In
geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
points, particularly points on a
projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, their cross ratio is defined as
:
where an orientation of the line determines the sign of each distance and the distance is measured as projected into
Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.)
The point ''D'' is the
harmonic conjugate
In mathematics, a real-valued function u(x,y) defined on a connected open set \Omega \subset \R^2 is said to have a conjugate (function) v(x,y) if and only if they are respectively the real and imaginary parts of a holomorphic function f(z) of ...
of ''C'' with respect to ''A'' and ''B'' precisely if the cross-ratio of the quadruple is −1, called the ''harmonic ratio''. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name ''anharmonic ratio''.
The cross-ratio is preserved by
linear fractional transformation
In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form
:z \mapsto \frac ,
which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfo ...
s. It is essentially the only projective
invariant of a quadruple of collinear points; this underlies its importance for
projective geometry.
The cross-ratio had been defined in deep antiquity, possibly already by
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
, and was considered by
Pappus, who noted its key invariance property. It was extensively studied in the 19th century.
Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the
Riemann sphere.
In the
Cayley–Klein model of
hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.
Terminology and history
Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
made implicit use of concepts equivalent to the cross-ratio in his ''Collection: Book VII''. Early users of Pappus included
Isaac Newton,
Michel Chasles
Michel Floréal Chasles (; 15 November 1793 – 18 December 1880) was a French mathematician.
Biography
He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coal ...
, and
Robert Simson. In 1986 Alexander Jones made a translation of the original by Pappus, then wrote a commentary on how the lemmas of Pappus relate to modern terminology.
Modern use of the cross ratio in projective geometry began with
Lazare Carnot in 1803 with his book ''Géométrie de Position''. The term used was ''le rapport anharmonique'' (Fr: anharmonic ratio). German geometers call it ''das Doppelverhältnis'' (Ger: double ratio).
Given three points on a line, a fourth point that makes the cross ratio equal to minus one is called the
projective harmonic conjugate
In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction:
:Given three collinear points , let be a point not lying on their join and let any line ...
. In 1847
Carl von Staudt called the construction of the fourth point a throw (Wurf), and used the construction to exhibit arithmetic implicit in geometry. His
Algebra of Throws provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry.
The English term "cross-ratio" was introduced in 1878 by
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in hi ...
.
Definition
The cross-ratio of a quadruple of distinct points on the
projectively extended real line with coordinates ''z''
1, ''z''
2, ''z''
3, ''z''
4 is given by
:
It can also be written as a "double ratio" of two division ratios of triples of points:
:
The cross-ratio is normally extended to the case when one of ''z''
1, ''z''
2, ''z''
3, ''z''
4 is
infinity this is done by removing the corresponding two differences from the formula. For example:
:
In the notation of
Euclidean geometry, if ''A'', ''B'', ''C'', ''D'' are collinear points, their cross ratio is:
:
where each of the distances is signed according to a consistent orientation of the line.
The same formulas can be applied to four different
complex numbers or, more generally, to elements of any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, and can also be extended as above to the case when one of them is the symbol ∞.
Properties
The cross ratio of the four collinear points ''A'', ''B'', ''C'', ''D'' can be written as
:
where
describes the ratio with which the point ''C'' divides the line segment ''AB'', and
describes the ratio with which the point ''D'' divides that same line segment. The cross ratio then appears as a ratio of ratios, describing how the two points ''C'', ''D'' are situated with respect to the line segment ''AB''. As long as the points ''A'', ''B'', ''C'' and ''D'' are distinct, the cross ratio (''A'', ''B''; ''C'', ''D'') will be a non-zero real number. We can easily deduce that
* (''A'', ''B''; ''C'', ''D'') < 0 if and only if one of the points ''C'', ''D'' lies between the points ''A'', ''B'' and the other does not
* (''A'', ''B''; ''C'', ''D'') = 1 / (''A'', ''B''; ''D'', ''C'')
* (''A'', ''B''; ''C'', ''D'') = (''C'', ''D''; ''A'', ''B'')
* (''A'', ''B''; ''C'', ''D'') ≠ (''A'', ''B''; ''C'', ''E'') ↔ ''D'' ≠ ''E''
Six cross-ratios
Four points can be ordered in ways, but there are only six ways for partitioning them into two unordered pairs. Thus, four points can have only six different cross-ratios, which are related as:
:
See ''
Anharmonic group
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
'' below.
Projective geometry
The cross-ratio is a projective
invariant in the sense that it is preserved by the
projective transformations of a projective line.
In particular, if four points lie on a straight line ''L'' in R
2 then their cross-ratio is a well-defined quantity, because any choice of the origin and even of the scale on the line will yield the same value of the cross-ratio.
Furthermore, let be four distinct lines in the plane passing through the same point ''Q''. Then any line ''L'' not passing through ''Q'' intersects these lines in four distinct points ''P''
''i'' (if ''L'' is
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of IB ...
to ''L''
''i'' then the corresponding intersection point is "at infinity"). It turns out that the cross-ratio of these points (taken in a fixed order) does not depend on the choice of a line ''L'', and hence it is an invariant of the 4-tuple of lines .
This can be understood as follows: if ''L'' and ''L''′ are two lines not passing through ''Q'' then the perspective transformation from ''L'' to ''L''′ with the center ''Q'' is a projective transformation that takes the quadruple of points on ''L'' into the quadruple of points on ''L''′.
Therefore, the invariance of the cross-ratio under projective automorphisms of the line implies (in fact, is equivalent to) the independence of the cross-ratio of the four
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
points on the lines from the choice of the line that contains them.
Definition in homogeneous coordinates
If four collinear points are represented in
homogeneous coordinates by vectors ''a'', ''b'', ''c'', ''d'' such that and , then their cross-ratio is ''k''.
Role in non-Euclidean geometry
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, Cayley enjoyed solving complex maths problems ...
and
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
found an application of the cross-ratio to
non-Euclidean geometry. Given a nonsingular
conic
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
''C'' in the real
projective plane, its
stabilizer ''G
C'' in the
projective group acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
transitively on the points in the interior of ''C''. However, there is an invariant for the action of ''G
C'' on ''pairs'' of points. In fact, every such invariant is expressible as a function of the appropriate cross ratio.
Hyperbolic geometry
Explicitly, let the conic be the
unit circle. For any two points ''P'', ''Q'', inside the unit circle . If the line connecting them intersects the circle in two points, ''X'' and ''Y'' and the points are, in order, . Then the hyperbolic distance between ''P'' and ''Q'' in the
Cayley–Klein model of the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ' ...
can be expressed as
:
(the factor one half is needed to make the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canon ...
−1). Since the cross-ratio is invariant under projective transformations, it follows that the hyperbolic distance is invariant under the projective transformations that preserve the conic ''C''.
Conversely, the group ''G'' acts transitively on the set of pairs of points in the unit disk at a fixed hyperbolic distance.
Later, partly through the influence of
Henri Poincaré, the cross ratio of four
complex numbers on a circle was used for hyperbolic metrics. Being on a circle means the four points are the image of four real points under a
Möbius transformation, and hence the cross ratio is a real number. The
Poincaré half-plane model and
Poincaré disk model are two models of hyperbolic geometry in the
complex projective line
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
.
These models are instances of
Cayley–Klein metric
In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance"Cayley (1859), ...
s.
Anharmonic group and Klein four-group
The cross-ratio may be defined by any of these four expressions:
:
These differ by the following
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
s of the variables (in
cycle notation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
):
:
We may consider the permutations of the four variables as an
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of the
symmetric group S
4 on functions of the four variables. Since the above four permutations leave the cross ratio unaltered, they form the
stabilizer ''K'' of the cross-ratio under this action, and this induces an
effective action
In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective ac ...
of the
quotient group on the orbit of the cross-ratio. The four permutations in ''K'' make a realization of the
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one ...
in S
4, and the quotient
is isomorphic to the symmetric group S
3.
Thus, the other permutations of the four variables alter the cross-ratio to give the following six values, which are the orbit of the six-element group
:
:
As functions of ''λ'', these are examples of
Möbius transformations, which under composition of functions form the Mobius group . The six transformations form a subgroup known as the anharmonic group, again isomorphic to S
3. They are the torsion elements (
elliptic transforms) in . Namely,
,
, and
are of order 2 with respective
fixed points −1, 1/2, and 2 (namely, the orbit of the harmonic cross-ratio). Meanwhile, the elements
and
are of order 3 in , and each fixes both values
of the "most symmetric" cross-ratio.
The anharmonic group is generated by and . Its action on gives an isomorphism with S
3. It may also be realised as the six Möbius transformations mentioned,
which yields a projective
representation of S3 over any field (since it is defined with integer entries), and is always faithful/injective (since no two terms differ only by 1/−1). Over the field with two elements, the projective line only has three points, so this representation is an isomorphism, and is the
exceptional isomorphism . In characteristic 3, this stabilizes the point