EM225
Comparisons of geometries
Introduction
During this module, we have met various different geometries such as Euclidean, nonEuclidean, projective, inversive and so on. The aim here is to summarise some of the features
of these various geometries and make some comparisons between them. In doing so, we shall
have in mind the question What is geometry?
Euclidean geometry
Euclidean geometry comprises points and ‘straight’ lines in the plane 42.
The notion of distance between points is understood to mean the distance along a straight
line.
The transformations of Euclidean geometry are rotations, translations, reflections and
combinations of these. These are the transformations that preserve distances; a distancepreserving mapping 42 → 42 is called an isometry of 42.
Thus an isometry is a mapping 42 → 42 such that for all x, y ∈ 42, the distance between
the points equals the distance between their images:
d ( x, y ) = d ( f ( x), f ( y ))
f(x)
x
d
d
f
f(y)
y
Every isometry is a translation, rotation, reflection or glide reflection. (A glide reflection is a
translation along a line followed by reflection in the line.) Every isometry is the composite of
(at most three) reflections. Thus, in a sense, the reflections are the ‘most basic’ isometries.
Isometries preserve magnitudes of angles (but may reverse orientation).
Two figures are congruent (or, more precisely, Euclidean congruent) if there exists an
isometry that maps one to the other.
f
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Parallels
Euclidean geometry satisfies the parallel axiom: given any line l and any point P not lying
on l, there exists a unique line m containing P that is parallel to l.
P
m
l
Triangle Theorem: the angles of a Euclidean triangle sum to 180°.
Spherical geometry
The points of spherical geometry are points on the surface of a sphere and the ‘lines’ are great
circles. A great circle on the sphere is a circle whose centre is the centre of the sphere; a great
circle is obtained by intersecting the sphere with a plane through its centre.
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The notion of distance between points is understood to mean the shortest distance along a
great circle. (Generally, long-haul flights follow great circles on the surface of the earth.)
Q
P
d
The transformations of spherical geometry are the mappings of the sphere to itself that
preserve ‘great circle distances’; they are called spherical isometries. They are:
•
rotations about great circles – in other words, rotations about axes that pass through the
centre of the sphere
•
reflections in great circles – in other words, reflections in planes through the centre of the
sphere
•
a composite of a reflection and a rotation
Again every spherical isometry is the composite of (at most three) reflections in great circles.
Thus, in a sense, the reflections in great circles are the ‘most basic’ spherical isometries.
Spherical isometries preserve distances (along great circles) and magnitudes of angles.
Two figures are spherical congruent if there exists an spherical isometry that maps one to
the other.
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Parallels
In spherical geometry every pair of lines (great circles) intersect so there are no parallel lines.
Hence, spherical geometry satisfies the following parallel axiom: given any line l and any
point P not lying on l, there does not exist a line m containing P that is parallel to l.
P
m
l
Triangle Theorem: the angle sum of a spherical triangle is strictly greater than 180°.
Inversive Geometry
The points of inversive geometry are points of the plane 42 and the lines are ordinary
Euclidean straight lines. However inversive transformations are
•
reflections in lines
•
inversions in circles.
+
P
Q
O
OP × OQ = r 2
r
Since isometries (transformations of Euclidean geometry) are composites of reflections,
inversive geometry includes all the ‘normal’ Euclidean transformations together with many
new transformations, namely those that include inversions in circles.
Inversive transformations
•
do not preserve distance
•
preserve (magnitudes) of angles.
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Two figures are inversive congruent if there exists an inversive transformation that maps one
to the other.
Fundamental Theorem of Inversive Geometry
Let p, q, r and p′, q′, r′ be two sets of three points in the plane. Then there exists an
inversive transformation that maps p = p′, q = q′, r = r ′ .
Consequences of the fundamental theorem
All triangles are inversive congruent.
All circles are inversive congruent – a circle is determined by three points.
Any line and any circle are inversive congruent.
p
q
r
p′
r′
q′
This means that ‘circles’ and ‘lines’ are the same in inversive geometry! Actually this is not
quite correct – a circle is a ‘closed’ curve but a line has two ends. Strictly speaking, we need
to ‘do’ inversive geometry on a sphere not a plane, with the correspondence given by
stereographic projection. Then a line in the plane corresponds to a circle through the north
pole; see the diagram below. Thus the points of inversive geometry are points on a sphere and
the lines of inversive geometry are circles – but not necessarily great circles – on the sphere.
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N
We may therefore regard points of spherical geometry as comprising points in the plane
together with an additional ‘point at infinity’ (corresponding to the north pole on the sphere).
Non-Euclidean (hyperbolic) geometry
We shall describe the Poincaré model of hyperbolic geometry. The points of hyperbolic
geometry are the points of a disc D not including its boundary. We shall take this to be the
unit disc:
D = {(x, y) ∈ 42 : x2 + y2 < 1}
The ‘lines’ of hyperbolic geometry are diameters of the disc D and arcs of circles that
intersect the boundary circle, C = {(x, y) ∈ 42 : x2 + y2 < 1}, at right angles.
D
C
Two hyperbolic lines are parallel if the ‘meet’ on C. (But since C is not part of the
hyperbolic ‘world’, the parallel lines do not meet in D but rather ‘at infinity’ in some sense.)
Two lines are ultra-parallel if they do not meet ‘even on C ’.
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ultra-parallel
D
C
parallel
Hyperbolic geometry satisfies the following parallel axiom: given any line l and any point
P not lying on l, then
•
there exists a exactly two lines containing P that are parallel to l and
•
there exist infinitely many lines containing P that are ultra-parallel to l.
ultra-parallels to l
parallels to l
D
D
P
P
l
l
Distance
Distance can be defined in hyperbolic geometry. Distance has the curious property that, seen
from the outside, distances appear to shrink the nearer one travels towards C. Thus a ruler of
constant length appears from the outside to be smaller when it is nearer to the boundary C.
D
C
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three rulers of
equal length
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Hyperbolic transformations
A hyperbolic reflection in a line is
either reflection in the line if the line is itself a diagonal
or
inversion in the circle of which the line is an arc.
D
reflection in the line
means inversion in
this circle
line
C
A general hyperbolic transformation is a composition of (at most three) hyperbolic
reflections. Hyperbolic transformations preserve distances in D so may be called hyperbolic
isometries.
Triangle Theorem: the angle sum of a hyperbolic triangle is strictly less than 180°.
D
Projective geometry
The Points of projective geometry are lines through the origin. If we choose a particular
embedding plane, we can think of the Points of projective geometry as points of the plane 42
together with all the ideal points (or ‘points at infinity’). In this sense projective geometry,
comprises points in the plane 42 together with a line of points ‘at infinity’ (all the ideal
points for the chosen embedding plane).
projective point with
no representative on
the embedding plane
embedding plane
projective point with
a representative on
the embedding plane
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Projective transformations
A perspectivity from O is a mapping of the projective plane to itself such that P maps to
P′ when O, P, P′ are collinear.
e
tiv
ec e
oj
Pr plan
O
ve
cti
oje
Pr lane
p
A projective transformation is a composition of at most three perspectivities.
Two figures are projectively congruent if there is a projective transformation that maps one
to the other.
Fundamental theorem of Projective Geometry
Let p, q, r, s be any four points no three of which are collinear and let p′, q′, r′, s′ be any
four points no three of which are collinear. Then there exists a unique projective
transformation that maps p = p ′, q = q ′, r = r ′, s = s ′ .
Consequence: all quadrilaterals are projectively congruent.
D
A
A′
C′
B
C
D′
B′
Parallels
Since any two Lines determine a unique Point of intersection, there are no parallel Lines in
projective geometry.
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Comparisons
The following table summarises some of the properties of the different geometries we have
encountered. (There is yet another geometry – affine geometry – that we have not met in this
module. Affine geometry sits somewhere between Euclidean and projective geometry in a
sense that will be discussed briefly below.)
Euclidean
Spherical
Hyperbolic
Inversive
Projective
Points
42
Sphere
or
42 with
point at
infinity
Disc
or
42
Sphere
or
42 with
point at
infinity
Lines in 43
through 0
or
42 with line
of points at
infinity
Distance
Yes
Yes
Yes
No
No
Parallel line
(given P
and l)?
Unique
parallel
No parallels
Many (ultra)
parallels
Triangles –
angle sum
180°
> 180°
< 180°
Variable (all
triangles
inversive
congruent)
No angles
1
1
3
4 (no 3 of
which
collinear)
How many
1
points can be
mapped?
No parallels
Kleinian View of Geometry
How are we to make sense of this diversity?
In 1872, Felix Klein (of Klein bottle fame but very much more besides) described a new way
of thinking about geometry. Klein presented this new view of geometry in his inaugural
lecture at the University of Erlangen and today it is frequently known as Klein’s Erlangen
Program or the Kleinian view of geometry.
Klein’s view was that the transformations are the key to geometry; the ‘allowable’
transformations essentially define the essence of a particular geometry.
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In any geometry, the collection of all allowable transformations satisfy the following
properties:
•
the composite of two transformations ( g o f , meaning ‘do f then g’) is another
allowable transformation
•
the associative law holds: h o ( g o f ) = (h o g ) o f
•
the identity mapping (‘fix all points’) is an allowable transformation
•
given any allowable transformation f, its inverse (undo the effect of f) is also an
allowable transformation.
We summarise these properties by saying that the set of allowable transformations of a
particular geometry forms a group. We can now state Klein’s view of geometry more
precisely.
A geometry consists of a set of points and a group of allowable
transformations. The properties of the geometry are those properties
preserved by the group of transformations.
Note that the greater the number of transformations the fewer the number of geometric
properties. For example, if there were no transformations apart from the identity then all
points would be ‘essentially different’ (because no point could be mapped to any other point).
Such a geometry would have lots of properties – each point would be essentially different –
but would not be very interesting.
As a second example, every Euclidean transformation is a composite of reflections. Since a
reflection is a special case of an inversive transformation, every Euclidean transformation is
an inversive transformation. But there are many inversive transformations that are not
Euclidean transformations – all the inversions in circles. Therefore the group of inversive
transformations is strictly bigger than the group of Euclidean transformations.
Inversive
Euclidean
In fact, it can be shown that the projective transformations form the largest group. In this
sense, projective geometry is the ‘most basic’ – all other geometries (that we have considered)
are special cases of projective geometry.
Reference
David Branan, Matthew Esplen, Jeremy Gray, Geometry, Cambridge University Press, 1999.
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