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Cross-ratio

From Wikipedia, the free encyclopedia

Invariant in projective geometry

Not to be confused withOdds ratio.

PointsA,B,C,D andA′,B′,C′,D′ are related by a projective transformation so their cross ratios,(A,B;C,D) and(A′,B′;C′,D′) are equal.

Ingeometry, thecross-ratio, also called thedouble ratio andanharmonic ratio, is a number associated with a list of fourcollinear points, particularly points on aprojective line. Given four pointsA,B,C,D on a line, their cross ratio is defined as

(A,B;C,D)={\frac {AC\cdot BD}{BC\cdot AD}}

where an orientation of the line determines the sign of each distance and the distance is measured as projected intoEuclidean 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 pointD is theharmonic conjugate ofC with respect toA andB precisely if the cross-ratio of the quadruple is−1, called theharmonic ratio. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the nameanharmonic ratio.

The cross-ratio is preserved bylinear fractional transformations. It is essentially the only projectiveinvariant of a quadruple of collinear points; this underlies its importance forprojective geometry.

The cross-ratio had been defined in deep antiquity, possibly already byEuclid, and was considered byPappus, who noted its key invariance property. It was extensively studied in the 19th century.^[1]

Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on theRiemann sphere.In theCayley–Klein model ofhyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.

Terminology and history

D is theharmonic conjugate ofC with respect toA andB, so that the cross-ratio(A,B;C,D) equals −1.

Pappus of Alexandria made implicit use of concepts equivalent to the cross-ratio in hisCollection: Book VII. Early users of Pappus includedIsaac Newton,Michel Chasles, andRobert 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.^[2]

Modern use of the cross ratio in projective geometry began withLazare Carnot in 1803 with his bookGéométrie de Position.^[3]^{[pages needed]} Chasles coined the French termrapport anharmonique [anharmonic ratio] in 1837.^[4] German geometers call itdas Doppelverhältnis [double ratio].

Carl von Staudt was unsatisfied with past definitions of the cross-ratio relying on algebraic manipulation of Euclidean distances rather than being based purely on synthetic projective geometry concepts. In 1847, von Staudt demonstrated that the algebraic structure is implicit in projective geometry, by creating an algebra based on construction of theprojective harmonic conjugate, which he called athrow (German:Wurf): given three points on a line, the harmonic conjugate is a fourth point that makes the cross ratio equal to−1. Hisalgebra of throws provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry.^[5]

The English term "cross-ratio" was introduced in 1878 byWilliam Kingdon Clifford.^[6]

Definition

IfA,B,C, andD are four points on an orientedaffine line, their cross ratio is:

(A,B;C,D)={\frac {AC:BC}{AD:BD}},

with the notation $W X : Y Z {\displaystyle WX:YZ}$ defined to mean the signed ratio of the displacement fromW toX to the displacement fromY toZ. For collinear displacements this is adimensionless quantity.

If the displacements themselves are taken to be signed real numbers, then the cross ratio between points can be written

(A,B;C,D)={\frac {AC}{BC}}{\bigg /}{\frac {AD}{BD}}={\frac {AC\cdot BD}{BC\cdot AD}}.

If ${\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}$ is theprojectively extended real line, the cross-ratio of four distinct numbers $x_{1},x_{2},x_{3},x_{4}$ in ${\widehat {\mathbb {R} }}$ is given by

(x_{1},x_{2};x_{3},x_{4})={\frac {x_{3}-x_{1}}{x_{3}-x_{2}}}{\bigg /}{\frac {x_{4}-x_{1}}{x_{4}-x_{2}}}={\frac {(x_{3}-x_{1})(x_{4}-x_{2})}{(x_{3}-x_{2})(x_{4}-x_{1})}}.

When one of $x_{1},x_{2},x_{3},x_{4}$ is thepoint at infinity( $\infty$ ), this reduces to e.g.

(\infty ,x_{2};x_{3},x_{4})={\frac {(x_{3}-\infty )(x_{4}-x_{2})}{(x_{3}-x_{2})(x_{4}-\infty )}}={\frac {(x_{4}-x_{2})}{(x_{3}-x_{2})}}.

The same formulas can be applied to four distinctcomplex numbers or, more generally, to elements of anyfield, and can also be projectively extended as above to the case when one of them is $\infty ={\tfrac {1}{0}}.$

The cross ratio can for example be defined for pencils of lines, circles, or conics. For instance, the cross-ratio of $4 {\displaystyle 4}$ coaxial circles can be defined in numerous equivalent ways:

Let $A {\displaystyle A}$ be a point of intersection of the circles on their radical axis, if it exists. Then the cross-ratio of the circles can be defined as the cross-ratio of the tangents to the circles through $A {\displaystyle A}$ .
More generally, given any point in the plane, the $4 {\displaystyle 4}$ polars of this point with respect to those circles are concurrent and their cross-ratio doesn't depend on the chosen point.
By taking the lines orthogonal to the tangents at $A {\displaystyle A}$ and projecting on the line on which lies the circle centers, we deduce it is equal to the cross-ratio of the circle centers.
It can be proven through an inversion that the cross-ratio of these circles can be equivalently defined as the cross-ratio of the second points of intersection different than $A {\displaystyle A}$ of a circle (and in a degenerate case a line) that passes through $A {\displaystyle A}$ .

Properties

The cross ratio of the four collinear pointsA,B,C, andD can be written as

(A,B;C,D)={\frac {AC:CB}{AD:DB}}

where ${\textstyle AC:CB}$ describes the ratio with which the pointC divides the line segmentAB, and ${\textstyle AD:DB}$ describes the ratio with which the pointD divides that same line segment. The cross ratio then appears as a ratio of ratios, describing how the two pointsC andD are situated with respect to the line segmentAB. As long as the pointsA,B,C, andD 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 pointsC orD lies between the pointsA andB 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 in4! = 4 × 3 × 2 × 1 = 24 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:

{\begin{aligned}&(A,B;C,D)=(B,A;D,C)=(C,D;A,B)=(D,C;B,A)=\lambda ,{\vphantom {\frac {1}{1}}}\\[4mu]&(A,B;D,C)=(B,A;C,D)=(C,D;B,A)=(D,C;A,B)={\frac {1}{\lambda }},\\[4mu]&(A,C;B,D)=(B,D;A,C)=(C,A;D,B)=(D,B;C,A)=1-\lambda ,{\vphantom {\frac {1}{1}}}\\[4mu]&(A,C;D,B)=(B,D;C,A)=(C,A;B,D)=(D,B;A,C)={\frac {1}{1-\lambda }},\\[4mu]&(A,D;B,C)=(B,C;A,D)=(C,B;D,A)=(D,A;C,B)={\frac {\lambda -1}{\lambda }},\\[4mu]&(A,D;C,B)=(B,C;D,A)=(C,B;A,D)=(D,A;B,C)={\frac {\lambda }{\lambda -1}}.\end{aligned}}

SeeAnharmonic group below.

Projective geometry

Further information:Projective geometry

Use ofcross-ratios inprojective geometry to measure real-world dimensions of features depicted in aperspective projection. A, B, C, D and V are points on the image, their separation given in pixels; A', B', C' and D' are in the real world, their separation in metres.
1. The width of the side street, W is computed from the known widths of the adjacent shops.
2. As avanishing point, V is visible, the width of only one shop is needed.

The cross-ratio is aprojectiveinvariant in the sense that it is preserved by theprojective transformations of a projective line.

In particular, if four points lie on a straight line ${\textstyle L}$ in ${\textstyle {\mathbf {R}}^{2}}$ 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 ${\textstyle \{L_{i}\mid 1\leq i\leq 4\}}$ be four distinct lines in the plane passing through the same point ${\textstyle Q}$ . Then any line ${\textstyle L}$ not passing through ${\textstyle Q}$ intersects these lines in four distinct points ${\textstyle P_{i}}$ (if ${\textstyle L}$ isparallel to ${\textstyle L_{i}}$ 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 ${\textstyle L}$ , and hence it is an invariant of the 4-tuple of lines ${\textstyle L_{i}.}$

This can be understood as follows: if ${\textstyle L}$ and ${\textstyle L'}$ are two lines not passing through ${\textstyle Q}$ then the perspective transformation from ${\textstyle L}$ to ${\textstyle L'}$ with the center ${\textstyle Q}$ is a projective transformation that takes the quadruple ${\textstyle \{P_{i}\}}$ of points on ${\textstyle L}$ into the quadruple ${\textstyle \{P_{i}'\}}$ of points on ${\textstyle L'}$ .

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 fourcollinear points ${\textstyle \{P_{i}\}}$ on the lines ${\textstyle \{L_{i}\}}$ from the choice of the line that contains them.

Definition in homogeneous coordinates

If four collinear points are represented inhomogeneous coordinates by vectors $\alpha ,\beta ,\gamma ,\delta$ such that $\gamma =a\alpha +b\beta$ and $\delta =c\alpha +d\beta$ , then their cross-ratio is $(b/a)/(d/c)$ .^[7]

Role in non-Euclidean geometry

Arthur Cayley andFelix Klein found an application of the cross-ratio tonon-Euclidean geometry. Given a nonsingularconic $C {\displaystyle C}$ in the realprojective plane, itsstabilizer $G_{C}$ in theprojective group $G=\operatorname {PGL} (3,\mathbb {R} )$ acts transitively on the points in the interior of $C {\displaystyle C}$ . However, there is an invariant for the action of $G_{C}$ onpairs of points. In fact, every such invariant is expressible as a function of the appropriate cross-ratio.^{[citation needed]}

Hyperbolic geometry

Explicitly, let the conic be theunit circle. For any two pointsP andQ, inside the unit circle . If the line connecting them intersects the circle in two points,X andY and the points are, in order,X,P,Q,Y. Then the hyperbolic distance betweenP andQ in theCayley–Klein model of thehyperbolic plane can be expressed as

d_{h}(P,Q)={\frac {1}{2}}\left|\log {\frac {|XQ||PY|}{|XP||QY|}}\right|

(the factor one half is needed to make thecurvature−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 conicC.

Conversely, the groupG acts transitively on the set of pairs of points(p,q) in the unit disk at a fixed hyperbolic distance.

Later, partly through the influence ofHenri Poincaré, the cross ratio of fourcomplex 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 aMöbius transformation, and hence the cross ratio is a real number. ThePoincaré half-plane model andPoincaré disk model are two models of hyperbolic geometry in thecomplex projective line.

These models are instances ofCayley–Klein metrics.

Anharmonic group and Klein four-group

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

(A,B;C,D)=(B,A;D,C)=(C,D;A,B)=(D,C;B,A).

These differ by the followingpermutations of the variables (incycle notation):

1,\ (\,A\,B\,)(\,C\,D\,),\ (\,A\,C\,)(\,B\,D\,),\ (\,A\,D\,)(\,B\,C\,).

We may consider the permutations of the four variables as anaction of thesymmetric groupS₄ on functions of the four variables. Since the above four permutations leave the cross ratio unaltered, they form thestabilizerK of the cross-ratio under this action, and this induces aneffective action of thequotient group $\mathrm {S} _{4}/K$ on the orbit of the cross-ratio. The four permutations inK provide a realization of theKlein four-group inS₄, and the quotient $\mathrm {S} _{4}/K$ is isomorphic to the symmetric groupS₃.

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 $\mathrm {S} _{4}/K\cong \mathrm {S} _{3}$ :

{\begin{aligned}(A,B;C,D)&=\lambda &(A,B;D,C)&={\frac {1}{\lambda }},\\[4mu](A,C;D,B)&={\frac {1}{1-\lambda }}&(A,C;B,D)&=1-\lambda ,\\[4mu](A,D;C,B)&={\frac {\lambda }{\lambda -1}}&(A,D;B,C)&={\frac {\lambda -1}{\lambda }}.\end{aligned}}

The stabilizer of{0, 1, ∞} is isomorphic to therotation group of thetrigonal dihedron, thedihedral groupD₃. It is convenient to visualize this by aMöbius transformationM mapping the real axis to the complex unit circle (the equator of theRiemann sphere), with0, 1, ∞ equally spaced.
Considering{0, 1, ∞} as the vertices of the dihedron, the other fixed points of the2-cycles are the points{2, −1, 1/2}, which underM are opposite each vertex on the Riemann sphere, at the midpoint of the opposite edge. Each2-cycles is a half-turn rotation of the Riemann sphere exchanging the hemispheres (the interior and exterior of the circle in the diagram).
The fixed points of the3-cycles areexp(±iπ/3), corresponding underM to the poles of the sphere:exp(iπ/3) is the origin andexp(−iπ/3) is thepoint at infinity. Each3-cycle is a1/3 turn rotation about their axis, and they are exchanged by the2-cycles.

As functions of $\lambda ,$ these are examples ofMöbius transformations, which under composition of functions form the Mobius groupPGL(2,C). The six transformations form a subgroup known as theanharmonic group, again isomorphic toS₃. They are the torsion elements (elliptic transforms) inPGL(2,C). Namely, ${\textstyle {\tfrac {1}{\lambda }}}$ , $1-\lambda \,$ , and ${\textstyle {\tfrac {\lambda }{\lambda -1}}}$ are of order2 with respectivefixed points $-1,$ ${\textstyle {\tfrac {1}{2}},}$ and $2,$ (namely, the orbit of the harmonic cross-ratio). Meanwhile, the elements ${\textstyle {\tfrac {1}{1-\lambda }}}$ and ${\textstyle {\tfrac {\lambda -1}{\lambda }}}$ are of order3 inPGL(2,C), and each fixes both values ${\textstyle e^{\pm i\pi /3}={\tfrac {1}{2}}\pm {\tfrac {\sqrt {3}}{2}}i}$ of the "most symmetric" cross-ratio (the solutions to $x^{2}-x+1$ , theprimitive sixthroots of unity). The order2 elements exchange these two elements (as they do any pair other than their fixed points), and thus the action of the anharmonic group on $e^{\pm i\pi /3}$ gives the quotient map of symmetric groups $\mathrm {S} _{3}\to \mathrm {S} _{2}$ .

Further, the fixed points of the individual2-cycles are, respectively, $-1,$ ${\textstyle {\tfrac {1}{2}},}$ and $2,$ and this set is also preserved and permuted by the3-cycles. Geometrically, this can be visualized as therotation group of thetrigonal dihedron, which is isomorphic to thedihedral group of the triangleD₃, as illustrated at right. Algebraically, this corresponds to the action ofS₃ on the2-cycles (itsSylow 2-subgroups) by conjugation and realizes the isomorphism with the group ofinner automorphisms, ${\textstyle \mathrm {S} _{3}\mathrel {\overset {\sim }{\to }} \operatorname {Inn} (\mathrm {S} _{3})\cong \mathrm {S} _{3}.}$

The anharmonic group is generated by ${\textstyle \lambda \mapsto {\tfrac {1}{\lambda }}}$ and ${\textstyle \lambda \mapsto 1-\lambda .}$ Its action on $\{0,1,\infty \}$ gives an isomorphism withS₃. It may also be realised as the six Möbius transformations mentioned,^[8] which yields a projectiverepresentation ofS₃ over any field (since it is defined with integer entries), and is always faithful/injective (since no two terms differ only by1/−1). Over the field with two elements, the projective line only has three points, so this representation is an isomorphism, and is theexceptional isomorphism $\mathrm {S} _{3}\approx \mathrm {PGL} (2,\mathbb {Z} _{2})$ . In characteristic3, this stabilizes the point $-1=[-1:1]$ , which corresponds to the orbit of the harmonic cross-ratio being only a single point, since ${\textstyle 2={\tfrac {1}{2}}=-1}$ . Over the field with three elements, the projective line has only 4 points and $\mathrm {S} _{4}\approx \mathrm {PGL} (2,\mathbb {Z} _{3})$ , and thus the representation is exactly the stabilizer of the harmonic cross-ratio, yielding an embedding $\mathrm {S} _{3}\hookrightarrow \mathrm {S} _{4}$ equals the stabilizer of the point $-1$ .

Exceptional orbits

For certain values of $\lambda$ there will be greater symmetry and therefore fewer than six possible values for the cross-ratio. These values of $\lambda$ correspond tofixed points of the action ofS₃ on the Riemann sphere (given by the above six functions); or, equivalently, those points with a non-trivialstabilizer in this permutation group.

The first set of fixed points is $\{0,1,\infty \}.$ However, the cross-ratio can never take on these values if the pointsA,B,C, andD are all distinct. These values are limit values as one pair of coordinates approach each other:

{\begin{aligned}(Z,B;Z,D)&=(A,Z;C,Z)=0,\\[4mu](Z,Z;C,D)&=(A,B;Z,Z)=1,\\[4mu](Z,B;C,Z)&=(A,Z;Z,D)=\infty .\end{aligned}}

The second set of fixed points is ${\textstyle {\big \{}{-1},{\tfrac {1}{2}},2{\big \}}.}$ This situation is what is classically called theharmonic cross-ratio, and arises inprojective harmonic conjugates. In the real case, there are no other exceptional orbits.

In the complex case, the most symmetric cross-ratio occurs when $\lambda =e^{\pm i\pi /3}$ . 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 article:Möbius transformation

The cross-ratio is invariant under theprojective transformations of the line. In the case of acomplex projective line, or theRiemann sphere, these transformations are known asMöbius transformations. A general Möbius transformation has the form

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

These transformations form agroup acting on theRiemann sphere, theMöbius group.

The projective invariance of the cross-ratio means that

(f(z_{1}),f(z_{2});f(z_{3}),f(z_{4}))=(z_{1},z_{2};z_{3},z_{4}).\

The cross-ratio isreal if and only if the four points are eithercollinear orconcyclic, reflecting the fact that every Möbius transformation mapsgeneralized circles to generalized circles.

The action of the Möbius group issimply transitive on the set of triples of distinct points of the Riemann sphere: given any ordered triple of distinct points, $(z_{2},z_{3},z_{4})$ , there is a unique Möbius transformation $f(z)$ that maps it to the triple $(0,1,\infty )$ . This transformation can be conveniently described using the cross-ratio: since $(z,z_{2};z_{3},z_{4})$ must equal $(f(z),1;0,\infty )$ , which in turn equals $f(z)$ , we obtain

f(z)=(z,z_{2};z_{3},z_{4}).

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 $z_{j}-z_{k}$ are invariant under thetranslations

z\mapsto z+a

where $a {\displaystyle a}$ is aconstant in the ground field $\mathbb {F}$ . Furthermore, the division ratios are invariant under ahomothety

z\mapsto bz

for a non-zero constant $b {\displaystyle b}$ in $\mathbb {F}$ . Therefore, the cross-ratio is invariant under theaffine transformations.

In order to obtain a well-definedinversion mapping

T:z\mapsto z^{-1},

the affine line needs to be augmented by thepoint at infinity, denoted $\infty$ , forming the projective line $\mathrm {P} ^{1}(\mathbb {F} )$ . Each affine mapping $f:\mathbb {F} \to \mathbb {F}$ can be uniquely extended to a mapping of $\mathrm {P} ^{1}(\mathbb {F} )$ into itself that fixes the point at infinity. The map $T {\displaystyle T}$ swaps $0 {\displaystyle 0}$ and $\infty$ . The projective group isgenerated by $T {\displaystyle T}$ and the affine mappings extended to $\mathrm {P} ^{1}(\mathbb {F} )$ . In the case $\mathbb {F} =\mathbb {C}$ , thecomplex plane, this results in theMöbius group. Since the cross-ratio is also invariant under $T {\displaystyle T}$ , it is invariant under any projective mapping of $\mathrm {P} ^{1}(\mathbb {F} )$ into itself.

Co-ordinate description

If we write the complex points as vectors ${\vec {x_{n}}}=[\Re (z_{n}),\Im (z_{n})]^{\mathrm {T} }$ and define $x_{nm}=x_{n}-x_{m}$ , and let $(a,b)$ be thedot product of $a {\displaystyle a}$ with $b {\displaystyle b}$ , then the real part of the cross ratio is given by:

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}}}

This is an invariant of the 2-dimensionalspecial conformal transformation such as inversion $x^{\mu }\rightarrow {\frac {x^{\mu }}{|x|^{2}}}$ .

The imaginary part must make use of the 2-dimensional cross product $a\times b=[a,b]=a_{2}b_{1}-a_{1}b_{2}$

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}}}

Ring homography

The concept of cross ratio only depends on thering 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 ahomography that takes three designated points to0, 1, and∞. Under restrictions having to do with inverses, it is possible to generate such a mapping with ring operations in theprojective 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 theSchwarzian derivative, and more generally ofprojective connections.

Higher-dimensional generalizations

Further information: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 spaces are more complicated, and distinctk-tuples of points are not ingeneral position.

While the projective linear group of the projective line is 3-transitive (any three distinct points can be mapped to any other three points), and indeed simply 3-transitive (there is aunique 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 ofn-space $\mathbf {P} ^{n}=\mathbf {P} (K^{n+1})$ has(n + 1)² − 1 dimensions (because it is $\mathrm {PGL} (n,K)=\mathbf {P} (\mathrm {GL} (n+1,K)),$ 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 ofn² 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 ingeneral 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 toRiemann surfaces of positivegenus exists, using theAbel–Jacobi map andtheta functions.

See also

Hilbert metric

Notes

^A theorem on the anharmonic ratio of lines appeared in the work ofPappus, butMichel Chasles, who devoted considerable efforts to reconstructing lost works ofEuclid, asserted that it had earlier appeared in his bookPorisms.
^Alexander Jones (1986)Book 7 of the Collection, part 1: introduction, text, translationISBN 0-387-96257-3, part 2: commentary, index, figuresISBN 3-540-96257-3,Springer-Verlag
^Carnot, Lazare (1803).Géométrie de Position. Crapelet.
^Chasles, Michel (1837).Aperçu historique sur l'origine et le développement des méthodes en géométrie. Hayez. p. 35. (Link is to the reprinted second edition, Gauthier-Villars: 1875.)
^Howard Eves (1972)A Survey of Geometry, Revised Edition, page 73,Allyn and Bacon
^W.K. Clifford (1878)Elements of Dynamic, books I,II,III, page 42, London: MacMillan & Co; on-line presentation byCornell UniversityHistorical Mathematical Monographs.
^Irving Kaplansky (1969).Linear Algebra and Geometry: A Second Course. Courier Corporation.ISBN 0-486-43233-5.
^Chandrasekharan, K. (1985).Elliptic Functions. Grundlehren der mathematischen Wissenschaften. Vol. 281.Springer-Verlag. p. 120.ISBN 3-540-15295-4.Zbl 0575.33001.

References

Lars Ahlfors (1953,1966,1979)Complex Analysis, 1st edition, page 25; 2nd & 3rd editions, page 78,McGraw-HillISBN 0-07-000657-1 .
Viktor Blåsjö (2009) "Jakob Steiner's Systematische Entwickelung: The Culmination of Classical Geometry",Mathematical Intelligencer 31(1): 21–9.
John J. Milne (1911)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.
I. R. Shafarevich & A. O. Remizov (2012)Linear Algebra and Geometry,SpringerISBN 978-3-642-30993-9.

External links

MathPages – Kevin Brown explains the cross-ratio in his article aboutPascal's Mystic Hexagram
Cross-Ratio atcut-the-knot
Weisstein, Eric W."Cross-ratio".MathWorld.
Ardila, Federico (6 July 2018)."The Cross Ratio"(video).youtube.Brady Haran.Archived from the original on 2021-12-12. Retrieved6 July 2018.

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