View source for Cross-ratio

{{short description|An invariant under projective transformations}}
{{Distinguish|Odds ratio}}
[[File:Projection geometry.svg|thumb|Points ''A'', ''B'', ''C'', ''D'' and ''A''′, ''B''′, ''C''′, ''D''′ are related by a projective transformation so their cross ratios, {{nowrap|(''A'', ''B''; ''C'', ''D'')}} and {{nowrap|(''A''′, ''B''′; ''C''′, ''D''′)}} are equal.]]
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, their cross ratio is defined as

: <math>(A,B;C,D) = \frac {AC\cdot BD}{BC\cdot AD}</math>

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 [[Projective harmonic conjugate|harmonic conjugate]] of ''C'' with respect to ''A'' and ''B'' precisely if the cross-ratio of the quadruple is&nbsp;&minus;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]]s.  It is essentially the only projective [[invariant (mathematics)|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]], and was considered by [[Pappus of Alexandria|Pappus]], who noted its key invariance property. It was extensively studied in the 19th century.<ref>A theorem on the anharmonic ratio of lines appeared in the work of [[Pappus of Alexandria|Pappus]], but [[Michel Chasles]], who devoted considerable efforts to reconstructing lost works of [[Euclid]], asserted that it had earlier appeared in his book ''Porisms''.</ref> 

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==
[[File:pappusharmonic.svg|thumb|right|300px|''D'' is the [[Projective harmonic conjugate|harmonic conjugate]] of ''C'' with respect to ''A'' and ''B'', so that the cross-ratio {{nowrap|(''A'', ''B''; ''C'', ''D'')}} equals&nbsp;−1.]]

[[Pappus of Alexandria]] made implicit use of concepts equivalent to the cross-ratio in his ''Collection: Book VII''. Early users of Pappus included [[Isaac Newton]], [[Michel Chasles]], 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.<ref>Alexander Jones (1986) ''Book 7 of the Collection'', part 1: introduction, text, translation {{ISBN|0-387-96257-3}}, part 2: commentary, index, figures {{ISBN|3-540-96257-3}}, [[Springer-Verlag]]</ref>

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 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 [[Karl von Staudt#Algebra of throws|Algebra of Throws]] provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry.<ref>[[Howard Eves]] (1972) ''A Survey of Geometry'', Revised Edition, page 73, [[Allyn and Bacon]]</ref>

The English term "cross-ratio" was introduced in 1878 by [[William Kingdon Clifford]].<ref>[[W.K. Clifford]] (1878) [http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=04370002 Elements of Dynamic, books I,II,III], page 42, London: MacMillan & Co; on-line presentation by [[Cornell University]] ''Historical Mathematical Monographs''.</ref>

==Definition==
The cross-ratio of a quadruple of distinct points on the [[real line]] with coordinates ''z''<sub>1</sub>,&nbsp;''z''<sub>2</sub>,&nbsp;''z''<sub>3</sub>,&nbsp;''z''<sub>4</sub> is given by

:<math>(z_1,z_2;z_3,z_4) = \frac{(z_3-z_1)(z_4-z_2)}{(z_3-z_2)(z_4-z_1)}.</math>

It can also be written as a "double ratio" of two division ratios of triples of points:

:<math>(z_1,z_2;z_3,z_4) = \frac{z_3-z_1}{z_3-z_2}:\frac{z_4-z_1}{z_4-z_2}.</math>

The cross-ratio is normally  extended to the case when one of ''z''<sub>1</sub>,&nbsp;''z''<sub>2</sub>,&nbsp;''z''<sub>3</sub>,&nbsp;''z''<sub>4</sub> is [[infinity]] <math> (\infty);</math> this is done by removing the corresponding two differences from the formula.

For example: if <math> z_1 = \infty</math> the cross ratio becomes: 
:<math>(z_1,z_2;z_3,z_4) = \frac{(z_3-\infty)(z_4-z_2)}{(z_3-z_2)(z_4-\infty)}=\frac{(z_4-z_2)}{(z_3-z_2)} .</math>

In geometry, if ''A'', ''B'', ''C'' and ''D'' are collinear points, then the cross ratio is defined similarly as
:<math>(A,B;C,D) = \frac {AC\cdot BD}{BC\cdot AD} ,</math> 
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 number]]s or, more generally, to elements of any [[field (mathematics)|field]] and can also be extended to the case when one of them is the symbol ∞, by removing the corresponding two differences from the formula.
The formula shows that cross-ratio is a [[function (mathematics)|function]] of four points, generally four numbers <math>z_1, \ z_2, \ z_3, \ z_4</math> taken from a field.

==Properties==
The cross ratio of the four collinear points ''A'', ''B'', ''C'', ''D'' can be written as
:<math>(A,B;C,D) = \frac {AC}{CB}:\frac {AD}{DB}</math> 
where <math>\frac {AC}{CB}</math> describes the ratio with which the point ''C'' divides the line segment ''AB'', and <math>\frac {AD}{DB}</math> 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 {{nowrap|1=[[factorial|4!]] =  4 × 3 × 2 × 1 = 24}} ways, but there are only six ways for partitioning them into two non-ordered pairs. Thus, four points can have only six different cross-ratios, which are related as:
 
: <math>
\begin{align}
& (A,B;C,D) = (B,A;D,C) = (C,D;A,B) = (D,C;B,A) = \lambda \\[6pt]
& (A,B;D,C) = (B,A;C,D) = (C,D;B,A) = (D,C;A,B) = \frac 1 \lambda \\[6pt]
& (A,C;B,D) = (B,D;A,C) = (C,A;D,B) = (D,B;C,A) = 1-\lambda \\[6pt]
& (A,C;D,B) = (B,D;C,A) = (C,A;B,D) = (D,B;A,C) = \frac 1 {1-\lambda} \\[6pt]
& (A,D;B,C) = (B,C;A,D) = (C,B;D,A) = (D,A;C,B) = \frac{\lambda-1} \lambda \\[6pt]
& (A,D;C,B) = (B,C;D,A) =  (C,B;A,D) = (D,A;B,C) = \frac \lambda {\lambda-1}
\end{align}
</math>

==Projective geometry==
{{further|Projective geometry}}
{{cross_ratio_metrology_example.svg}}
Cross-ratio is a '''projective [[Invariant (mathematics)|invariant]]''' in the sense that it is preserved by the [[projective transformation]]s of a projective line.

In particular, if four points lie on a straight line ''L'' in '''R'''<sup>2</sup> 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 {{nowrap|{''L''<sub>''i''</sub> {{!}} 1 ≤ ''i'' ≤ 4} }} 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''<sub>''i''</sub> (if ''L'' is [[Parallel (geometry)|parallel]] to ''L''<sub>''i''</sub> 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 {''L''<sub>''i''</sub>}. 

This can be understood as follows: if ''L'' and ''L''&prime; are two lines not passing through ''Q'' then the perspective transformation from ''L'' to ''L''&prime; with the center ''Q'' is a projective transformation that takes the quadruple {''P''<sub>''i''</sub>} of points on ''L'' into the quadruple {''P''<sub>''i''</sub>&prime;} of points on ''L''&prime;.

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]] points {''P''<sub>''i''</sub>} on the lines {''L''<sub>''i''</sub>} from the choice of the line that contains them.

{{clear}}

==Definition in homogeneous coordinates==
If four collinear points are represented in [[homogeneous coordinate]]s by vectors ''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;''d'' such that {{nowrap|1=''c'' = ''a'' + ''b''}} and {{nowrap|1=''d'' = ''ka'' + ''b''}}, then their cross-ratio is&nbsp;''k''.<ref>{{cite book|author=Irving Kaplansky |title=Linear Algebra and Geometry: A Second Course |year=1969 |isbn=0-486-43233-5 }}</ref>

==Role in non-Euclidean geometry==
[[Arthur Cayley]] and [[Felix Klein]] found an application of the cross-ratio to [[non-Euclidean geometry]]. Given a nonsingular [[Conic section|conic]] ''C'' in the real [[projective plane]], its [[Stabilizer subgroup#Orbits and stabilizers|stabilizer]] ''G<sub>C</sub>'' in the [[projective group]] {{nowrap|1=''G'' = PGL(3, '''R''')}} [[Group action (mathematics)|acts]] [[transitive action|transitively]] on the points in the interior of ''C''. However, there is an invariant for the action of ''G<sub>C</sub>'' on ''pairs'' of points. In fact, every such invariant is expressible as a function of the appropriate cross ratio.{{citation needed|date=November 2010}}

===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, {{nowrap|''X'', ''P'', ''Q'', ''Y''}}. Then the hyperbolic distance between ''P'' and ''Q'' in the [[Cayley–Klein model]] of the [[hyperbolic geometry|hyperbolic plane]] can be expressed as

:<math> d_h(P,Q)=\frac{1}{2} \left| \log \left( \frac{|XQ||PY|}{|XP||QY|}\right ) \right| </math>

(the factor one half is needed to make the [[Gaussian curvature|curvature]] &minus;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 {{nowrap|(''p'', ''q'')}} in the unit disk at a fixed hyperbolic distance.

Later, partly through the influence of [[Henri Poincaré]], the cross ratio of four [[complex number]]s 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]].

These models are instances of [[Cayley–Klein metric]]s.

== The anharmonic group ==
The cross-ratio may be defined by any of these four expressions:

: <math> (A,B;C,D) = (B,A;D,C) = (C,D;A,B) = (D,C;B,A). \, </math>

These differ by the following [[permutation]]s of the variables:

: <math>
\begin{align}
(A, B) (C, D) \\[6pt]
(A, C) (B, D) \\[6pt]
(A, D) (B, C) 
\end{align}
</math>

These three and the identity permutation leave the cross ratio unaltered. They make up a realization of the [[Klein four-group]], a [[group (mathematics)|group]] of order&nbsp;4 in which the order of every non-identity element is&nbsp;2.

Other permutations of the four variables alter the cross-ratio so that it may take any of the following six values.

: <math>
\begin{align}
(A, B; C, D) & = \lambda & (A, B; D, C) & = \frac 1 \lambda \\[6pt]
(A, C; D, B) & = \frac 1 {1-\lambda} & (A, C; B, D) & = 1-\lambda \\[6pt]
(A, D; C, B) & = \frac \lambda {\lambda-1} & (A, D; B, C) & = \frac{\lambda-1} \lambda
\end{align}
</math>

As functions of ''&lambda;'', these form a non-abelian group of order&nbsp;6 with the operation of composition of functions.  This is the '''anharmonic group'''.  It is a subgroup of the group of all [[Möbius transformation]]s. The six cross-ratios listed above represent torsion elements (geometrically, [[elliptic transform]]s) of {{nowrap|[[Projective linear group|PGL]](2, '''Z''')}}. Namely, <math>\frac{1}{\lambda}</math>, <math>1-\lambda\,</math>, and <math>\frac{\lambda}{\lambda-1}</math> are of order 2 in {{nowrap|PGL(2, '''Z''')}}, with [[Fixed point (mathematics)|fixed points]], respectively, −1, 1/2, and 2 (namely, the orbit of the harmonic cross-ratio). Meanwhile, elements
<math>\frac{1}{1-\lambda}</math> and <math>\frac{\lambda-1}{\lambda}</math> are of order 3 in {{nowrap|PGL(2, '''Z''')}} – in {{nowrap|PSL(2, '''Z''')}} (this corresponds to the subgroup [[Alternating group|A<sub>3</sub>]] of even elements). Each of them fixes both values <math>e^{\pm i\pi/3}</math> of the "most symmetric" cross-ratio.

The anharmonic group is generated by {{nowrap|''λ'' ↦ 1/''λ''}} and {{nowrap|''λ'' ↦ 1 − ''λ''}}. Its action on {{nowrap|{0, 1, ∞}{{null}}}} gives an isomorphism with S<sub>3</sub>. It may also be realised as the six Möbius transformations mentioned,<ref>{{cite book | last=Chandrasekharan | first=K. | authorlink=K. S. Chandrasekharan | title=Elliptic Functions | series=Grundlehren der mathematischen Wissenschaften | volume=281 | publisher=[[Springer-Verlag]] | year=1985 | isbn=3-540-15295-4 | zbl=0575.33001 | page=120 }}</ref> which yields a projective [[Dihedral group of order 6#Representation theory|representation of S<sub>3</sub>]] 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 [[Projective linear group#Exceptional isomorphisms|exceptional isomorphism]] <math>\mathrm{S}_3 \approx \mathrm{PGL}(2, 2)</math>. In characteristic 3, this stabilizes the point <math>-1 = [-1:1]</math>, which corresponds to the orbit of the harmonic cross-ratio being only a single point, since <math>2 = 1/2 = -1</math>. Over the field with 3 elements, the projective line has only 4 points and <math>\mathrm{S}_4 \approx \mathrm{PGL}(2, 3)</math>, and thus the representation is exactly the stabilizer of the harmonic cross-ratio, yielding an embedding <math>\mathrm{S}_3 \hookrightarrow \mathrm{S}_4</math> equals the stabilizer of the point <math>-1</math>.

===Role of Klein four-group===
In the language of [[group theory]], the [[symmetric group]] S<sub>4</sub> acts on the cross-ratio by permuting coordinates. The [[kernel (group theory)|kernel]] of this action is isomorphic to the [[Klein four-group]] K. This group consists of 2-cycle permutations of type <math> (AB)(CD) </math> (in addition to the identity), which preserve the cross-ratio. The effective symmetry group is then the [[quotient group]] <math>\mathrm{S}_4/\mathrm{K}</math>, which is isomorphic to S<sub>3</sub>.

===Exceptional orbits===
For certain values of ''λ'' there will be greater symmetry and therefore fewer than six possible values for the cross-ratio. These values of ''λ'' correspond to [[fixed point (mathematics)|fixed points]] of the action of S<sub>3</sub> on the Riemann sphere (given by the above six functions); or, equivalently, those points with a non-trivial [[stabilizer (group theory)|stabilizer]] in this permutation group.

The first set of fixed points is {{nowrap|{0, 1, ∞}.}} However, the cross-ratio can never take on these values if the points ''A'', ''B'', ''C'' and ''D'' are all distinct. These values are limit values as one pair of coordinates approach each other:
:<math>(Z,B;Z,D) = (A,Z;C,Z) = 0</math>
:<math>(Z,Z;C,D) = (A,B;Z,Z) = 1</math>
:<math>(Z,B;C,Z) = (A,Z;Z,D) = \infty.</math>

The second set of fixed points is {{nowrap|{&minus;1, 1/2, 2}.}} This situation is what is classically called the '''{{visible anchor|harmonic cross-ratio}}''', and arises in [[projective harmonic conjugate]]s.  In the real case, there are no other exceptional orbits.

In the complex case, the most symmetric cross-ratio occurs when <math>\lambda = e^{\pm i\pi/3}</math>. These are then the only two values of the cross-ratio, and these are acted on according to the sign of the permutation.

==Transformational approach==
{{Main|Möbius transformation}}

The cross-ratio is invariant under the [[projective transformation]]s of the line. In the case of a [[complex number|complex]] projective line, or the [[Riemann sphere]], these transformations are known as [[Möbius transformation]]s. A general Möbius transformation has the form

:<math>f(z) = \frac{az+b}{cz+d}\;,\quad \mbox{where } a,b,c,d\in\mathbb{C} \mbox{ and } ad-bc \ne 0.</math>

These transformations form a [[group (mathematics)|group]] [[Group action (mathematics)|acting]] on the [[Riemann sphere]], the [[Möbius group]].

The projective invariance of the cross-ratio means that

:<math>(f(z_1), f(z_2); f(z_3), f(z_4)) = (z_1, z_2; z_3, z_4).\ </math>

The cross-ratio is [[real number|real]] if and only if the four points are either [[line (geometry)#Collinear points|collinear]] or [[concyclic]], reflecting the fact that every Möbius transformation maps [[generalized circle]]s to generalized circles.

The action of the Möbius group is [[simply transitive]] on the set of triples of distinct points of the Riemann sphere: given any ordered triple of distinct points, {{nowrap|(''z''<sub>2</sub>, ''z''<sub>3</sub>, ''z''<sub>4</sub>)}}, there is a unique Möbius transformation ''f''(''z'') that maps it to the triple {{nowrap|(1, 0, ∞)}}. This transformation can be conveniently described using the cross-ratio: since {{nowrap|(''z'', ''z''<sub>2</sub>, ''z''<sub>3</sub>, ''z''<sub>4</sub>)}} must equal {{nowrap|(''f''(''z''), 1; 0, ∞)}}, which in turn equals ''f''(''z''), we obtain

:<math>f(z)=(z, z_2; z_3, z_4) .</math>

An alternative explanation for the invariance of the cross-ratio is based on the fact that the group of projective transformations of a line is generated by the translations, the homotheties, and the multiplicative inversion. The differences {{nowrap|''z''<sub>''j''</sub> &minus; ''z''<sub>''k''</sub>}} are invariant under the [[translation (mathematics)|translations]]

: <math>z \mapsto z + a</math>

where ''a'' is a [[constant (mathematics)|constant]] in the ground field ''F''. Furthermore, the division ratios are invariant under a [[Homothetic transformation|homothety]]

:<math>z \mapsto b z</math>

for a non-zero constant ''b'' in ''F''. Therefore, the cross-ratio is invariant under the [[affine transformation]]s.

In order to obtain a well-defined [[multiplicative inverse|inversion mapping]]

:<math>T : z \mapsto z^{-1},</math>

the affine line needs to be augmented by the [[point at infinity]], denoted ∞, forming the projective line ''P''<sup>1</sup>(''F''). Each affine mapping {{nowrap|''f'' : ''F'' → ''F''}} can be uniquely extended to a mapping of ''P''<sup>1</sup>(''F'') into itself that fixes the point at infinity. The map ''T'' swaps 0 and ∞. The projective group is [[generating set of a group|generated by]] ''T'' and the affine mappings extended to ''P''<sup>1</sup>(''F''). In the case {{nowrap|1=''F'' = '''C'''}}, the [[complex plane]], this results in the [[Möbius group]]. Since the cross-ratio is also invariant under ''T'', it is invariant under any projective mapping of ''P''<sup>1</sup>(''F'') into itself.

===Co-ordinate description===
If we write the complex points as vectors <math>\overrightarrow{x}_n = [\Re(z_n),\Im(z_n)]^{\rm T}</math> and define <math>x_{nm}=x_n-x_m</math>. Let <math>(a,b)</math> be the dot product of <math>a</math> with <math>b</math> then the real part of the cross ratio is given by:
::<math>
C_1 = \frac{ (x_{12},x_{14}) (x_{23},x_{34}) - (x_{12},x_{34}) (x_{14},x_{23}) + (x_{12},x_{23}) (x_{14},x_{34}) } {|x_{23}|^2 |x_{14}|^2  }
</math>

This is an invariant of the 2D [[special conformal transformation]] such as inversion <math>x^\mu \rightarrow \frac{x^\mu}{|x|^2} </math>.

The imaginary part must make use of the 2-dimensional cross product <math>a\times b = [a,b] = a_2 b_1 - a_1 b_2</math>

::<math>
C_2 = \frac{ (x_{12},x_{14}) [x_{34},x_{23}] - (x_{43},x_{23}) [ x_{12},x_{34} ]  } {|x_{23}|^2 |x_{14}|^2  }
</math>

==Ring homography==
The concept of cross ratio only depends on the [[ring (mathematics)|ring]] operations of addition, multiplication, and inversion (though inversion of a given element is not certain in a ring). One approach to cross ratio interprets it as a [[homography]] that takes three designated points to 0, 1, and infinity. Under restrictions having to do with inverses, it is possible to generate such a mapping with ring operations in the [[projective line over a ring#Cross-ratio|projective line over a ring]]. The cross ratio of four points is the evaluation of this homography at the fourth point.

==Differential-geometric point of view==
The theory takes on a differential calculus aspect as the four points are brought into proximity. This leads to the theory of the [[Schwarzian derivative]], and more generally of [[projective connection]]s.

==Higher-dimensional generalizations==
{{Further|General position}}
The cross-ratio does not generalize in a simple manner to higher dimensions, due to other geometric properties of configurations of points, notably collinearity – [[Configuration space (mathematics)|configuration space]]s are more complicated, and distinct ''k''-tuples of points are not in [[general position]].

While the projective linear group of the plane is 3-transitive (any three distinct points can be mapped to any other three points), and indeed simply 3-transitive (there is a ''unique'' projective map taking any triple to another triple), with the cross ratio thus being the unique projective invariant of a set of four points, there are basic geometric invariants in higher dimension. The projective linear group of ''n''-space <math>\mathbf{P}^n=\mathbf{P}(K^{n+1})</math> has (''n''&nbsp;+&nbsp;1)<sup>2</sup>&nbsp;&minus;&nbsp;1 dimensions (because it is <math>\mathrm{PGL}(n,K) = \mathbf{P}(\mathrm{GL}(n+1,K)),</math> projectivization removing one dimension), but in other dimensions the projective linear group is only 2-transitive – because three collinear points must be mapped to three collinear points (which is not a restriction in the projective line) – and thus there is not a "generalized cross ratio" providing the unique invariant of ''n''<sup>2</sup> points.

Collinearity is not the only geometric property of configurations of points that must be maintained – for example, [[five points determine a conic]], but six general points do not lie on a conic, so whether any 6-tuple of points lies on a conic is also a projective invariant. One can study orbits of points in [[general position]] – in the line "general position" is equivalent to being distinct, while in higher dimensions it requires geometric considerations, as discussed – but, as the above indicates, this is more complicated and less informative.

However, a generalization to [[Riemann surfaces]] of positive [[genus]] exists, using the [[Abel–Jacobi map]] and [[theta functions]].

==See also==
* [[Hilbert metric]]

== Notes ==
{{Reflist}}

== References ==
* [[Lars Ahlfors]] (1953,1966,1979) ''Complex Analysis'', 1st edition, page 25; 2nd & 3rd editions, page 78, [[McGraw-Hill]] {{ISBN|0-07-000657-1}} .
* Viktor Blåsjö (2009) "[http://www.springerlink.com/content/p01527115762n730/ Jakob Steiner’s Systematische Entwickelung: The Culmination of Classical Geometry]", [[Mathematical Intelligencer]] 31(1): 21&ndash;9.
* John J. Milne (1911) [https://books.google.com/books?id=BOk8AAAAIAAJ&pg=PA11&dq=Milne+Cross-ratio&hl=en&ei=_rJITaCeH4W0sAPj8fTfCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCYQ6AEwAA#v=onepage&q&f=false An Elementary Treatise on Cross-Ratio Geometry with Historical Notes], [[Cambridge University Press]].
* [[Dirk Struik]] (1953) ''Lectures on Analytic and Projective Geometry'', page 7, [[Addison-Wesley]].
* [[Igor Shafarevich|I. R. Shafarevich]] & A. O. Remizov (2012) ''Linear Algebra and Geometry'', [[Springer Science+Business Media|Springer]] {{ISBN|978-3-642-30993-9}}.

==External links==
* [http://www.mathpages.com/home/kmath543/kmath543.htm MathPages – Kevin Brown explains the cross-ratio in his article about ''Pascal's Mystic Hexagram'']
* [http://www.cut-the-knot.org/pythagoras/Cross-Ratio.shtml Cross-Ratio] at [[cut-the-knot]]
* {{mathworld|CrossRatio}}
* {{cite web |last1=Ardila |first1=Federico |title=The Cross Ratio |url=https://www.youtube.com/watch?v=ffvojZONF_A |website=youtube |publisher=[[Brady Haran]] |accessdate=6 July 2018 |format=video}}

{{DEFAULTSORT:Cross-Ratio}}
[[Category:Projective geometry]]
[[Category:Ratios]]