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| [[Image: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 ''(A, B; C, D)'' equals -1.]]
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| In [[geometry]], the '''cross-ratio''', also called '''double ratio''' and '''anharmonic ratio''', is a special number associated with an ordered quadruple of [[collinear]] points, particularly points on a [[projective line]]. Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the [[Riemann sphere]].
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| The cross-ratio is preserved by the [[fractional linear transformation]]s and
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| it is essentially the only projective [[invariant (mathematics)|invariant]] of a quadruple of points, which underlies its importance for [[projective geometry]]. In the [[Cayley–Klein model]] of [[hyperbolic geometry]], the distance between points is expressed in terms of a certain cross-ratio.
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| 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>
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| ==Definition==
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| The cross-ratio of a 4-tuple of distinct points on the [[real line]] with coordinates ''z''<sub>1</sub>, ''z''<sub>2</sub>, ''z''<sub>3</sub>, ''z''<sub>4</sub> is given by
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| :<math>(z_1,z_2;z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)}.</math>
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| It can also be written as a "double ratio" of two division ratios of triples of points:
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| :<math>(z_1,z_2;z_3,z_4) = \frac{z_1-z_3}{z_2-z_3}:\frac{z_1-z_4}{z_2-z_4}.</math>
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| 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.
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| 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.
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| In geometry, if A, B, C and D are collinear points, then the cross ratio is defined similarly as
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| :<math>(A,B;C,D) = \frac {AC\cdot BD}{BC\cdot AD}.</math>
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| where each of the distances is signed according to a fixed orientation of the line.
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| ==Terminology and history==
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| [[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.
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| 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). However, in 1847 [[Karl von Staudt]] introduced the term '''Throw''' (''Wurf'') to avoid the metrical implication of a ratio. His construction of the [[Karl von Staudt#Algebra of throws|Algebra of Throws]] provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry.
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| The English term "cross-ratio" was introduced in 1878 by [[William Kingdon Clifford]].<ref>W.K. Clifford (1878) [http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=04370002 Elements of Dynamic], page 42, London: MacMillan & Co; online presentation from [[Cornell University]] ''Historical Mathematical Monographs''</ref>
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| ==Projective geometry==
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| [[Image:Birapport et projection.png|thumb|300px|Points A, B, C, D and A', B', C', D' are related by a projective transformation so their cross ratios, (A, B; C, D) and (A', B'; C', D') are equal.]]
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| Cross-ratio is a '''projective 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 {''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''′ 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 {''P''<sub>''i''</sub>} of points on ''L'' into the quadruple {''P''<sub>''i''</sub>′} of points on ''L''′. Therefore, the
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| 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.
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| ==Definition in homogeneous coordinates==
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| If the four points are represented in homogeneous coordinates by vectors a,b,c,d such that c=a+b and d=ka+b, then their cross-ratio is k.
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| ==Role in non-Euclidean geometry==
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| [[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|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}}
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| Explicitly, let the conic be the [[unit circle]]. For any two points in the unit disk, ''p'', ''q'', the line connecting them intersects the circle in two points, ''a'' and ''b''. The points are, in order, {{nowrap|1=''a'', ''p'', ''q'', ''b''}}. Then the distance between ''p'' and ''q'' in the [[Cayley–Klein model]] of the plane [[hyperbolic geometry]] can be expressed as
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| :<math>d(p,q)=\frac{1}{2} \log \frac{|q-a||b-p|}{|p-a||b-q|}</math>
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| (the factor one half is needed to make the [[Gaussian curvature|curvature]] −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 (''p'',''q'') in the unit disk at a fixed hyperbolic distance.
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| ==Six cross-ratios==
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| There is a number of definitions of the cross-ratio. However, they all differ from each other by a suitable [[permutation]] of the coordinates. In general, there are six possible different values the cross-ratio <math>(z_1,z_2;z_3,z_4)</math> can take depending on the order in which the points ''z''<sub>''i''</sub> are given.
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| ===Action of symmetric group===
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| Since there are 24 possible permutations of the four coordinates, some permutations must leave the cross-ratio unaltered. In fact, exchanging any two pairs of coordinates preserves the cross-ratio:
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| :<math>(z_1,z_2;z_3,z_4) = (z_2,z_1;z_4,z_3) = (z_3,z_4;z_1,z_2) = (z_4,z_3;z_2,z_1).\,</math>
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| Using these symmetries, there can then be 6 possible values of the cross-ratio, depending on the order in which the points are given. These are:
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| {| style="margin-left: 2em;"
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| | <math>(z_1, z_2; z_3, z_4) = \lambda\,</math>
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| | width=50px |
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| | <math>(z_1, z_2; z_4, z_3) = {1\over\lambda}</math>
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| |-
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| | <math>(z_1, z_3; z_4, z_2) = {1\over{1-\lambda}}</math> ||
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| | <math>(z_1, z_3; z_2, z_4) = 1-\lambda\,</math>
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| |-
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| | <math>(z_1, z_4; z_3, z_2) = {\lambda\over{\lambda-1}}</math> ||
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| | <math>(z_1, z_4; z_2, z_3) = {{\lambda-1}\over\lambda}</math>
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| |}
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| ===Six cross-ratios as Möbius transformations===
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| Viewed as [[Möbius transformation]]s, the six cross-ratios listed above represent torsion elements of [[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 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
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| <math>\frac{1}{1-\lambda}</math> and <math>\frac{\lambda-1}{\lambda}</math> are of order 3 in PGL(2,Z) – in fact in PSL(2,Z). Each of them fixes both values <math>e^{\pm i\pi/3}</math> of the "most symmetric" cross-ratio.
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| The '''anharmonic group''' is the group of order 6 generated by λ→1/λ and λ→1−λ. It is abstractly isomorphic to S<sub>3</sub> and may 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>
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| ===Role of Klein four-group===
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| 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>S_4/K</math>, which is isomorphic to ''S''<sub>3</sub>.
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| ===Exceptional orbits===
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| For certain values of λ there will be an enhanced 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.
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| The first set of fixed points is {0, 1, ∞}. However, the cross-ratio can never take on these values if the points {''z''<sub>''i''</sub>} are all distinct. These values are limit values as one pair of coordinates approach each other:
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| :<math>(z,z_2;z,z_4) = (z_1,z;z_3,z) = 0\,</math>
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| :<math>(z,z;z_3,z_4) = (z_1,z_2;z,z) = 1\,</math>
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| :<math>(z,z_2;z_3,z) = (z_1,z;z,z_4) = \infty.</math>
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| The second set of fixed points is {−1, 1/2, 2}. This situation is what is classically called the '''harmonic cross-ratio,''' and arises in [[projective harmonic conjugates]]. In the real case, there are no other exceptional orbits.
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| The most symmetric cross-ratio occurs when <math>\lambda = e^{\pm i\pi/3}</math>. These are then the only two possible values of the cross-ratio.
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| ==Transformational approach==
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| {{Main|Möbius transformation}}
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| 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 transformation are known as [[Möbius transformation]]s. A general Möbius transformation has the form
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| :<math>f(z) = \frac{az+b}{cz+d}\;,\quad \mbox{where } a,b,c,d\in\Bbb{C} \mbox{ and } ad-bc \ne 0.</math>
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| These transformations form a [[group (mathematics)|group]] [[group action|acting]] on the [[Riemann sphere]], the [[Möbius group]].
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| The projective invariance of the cross-ratio means that
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| :<math>(f(z_1), f(z_2); f(z_3), f(z_4)) = (z_1, z_2; z_3, z_4).\ </math>
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| 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.
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| 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, (''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 (1,0,∞). This transformation can be conveniently described using the cross-ratio: since (''z'',''z''<sub>2</sub>,''z''<sub>3</sub>,''z''<sub>4</sub>) must equal (''f''(''z''),1;0,∞) which in turn equals ''f''(''z''), we obtain
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| :<math>f(z)=(z, z_2; z_3, z_4). \, </math>
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| 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 ''z''<sub>''j''</sub> - ''z''<sub>''k''</sub> are invariant under the [[translation (mathematics)|translations]]
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| : <math>z \mapsto z + a</math>
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| where ''a'' is a [[constant (mathematics)|constant]] in the ground field ''F''. Furthermore, the division ratios are invariant under a [[homothety]]
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| :<math>z \mapsto b z</math>
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| for a non-zero constant ''b'' in ''F''. Therefore, the cross-ratio is invariant under the [[affine transformation]]s.
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| In order to obtain a well-defined [[multiplicative inverse|inversion mapping]]
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| :<math>T : z \mapsto z^{-1},</math>
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| 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 ''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 ''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.
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| ==Differential-geometric point of view==
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| 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.
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| ==Higher-dimensional generalizations==
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| {{Further|General position}}
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| 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]]s are more complicated, and distinct ''k''-tuples of points are not in [[general position]].
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| 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'' + 1)<sup>2</sup> − 1 dimensions (because it is <math>PGL(n,K) = \mathbf{P}(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.
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| 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.
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| ==See also==
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| * [[Homography]]
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| ==Notes and references==
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| <references/>
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| * [[Lars Ahlfors]] (1953,1966,1979) ''Complex Analysis'', 1st edition, page 25; 2nd & 3rd editions, page 78, [[McGraw-Hill]] ISBN 0-07-000657-1 .
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| * Viktor Blåsjö (2009) "[http://www.springerlink.com/content/p01527115762n730/ Jakob Steiner’s Systematische Entwickelung: The Culmination of Classical Geometry]", [[Mathematical Intelligencer]] 31(1): 21–9.
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| * 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]] .
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| * John J. Milne (1911) [http://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]].
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| * [[Dirk Struik]] (1953) ''Lectures on Analytic and Projective Geometry'', page 7, [[Addison-Wesley]].
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| * [[Igor Shafarevich|I. R. Shafarevich]] & A. O. Remizov (2012) ''Linear Algebra and Geometry'', [[Springer Science+Business Media|Springer]] ISBN 978-3-642-30993-9.
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| ==External links==
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| * [http://www.mathpages.com/home/kmath543/kmath543.htm MathPages - Kevin Brown explains the cross-ratio in his article about ''Pascal's Mystic Hexagram'']
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| * [http://www.users.bigpond.com/pmurray/Java/MoebApplet.html Java Applet] demonstrating the invariance of the cross ratio under a bilinear transformation
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| * [http://www.cut-the-knot.org/pythagoras/Cross-Ratio.shtml Cross-Ratio] at [[cut-the-knot]]
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| * {{mathworld|CrossRatio}}
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| {{DEFAULTSORT:Cross-Ratio}}
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| [[Category:Projective geometry]]
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| [[Category:Ratios]]
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