Next: 3 Riemannian Geometry
Up: Axiomatic and Coordinate Geometry
Previous: 1 Introduction
2 The axiomatic approach
In order to avoid the many choices for the axiom of parallels
we look at those axioms that are satisfied by a convex region of
space (see Figure 1) in any axiomatic
geometry--Euclidean or not. The basic undefined entities are points,
lines and planes and the undefined relations among them are incidence
(to-lie-on) and separation (between-ness). The entities and relations
are made clear by the following axioms.
The geometry of a convex region
We begin with the axioms of incidence:
- Any two points lie on exactly one line and any line has at least
two points on it.
- Any three points not all on one line lie on exactly one plane
and any plane has at least three non-collinear points on it.
- If two points of a line are on a plane then every point of the
line is on the plane (we then say that the line is on the plane).
Then we have the axioms of dimension:
- There are at least four points which are non-collinear and
- If two planes meet then they meet in at least a pair of points.
The axioms of order and separation:
Of these the last axiom is rather special and is called Pasch's
axiom (see Figure 2). We shall see later that this is the one
property characterising planes in axiomatic geometry which shows that
there are no planes in general Riemannian geometry.
- If a point B is between the points A and C then A, B,
C are collinear and B is between C and A.
- If A, B, C are distinct collinear points then exactly one is
between the other two.
- If A and B are distinct points there is at least one point
C so that B is between A and C and a point D so that D is
between A and B.
- If A, B, C are non-collinear points and l is a line in
the plane of A, B, C so that none of these points lie on l,
then if l contains a point between A and B then it must contain
a point between A and C or a point between B and C.
Pasch's axiom of separation
Finally, we have the Archimedean least upper bound principle which gives
us the least upper bound property for points on a line:
- Given a sequence of points An and a point B such that
An + 1 is between B and An for all n, there is a point C
such that C is between B and An for all n and for all points
D lying between B and An for all n, D is between B and
Archimedean least upper bound principle
A much briefer list of equivalent axioms was provided by Veblen using
only the notion of points and the relation of between-ness; we use the
symbol [ABC] to denote B lies between A and C.
To check that a geometry conforms to these axioms is
obviously easier than to check the same for Hilbert's axioms; but the
equivalence of the two systems shows us that the former is enough.
- There are at least two distinct points.
- For any two distinct points A and B there is a point C
so that [ABC].
- If [ABC] then A, B and C are distinct.
- If [ABC] then [CBA] but not [BCA].
- We say that C lies on the line l (AB) if C = A or
C = B or [ABC] or [ACB] or [CAB]. A pair of lines is said to
meet if they have a point in common. If A, B, C all lie on a line
we say that they are collinear.
- If C and D are distinct points on the line l (A, B) then
A lies on l (C, D).
- There is at least one point not on l (A, B).
- If A, B, C are non-collinear and D,E are points so
that [BCD] and [AEC] then there is a point F so that [AFB] and
F lies on the line l (D, E).
- If A, B, C are non-collinear points then a point D
is said to be coplanar with A, B, C if it lies on a line
which meets two out of the three lines l (A, B), l (B, C),
l (A, C). The locus of all such points is called the plane determined
by the three non-collinear points A, B, C.
- There is at least one point not on the plane determined by
three non-collinear points A, B, C.
- Any two planes which meet have at least two distinct points
- Let the points of a line be divided into two disjoint classes each
of which satisfy: if A and B lie in the class the so does every
point C such that [ACB]. Then there is a point O on the line and a
pair of points P and N such that [NOP] and the two classes
consists of all points between N (respectively P) and O. (In
addition O lies in one of the classes.)
Now we will outline how the points, lines and planes of
such a geometry can be realised as the points, lines and planes of a
convex region in coordinate 3-space preserving all the relations
of incidence and between-ness; that is to say we have an embedding of our geometry into that of coordinate 3-space.
The first step is to construct a geometry consisting of ``ideal''
points, lines and planes; these would have ``existed'' if our
geometry were not confined to a region.
The collection of all lines passing through a fixed point has the
Thus we define an ``ideal point'' or Point to be a collection of lines
with the above two properties (see Figure 4). Any pair
(L1, L2) of distinct coplanar lines then determines a Point
constructed as follows:
- Any pair of lines in this collection are coplanar.
- For every point there is a line from this collection that
We leave it to the reader to check that such a collection indeed
satisfies the two properties mentioned above. Moreover, any point
clearly gives us a Point as the collection of all lines containing
- We take any plane p1 containing L1 and a different plane
p2 containing L2 which meets p1. Then we add the line of
L3 = p1 p2 to our collection.
- Let p be the plane containing L1 and L2 and p' be any
other plane containing a line L3 constructed as above, such that
p' meets p. We add the line of intersection
L4 = p p' to out
A collection of lines determines a Point
Given a pair of distinct Points
B we consider
those planes which contain a line each from the collections
B. The collection of all these
planes determine an ``ideal line'' or Line. Any pair (p1, p2) of
distinct planes then determines a Line constructed as follows:
We leave it to the reader to check that this construction is
independent of the choice of the pair (q1, q2). Moreover, we also
note that the collection of all planes containing a line also
satisfies the required conditions and thus determines a Line.
- We choose a pair of planes (q1, q2) such that each qi meets
- Let Ai be the Point determined by the pair of lines
(qi p1, qi p2).
- A plane p belongs to our collection if it contains a line
in each of the collections A1 and A2.
A collection of planes determines a Line
We say that a Point lies on a Line if every plane that belongs to the
collection determining the Line contains a line that belongs to the
collection determining the Point. This specifies the notion of incidence
for Points and Lines. It is clear that if a Line
determined by the line L' then
L contains the Point
A if and only if L' lies in the collection corresponding to
A. In particular, if the Point
A is that
determined A', then
A lies on
L if and only if A'
lies on L'.
A Line contains a Point
Thus we have embedded our given geometry into an ``ideal'' geometry of
Points and Lines with relations of incidence and separation. What are
the axioms satisfied by this ``ideal'' geometry? We claim that the
following axioms are satisfied:
The first three axioms are the axioms of incidence and the latter two
are the axioms of dimension for Projective geometry of 3 dimensions.
It is a well-known theorem (see for example ) that such
a geometry is isomorphic to the coordinate Projective geometry over
a skew-field (that is multiplication need not be commutative).
- Any pair of Points
B lie on exactly one
Line. We denote this line by
- Given four Points
D, the Line
AB meets the Line
CD if and only
if the Line
AD meets the Line
BC. In this case we
call the four Points coplanar.
- There are five Points so that no four of these are coplanar.
- Given any five Points
E, there is a point
F on the Line
AB so that the Points
F are coplanar.
The relation of separation is a little more intricate. We will try to
define the relation picturised in Figure 7. Given two pairs
(C,D) of Points lying on a
L; we say that the pairs separate each other if for
some plane p in the collection
L and some point O in the
plane p, there is a line L' in p and four points A', B', C'
and D' on L' so that:
We use the notation
AB||CD to denote this
relation. Now in case the Points
D correspond to points A', B', C' and D' respectively,
then the two relations of separation coincide;
AB||CD if and only these points are all
collinear, and either C is between A
and B while D is not or D is between A and B and C is not.
- The lines OA', OB', OC' and OD' are in the collections
- B' and C' are between A' and D', and B' is between A'
AB||CD on a Projective Line
The axioms satisfied by the relation of separation are:
The first three are the axioms of separation for Projective geometry and
the last is a form of the Archimedean least upper bound principle.
These axioms assure us that the base skew-field of our projective space
geometry is a complete ordered skew-field--which is then a
(commutative) field. Moreover, under some set theoretic assumptions
about cardinality this is just the field of real numbers.
D are collinear
and distinct then exactly one of the relations
C are collinear then there
is a Point
D so that
O is a Point,
L any Line so that the Points
D' of intersection of
L with the Lines
OD respectively are
- Given a sequence of Points
An and Points
O such that
AnB||An + 1O
holds for all n, then there is a Point
AnB||CO holds for all n; and so
for all points
D so that
holds for all n, we have
Thus our ``ideal'' geometry is none other than coordinate Projective
geometry of of dimension 3 over the field of real numbers. Hence we
have an embedding of the given geometry into real Projective space
The collection of all Points on a Line thus forms a real Projective
line which cannot be totally ordered in a way so as to satisfy the
Archimedean least upper bound principle. Thus for any line there is at
least one Point
A which lies on the corresponding Line that
does not correspond to a point. Moreover, if C and D are points on
the line then every Point
B so that
AB||CD corresponds to a point B. In
other words the geometry we have is that of a convex subset of
Projective space. For any such subset one can find a Projective plane
that does not meet it. Thus the convex subset is actually contained in
real Affine space; in other words we have a coordinatisation of our
Next: 3 Riemannian Geometry
Up: Axiomatic and Coordinate Geometry
Previous: 1 Introduction
Kapil Hari Paranjape