(ABCD) = ((a-c)/(b-c))/((a-d)/(b-d)).

The right side of this equation is defined as the

The definition is independend of the location of O and E i.e. independent of the special line coordinate system. In fact, changing the coordinate system to some other (defined by two other O', E'), the new coordinate x' is related to the old by a relation of the form x = H(x) = m*x'+n, m and n being appropriate constants. The clue is that by substituting this expression of x into the formula we get the same number expressed in the other coordinates i.e.

More generally the cross ratio of four numbers (abcd) = ((a-c)/(b-c))/((a-d)/(b-d)) is invariant under the broken linear functions x = h(x) = (m*x'+n)/(p*x'+q) and this makes it possible to generalize the cross ratio for four points A, B, C and D lying on a conic (see CrossRatio.html ).

Relations between two variables of the above form: x = h(x) = (m*x'+n)/(p*x'+q), with non-zero

The definition allows for a number to be taken at infinity. F.e. taking d at infinity the cross ratio reduces to (a-c)/(b-c). Taking further c=0, this amounts to a/b. Thus, the ratio of two numbers is the cross ratio (ab0I), I standing for

It is astonishing how many geometric facts depend on the cross-ratio and its properties. In fact, it can be proved that the cross-ratio, considered as a function of four points (variables), is the unique

Below is given a list of properties of the cross ratio.

[2] Given three pairs of real numbers (x

[3] Every permutation which is product of two transpositions of the letters a,b,c,d leaves the cross ratio invariant i.e.

(abcd) = (badc) = (cdab) = (dcba). Hence from the 4! in total permutations of the 4 letters, only 6 give different values.

[4] (abdb) = (abcd)

[5] If (abcd) = k, then

(abdc) = k

(acbd) = 1-k,

(acdb) = 1/(1-k),

(adbc) = (k-1)/k,

(adbc) = k/(k-1).

[4] The four numbers a,b,c,d are pairwise different if and only if (abcd) has a value different from 1, 0 and I (infiinity).

[5] In addition to these formulas, important formulas arise in the case (abcd) = -1 i.e. when the numbers build a

See the file CrossRatioLines.html for the version of cross ratio for a bundle of four concurring lines.

Chasles_Steiner_Envelope.html

Complex_Cross_Ratio.html

Complex_Cross_Ratio2.html

CrossRatio.html

CrossRatioLines.html

FourTangentsCrossRatio.html GoodParametrization.html

Harmonic.html

Harmonic_Bundle.html

HarmonicQuad.html

HomographicRelation.html

HomographicRelationExample.html

ParabolaProperty.html

RectHypeRelation.html

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