(two planes X,Y) are said to be conjugate when each is in-

cident with the other's polar (pole) and

x T y = 0 (5)

XT Y = 0. (6)

Sometimes conjugate points and conjugate planes are referred

to respectively as H-orthogonal or 'r-orthogonal. The Absolute

quadricc) is defined as the locus of self-conjugate points

and the envelope of self-conjugate planes,

T ~
x H x = 0 (7)

XT i X = 0. (8)

The polarity of points and planes induces a correspond-

ing polarity of lines. In Fig. 3.1.1, the join of points x,y

is the line p and the meet of their polar planes X Y defines

the polar line P Alternatively, the meet of planes T,U
also defines the same line P and the join of their poles t ,u

also defines the same polar line p Therefore P and p ,

the axis and ray coordinates of the polar line, are given

respectively by the nonsquare determinants (see Section 2.2),

P = IHx Hy|


Figure 3.1.1 A pair of polar lines.

P = IX TTY (10)

Substituting, (1), C2) in (9), (10) and expanding yields

the induced polar relations

P = Fp (11)

p = YP (12)


F = (13)

3 3

Y = (14)

I3 -

In (13) the common factor c has been removed which is necessary

for the Euclidean case where E 0.

Alternatively, (13) may be derived by substituting the

relations between ray and axis coordinates (2.2.48), (2.2.49)

in (14) to yield

r = A Y A. (15)

Conversely, (15) can be rearranged as

~ ~T ~ ~
Y = A r A. (16)

Since r, y are symmetrical and are adjoints, then to a scalar

multiple E,

F Y = 16 (17)

which may be used with (15), (16) to yield the tetrahedron

relationships (see Section 2.3),

A F = A (18)

Y A y = A. (19)

Two lines p,q (P,Q) are said to be conjugate when each

is incident with the other's polar line and

T 0
p F q =0 (20)

T ~
P Y Q = 0. (21)

Alternatively, lines which are conjugate are sometimes referred

to as F-orthogonal or y-orthogonal. Lines which are self-

conjugate are incident with their own polars and their assem-

blage forms the tangent lines to the Absolute. In line

coordinates, the Absolute is given by the quadratic complex,

Jessop [1903],

T ~
p F p = 0 (22)

T (
P Y P = 0. (23)

As previously defined, a linear combination of lines is

in general a screw

. A r P = 1 Q + .
n 1

n R. (2.2.65)

p = \ q + .


Analogous to (11), (12), the polar of a screw is given by

P = q + X r ,

p =X1 y Q + n y R (24)

and two screws which satisfy a bilinear relation of the

form (20) or (21) are also said to be conjugate.

In describing screws in elliptic space, Clifford [1873,

pp. 193] asserted that a screw can be expressed uniquely as

the sum of a line and its polar line and that this polar pair

represents the axes of the screw. Here this result is

generalized and, using ray coordinates, a screw p is expressed

as a linear combination of a unique line q and its polar q'

i i
p = Xq + X q (25)

where it is necessary to determine the scalars X,X and the
i I
pair of polar lines q,q The line q may be expressed in

ray coordinates by

q = A Q = A F q (26)

which is substituted in (.25) to yield

p = (XI6 + X A F)q. (27)

Provided that the matrix in (27) is nonsingular, then using

(13) q is easily solved for and

q = (XI X A F)p. (28)
(X2_X,2E) 6

Since q is a line, it must satisfy the identical relation

q A q = 0 (29)

which is used to eliminate q. Substituting (28) in (29)

yields, after some rearranging,

1 T ~ 2 ~
[(PT A p)X 2(pT p)XX + e(p Tp)A ]= 0 (30)
(\2 X 2 E)2

For e = -1,0,1, (30) may be solved for the ratio X:X which

is then substituted back in (28) to determine q and subse-

quently q' from (26). By extension of Clifford's definition,

the ratios X :X resulting from (30) are called the pitches

of the screw with respect to its axes. In the following

sections the solution is detailed for both elliptic and

Euclidean geometries.

Metrical collineations are defined as those which leave

the form of the Absolute invariant. For points and planes,

the projective collineations of space (see Section 2.3) are

given by

y = Kx Y = kX. (31)

In the image space, the Absolute must be expressible in the

same form as (7), (8) and

y H y = 0 (32)

Y iT Y = 0.


Substituting (31) in (32) and (33) yields

T T~
x (K 1 K)x = 0 (34)

X (k T k)X = 0. (35)

Comparing (34), (35) with (7), (8) for general x,X gives

K TK = p H (36)

k 7rk = TT (37)

where p is a nonzero scalar multiple. Equations (36), (37)

express the required conditions for a nonsingular collineation

to leave the Absolute invariant. Collineations which satisfy

(36) or (37) are referred to as metrical collineations and

they form a subgroup within the general group of projective

collineations. A projective collineation is expressed using

16 elements and since only the ratios are significant, it is

thus determined by 15 independent parameters. However, a

metrical collineation must also satisfy (36) or (37), and

by symmetry either matrix equation represents a set of 10

nonhomogeneous scalar equations from which p may be eliminated

by considering the 9 ratios of equations. Thus the 15

parameters of a general projective collineation are related

by 9 constraint equations to yield 15-9=6 independent para-

meters for the specification of a metrical collineation.

Briefly, for the induced collineations of lines or

screws, let K and k be respectively the ray and axis


q = Kp (38)

Q = kP. (39)

The equations of the Absolute are

T~ T ^T ^4
q Fq = p (K r K)p = 0 (40)

QT Q = P(k y k)P = 0 (41)

and comparing with (22), (23) yields

K r K = iF (42)

k y k = vy (43)

which are the conditions for an induced collineation to be

a metrical collineation.

The establishment of an Absolute enables the introduction

of a projective norm which is useful in the development of

metrical coordinates. For points and planes respectively,

the norms are given by the scalar functions

j|x 1| = (x T H x)2 (44)

|X 1 = (X 7T X)2 (45)

and for both lines and screws, the norms in terms of ray and

axis coordinates are

Hp 11= CpT r p)



IlP |I = (pT Y P) (47)

Normed elements are defined as elements whose norms are


Once an Absolute is established, it is possible to re-

move the restriction from homogeneous coordinates that only

the ratios are significant. This transition from projective

homogeneous coordinates to metrical homogeneous coordinates

is initiated by first normalizing all elements. For example,

the projective homogeneous coordinates of a point x are

normalized by

x_ x (48)
iIx c(x x)

Replacing x with a scalar multiple Xx, which also designates

the same point in projective coordinates, yields

Xx Xx = + x (49)
11 x 1 1 I | X ii | x 1i x

where the norm of the scalar is

XI II = (X ) (50)

Therefore in general, the sets of projective homogeneous

coordinates which correspond to a single element and differ

by a nonzero scalar factor, are transformed by the normaliza-

tion into two sets of homogeneous metrical coordinates which

designate the same element yet differ in sign. By including

further constraints, it is possible to resolve this

ambiguity of signs completely for hyperbolic geometry,

only partially for Euclidean geometry and not at all for

elliptic geometry, Busemann and Kelly [1953]. For the

case of elliptic geometry, this situation has interesting

consequences which are detailed in Section 3.2 together

with the Euclidean case in Section 3.3.

It is convenient to sometimes refer to elements of

projective space, i.e. points, planes, lines and screws, as

unit or unweighted elements since in terms of metrical

coordinates they have unity norms. Since all the elements

of projective space have a representation in terms of

normalized metrical coordinates, it is reasonable to assign

a new meaning to metrical coordinates with a nonunity norm.

Such coordinates are said to represent weighted elements,

Forder [1940], which are purely metrical in nature, i.e.

they have no projective representation and are thus a new

species of space element. Every weighted element may be

represented as a scalar multiple of an unweighted element,

e.g. for a weighted point

x = Ix (51)
1Ix I1

where the weight is simply the norm. Oftentimes the terms

weight and magnitude are used synonymously. A common physi-

cal example of a weighted element is a point mass.

Metrical collineations have already been defined as

a subgroup of collineations which preserve the form of the

Absolute and have been formulated using projective coordi-

nates in (36), (37), (42), (43). However, these formula-

tions only define collineations uniquely to a scalar multi-

ple and are thus not suitable when it is desired to employ

metrical coordinates with associated weights. For this

purpose, it is required that a collineation also preserves

the norms of weighted elements. Letting K and k be respec-

tively point and plane collineations, then it is required

that the relations

K I K = T (52)

k T k = 7 (53)

be satisfied identically, not just to a scalar factor.

For distinctiveness, a metrical collineation which

preserves the norm may be referred to as a unit or un-

weighted collineation or if the context is clear, simply as

a collineation. Weighted collineations do not preserve the

norm and are not considered here further. Since a unit

collineation may not be multiplied by an arbitrary nonzero

scalar factor, it represents 16 parameters. Further, because

(52) and (53) are symmetrical, either relation represents 10

independent scalar equations and thus, a metrical collineation

is specified by 16-10=6 independent parameters, a result

that agrees with the previous one employing projective


For the induced unit collineations of lines and

screws, let K and k be respectively the ray and axis trans-

formations then it is required that the relations

KT ~ ^
K F K =



be satisfied identically, not just to a scalar factor. It

may be demonstrated that the collineations induced from unit

collineations are also unit collineations. First, taking

the determinants of (52) (54) yields

k y k = y

IKI2 = 1

"K2 = 1

which are the


the identical

expressed as

^T ~ ^
K AK =

k Ak =k

Ik2 = 1

|k = 1



necessary and sufficient conditions for the

to be unweighted. Using (2.3.2) and (2.3.4),

relations (2.3.31) and (2.3.32) may be


k A

and taking the determinants of these equations yields

K 2 6 K

kl2 6
k = *k





Substituting (56) in (60), (61) yields the desired relations


The properties of hyperbolic, Euclidean and elliptic

space vary considerably and the preceding general analysis

only uncovers relations which are common to all. In the

following two sections, specific properties are detailed

which, in particular, distinguish elliptic and Euclidean


Section 3.2 Elliptic Geometry

Consider any vertical line, and a series of hori-
zontal planes cutting it at right angles. In
ordinary or Euclidean geometry these planes
intersect on the horizon, which is a straight
line infinitely distant. In the geometry of
a space of constant positive curvature, or
elliptic geometry, the horizon is at a certain
finite distance in all directions from the
vertical line with which we started; it belongs
to that particular line, which is called its polar,
and is not the same for all vertical lines.
Although it appears to be a great circle when
viewed from the neighborhood of its polar, yet
if we were to go to it and examine it we should
find it straight. Points of it which are in
opposite directions from a point on the polar
are really identical; and every straight line in
this space resembles a circle in being of finite
length, so that if we travel far enough along it
we shall arrive at our starting point. Every
straight line has a polar line, which is the in-
tersection of all planes at right angles to it.

Let us take a very small circle on a sphere, and
suppose it to expand, keeping always the same
centre. At the beginning the circle will be
concave inside and convex outside; but when the
expansion has gone on far enough it will become
a great circle of the sphere, which is of the
same shape on both sides, or is straight so far
as the surface of the sphere is concerned. So
if in Euclidian space we take a sphere and
suppose it to expand, keeping always the same
centre, it will continue to be concave inside
and convex outside so long as it is finite; but

when the radius has become infinite, the inside
in one direction is the same as the outside in
the opposite direction, opposite points being
identical; thus the sphere is of the same shape
on both sides, or is a plane, viz., the plane at
infinity. In elliptic space, just as in geometry
on the surface of a sphere, this takes place for
a finite length of the radius, not for an
infinite length; for every point there is a sphere
having its centre at that point, which is also a
plane. Or, which is the same thing, every point
has a polar plane which is the locus of all
points situate at a certain distance from it;
this distance is called a quadrant. So also
every plane has a certain point, called its pole,
which is distant a quadrant from every point in
the plane. All lines and planes perpendicular to
the plane pass through its pole, and conversely.
The polar lines of all lines in the plane pass
through its pole, and so do the polar planes of
all points in the plane.

When two lines are polars of one another, every
point of one is distant a quadrant from every
point of the other; hence the polar planes of
all points on one pass through the other.
Every line which is at right angles to one
meets the other, and conversely. W.K. Clifford
[1876, pp. 390-391]

In relation to hyperbolic and Euclidean geometries,

elliptic geometry has the simplest and most symmetrical

properties. The results of the previous section are

recounted for elliptic geometry by setting =l.

The elliptic polarity for points and planes is

nI = 14 (1)

T1 = I4 (.2)

and the polar relations are given by

X = HIx = I4x (3)

x = T1 X = I X.

Two points or planes which are elliptic conjugates satisfy


x 1 y = x 14 y = 0 (5)

xT iT Y = XT I4 Y = 0. (6)

The locus of self-conjugate points and the envelope of self-

conjugate planes define the elliptic Absolute

x H x = x I x = 0 (7)

X T X = X I X = 0. (8)
1 4

For ray and axis coordinates, the induced elliptic

polarity is given by

r1 = 16 (9)

yl= 16 (10)

and the polar relations are

P = F1 p = 16 p (11)

p = Y P = 6 P. (12)

Two lines which are elliptic conjugates satisfy

PT 1 q pT i6 q = 0 (13)

T T (
P Y Q = 6 Q = 0 (14)

and the line equations of the elliptic Absolute are

T T (
p 1 p = p 6 p = 0 (15)

P Yl P = P i6 P = 0. (16)

By linearity, the elliptic polarity of screws is given by

P = 11 q + n r r = A I q + n 16 r (17)

P= X1 Y1 Q + X n Y R = A 16 Q + n 16 R (18)

and two screws which satisfy (13) or (14) are also said to

be elliptic conjugates.

For points, planes, lines and screws the elliptic

polarity is specified by a correlation which is the identity

matrix, either 14 in (1), (2) or 16 in (9), (10). Thus,

the elliptic polar of an element is determined by interpret-

ing its coordinates in terms of dual coordinates, (3), (4),

(11), (12), (17), (18). Two elements which are elliptic

conjugates (5), (6) (13), (14), are also said to be either

elliptic-orthogonal or more specifically prefixed by the

polarity such as for points, IHI-orthogonal or I 4-orthogonal.

From the equations of the Absolute, (7), (8), (15), (16) it

is clear that the points, planes and lines which form the


Absolute are not real and thus it is often referred to as

an imaginary or virtual quadric surface. The elements

forming the Absolute are said to be elliptic self-conjugate

or alternatively, elliptic-isotropic.

For lines and screws, it is very important to carefully

distinguish projective relations from the metrical elliptic

relations which are similar in appearance due to the ele-

mentary form of the elliptic polarity. The projective

correlation between ray coordinates and axis coordinates

P = Ap (2.2.60)

p = AP (2.2.61)

may be substituted in (11), (12) to express the elliptic

polarity as a collineation

P = 16 p = 16 A P = AP (19)

p = 16 P = I6 A p = Ap (20)

where the product of correlations I6A yields the collineation

A = (21)

Further, substituting (2.2.60) and (2.2.61) in (13), (14)

yields alternative formulations for elliptic-orthogonality,

T ~ T T~ ~ ^
p 16 q A 6 q = P A q = 0 (22)

P I6 Q = p A I6 Q = p A Q = 0. (23)

Table 3.2.1 summarizes the projective relations along

side the elliptic relations which are very similar in form.

As throughout, a tilda is used to signify that a transforma-

tion is a correlation, e.g. I 6, whereas a caret denotes a

collineation, e.g. 16. In the table, each expression con-

tains either a correlation or a collineation and its

correspondent contains the other. Further, for each pair

of expressions, one of the two screws (or lines) which

correspond are in dual coordinates while the other two are

in the same coordinates. Without a clear notational distinc-

tion between correlations and collineations and between ray

and axis coordinates these relations are easily confused

and misinterpreted.

It was first discovered by Clifford [1873], that in

elliptic space a screw p can in general be expressed as a

linear combination of a unique line q and its polar q ,

p = Xq + X q (3.1.25)

both of which are called the axes of the screw. The ratios
i i
X :X and X:X are respectively the pitches of the screw

with respect to q and q Setting E=l in the general formu-

lations (3.1.27), (3.1.28) and (.3.1.30) yields

p = (XI6 + X A)q (24)

Table 3.2.1 Projective and metrical relations
that are similar in appearance


1. Transformations of ray
and axis coordinates

P = Ap

p = AP

p = I6 p

P = I P


1. Transformations of the
elliptic polarity

i -
p = Ap

P = AP

P = I6 p

p = I6 P

2. Reciprocity

2. Elliptic-orthogonality

p A q = 0

P A Q = 0

T ^
p I6 Q = 0

T 6 = 0
P1 I6 q = 0

T ^

T ^
p A Q= 0

T ~
P I6 q = 0

q = 6 [ X A]p
(A2 p) 2)2 I p

1 [(pTAp)X 2 2(T I6 p)XX + (pTAp)X1 ]= 0
( -X2 2)

2 i2
Assuming first that XA \

be expressed by

2 1
A -2bAA + X

= 0

T p 0 then (26) may
and p A p 3 0 then (26) may



T ~
p 16 p
b = .
p Ap

The solutions to (27) are


2 12
X = b + (b -1) 2

or equivalently

1 = i, 1


2 1)
u = b + (b 1)2

2 1
, b (b 1)2

Substituting in turn the roots (30) back in (25) yields the

corresponding pair of axes


2) 6
(1-- ) 6







q/P 7 2 E16 PAJp (33)

and since

(A)q = ql/ (.34)

the two axes are elliptic polars where the subscripts denote

the associated pitches.

When p A p = 0, p is a line and assuming that
p I6 p i 0 in (26), then either X=0 or \ =0 and thus from
(3.1.25), respectively either p=X q or p=Xq.
2 !2
The exceptional case occurs when X =X and thus the

pitch of the screw is either +1 or -1. The form of this

screw p may be determined by first taking the polar of


p = Xq + A q (35)

and then subtracting and adding in turn (3.1.25) and (35)


(p -p) = (X-X ) (q -q) (36)

(p +p) = (+X ) (q 4-q) (37)

When the pitch is +1 then (.36) yields

p=p or p=Ap


and thus the first and last three components of p are the

same. When the pitch is -1 then (37) yields

p = -p or p = -Ap

and the first and last three components are the same except

that they have opposite signs. As a consequence of these

special forms, screws of pitch +1 may be expressed in an

infinite number of ways as the sum of a line and its polar.

The axes of such screws are thus indeterminant and for this

reason Clifford referred to them as vectors, a right vector

when the pitch is +1 and a left vector when the pitch is -1.

Clifford also made the observation that a screw p, of any

pitch, may be expressed as the sum of a right vector and left

vector uniquely,

p = (P+ + P-P (39)
2 2

Vectors are closely related to right and left parallel lines

in elliptic space which are also called paratactics,

Sommerville [1929], or Clifford parallels, but are not

developed here.

The transition from projective coordinates to elliptic

metrical coordinates is made by first introducing elliptic

norms. By setting =l in (3.1.44) (3.1.47), the elliptic

norms for points, planes and lines or screws become

x 1 = (xT 4 x) (40)

1|X I, = (XT I X) (41)

||p 1| = (pT I6 p) (42)

11P I = (PT I6 P) (43)

As discussed in Section 3.1, by normalizing projective coordi-

nates, elements in projective space become unit or unweighted

elements in terms of metrical coordinates. In general, an

unweighted element has two representations in metrical

coordinates which differ only in sign. For elliptic space,

there is no additional criteria that can be imposed which

enables a definitive selection of sign without ambiguity.

The resolution to this predicament is merely to associate

the pair of coordinates with each element, e.g. x and -x for

a unit point. For the elliptic plane this has an interesting

consequence since it enables modeling in Euclidean space by

a unit sphere. That is, a point on the elliptic plane is

modeled either by a pair antipodal points on a unit sphere

or equivalently by a diametrical line through these points.

A line on the elliptic plane is represented by a great

circle. The analogy in space is somewhat more involved,

however Clifford's description given at the beginning of this

section is highly suggestive.

Elliptic collineations are defined as the subgroup of

projective collineations which leave the elliptic Absolute

invariant. In terms of metrical coordinates, the unweighted

collineations for points, planes and lines or screws satisfy

the relations given by setting E=l in (3.1.52) (3.1.55),

K I1 K = I4 (44)

kT 14 k = I4 (45)

K I K = I (46)
6 6

k I k = I (47)
6 6

Analogous to the convention for elements, a collineation

has two distinct representations which differ by sign,

e.g. K and -K for point collineations, see Busemann and

Kelly [1953]. From the above relations it is easy to de-

duce that the determinant of an elliptic collineation is

+1 or -1. However, both representations of a collineation

have the same determinant since the matrices are of an even

order, e.g. -KI = (-1)41K = K|. Further, it is readily

deduced from (44) (47) that the inverse of an elliptic
-1 T
collineation is equal to its transpose, e.g. K = K In

the literature such matrices are usually referred to as

orthogonal or orthonormal. However, for distinctiveness

elliptic collineations are referred to here as elliptic-

orthogonal or prefixed with the polarity such as 14


Section 3.3 Euclidean Geometry

We shall find throughout this period, that almost
every important proposition, though misleading
in its obvious interpretation, has nevertheless,
when rightly interpreted, a wide philosophical
bearing. So it is with the work of Cayley, the
pioneer of the projective method.

The projective formula for angles, in Euclidean
Geometry, was first obtained by Laguerre, in
1853. This formula had, however, a perfectly
Euclidean character, and it was left for
Cayley to generalize it so as to include both
angles and distances in Euclidean and non-
Euclidean systems alike.

Cayley was, to the last, a staunch supporter of
Euclidean space, though he believed that non-
Euclidean Geometries could be applied, within
Euclidean space, by a change in the definition
of distance. He has thus, in spite of his
Euclidean orthodoxy, provided the believers
in the possibility of non-Euclidean spaces
with one of their most powerful weapons. In
his "Sixth Memoir upon Quantics" (1859), he set
himself the task of "establishing the notion of
distance upon purely descriptive principles."
He showed that, with the ordinary notion of
distance, it can be rendered projective by
reference to the circular points and the line
at infinity, and that the same is true of angles.
Not content with this, he suggested a new
definition of distance, as the inverse sine or
cosine of a certain function of the coordinates;
with this definition, the properties usually
known as metrical become projective properties,
having reference to a certain conic, called by
Cayley the Absolute. (The circular points are,
analytically, a degenerate conic, so that
ordinary Geometry forms a particular case of the
above.) He proves that, when the Absolute is an
imaginary conic, the Geometry so obtained for two
dimensions is spherical Geometry. The correspon-
dence with Lobatchewsky, in the case where the
Absolute is real, is not worked out; indeed there
is, throughout, no evidence of acquaintance with
non-Euclidean systems. The importance of the
memoir, to Cayley, lies entirely in its proof
that metrical is only a branch of descriptive

The connection of Cayley's Theory of Distance
with Metageometry was first pointed out by Klein.
Klein showed in detail that, if the Absolute be
real, we get Lobatchewsky's (hyperbolic) system;
if it be imaginary, we get either spherical
Geometry or a new system, analogous to that of
Helmholtz, called by Klein elliptic; if the
Absolute be an imaginary point-pair, we get
parabolic Geometry, and if, in particular, the
point-pair be the circular points, we get ordinary
Euclid. .

Since these systems are all obtained from a
Euclidean plane, by a mere alteration in the
definition of distance, Cayley and Klein tend
to regard the whole question as one, not of
the nature of space, but of the definition of
distance. Since this definition, on their
view, is perfectly arbitrary, the philosophical
problem vanishes--Euclidean space is left in
undisputed possession, and the only problem
remaining is one of convention and mathematical
convenience. Bertrand Russell [1897, pp. 29-30]

In contrast to elliptic space, for which the elemen-

tary form of the polarity yields many symmetrical relations

amongst dual elements, the singular nature of the Euclidean

Absolute yields highly unsymmetrical dualistic properties.

As a consequence, various general formulas which may be

specialized for elliptic and hyperbolic geometry, do not

appear applicable to Euclidean geometry unless they are

treated as limiting cases where c-0. One example is the

formula for the distance between two points which yields

an indeterminant result unless an infinitesmal device is

utilized as in Klein [1908]. However, for most of the

developments here it is possible to set E=0 in the general

formulations of Section 3.1.

The Euclidean polarity for points and planes is given


I .
S = 1 (1)

T0 (2)

(As throughout, zero elements in an array are often denoted

by periods and I1 is used here in place of 1 for greater

symmetry.) The polar relations are

X = 10x = x =

x = 7TX = X =

where the notation X and x is introduced,

x = [xI x2 xT

x = [xi x2 x 3 .

From (3), for a point x not on the plane at infinity,

x # [0 xI x2 x3 T, its polar plane X is in fact the plane

at infinity, X = [x 0 0 0]T For a point x on the plane

at infinity, x = [0 x1 x2 x3] its polar plane X is in-

determinate since all of its components vanish, X = [0 0 0 0]T.

From (4), for a plane X which itself is not the plane

at infinity, X 3 [X0 0 0 0]T then its pole x is a point on

the plane at infinity, x = [0 X1 X2 X3] For X itself

being the plane at infinity, X = [x 0 0 0] its pole x is

indeterminate x = [0 0 0 0]T

Two points or planes which are Euclidean conjugates

satisfy respectively,

T ~ T 1
x 10 y = x y= x0 0 = 0 (7)

X T y=T 0 T Y = XT Y = 0. (8)


In (7) two points are Euclidean conjugates if either lies

on the plane at infinity. In (8), two planes which are

Euclidean conjugates are said to be Euclidean-orthogonal

and thus meet at right angles.

The angles of self-conjugate points and the envelope of

self-conjugate planes define the Euclidean Absolute

T ~ T 1 2
x I0 x = x x = x0 = 0 (9)

T ~ T 2 2 2
X TX 0 X = X = XX = X + X + X (10)

Equation (9) represents the plane at infinity taken twice

which is a rank one quadric locus. Equation (10) represents

what is referred to as the imaginary spherical circle since

it is the intersection of every sphere with the plane at

infinity and further, because it is a rank three quadric

envelope it represents a conic, see Klein [1908], Sommerville

[1934]. Thus unlike elliptic geometry, the locus and enve-

lope of the Euclidean Absolute are two distinct figures which

is a consequence of the Euclidean polarity being singular.

In terms of ray and axis coordinates, the induced

Euclidean polarity is respectively given by,

Fo = 3 (11)

YO = (12)

and the corresponding polar relations are

P = 0 p = =(13)

,. .l P 0

p = YO P = = (14)
I3 P P

where the 3xl arrays p, p P, P,0 0 are introduced,
I OT (15)

S= 01 02 03 = 23 P31 12T (15)

[T T T6)
P = 23 P31 P12 =-0 01 02 031 (16)

0 = [0 0 0Q]T. (17)

Considering firstly (14), it may be shown that the polar of

any line P not on the plane at infinity is in fact a line

on the plane at infinity, p = [0 0 0 P23 P31 P12]T Let the

line P be the meet of two finite planes, P = JXYI. The polar

line p is equivalent to the join of the two poles of X and Y

which are points on the plane at infinity, p = X Y and

thus represents a line on the plane at infinity.

This type of reasoning is not effective for (13). Let

line p be the join of two points p = |xyl and the polars

of x and y are both the plane at infinity. Since the meet

of any plane with itself vanishes identically, this does

not yield the line given by (13). It is not clear why this

synthetic argument fails. However, it should be noted that

(13) was derived from (3.1.13) where it was necessary to

delete a common factor E before e was set to zero to yield

(13). Alternatively, (13) may be derived directly from

(14) be multiplying throughout by A,

Ap = (A 18)

which yields the desired result

P = (19)

It also follows from both (13) and (.14) that the polar

of any line on the plane at infinity is indeterminate since

its coordinates vanish. Thus the Euclidean polar of the

Euclidean polar of a line or screw vanishes. This is exactly

equivalent to a double application of the Euclidean polar

operator w, where w =0, to a rotor or motor (line or screw)

and was invented by Clifford [1873] in the development of

biquaternions. For elliptic space, Clifford used the polar

operator w, where W2=1, which is equivalent to (.3.2.19) and

(3.2.20). Clifford did not discuss hyperbolic geometry in