The value of a cross ratio in general is not unique, as it depends on the order of selection of segments ( and there are six such selections possible ).
12.
If these segments are now endowed with the ordinary metric interpretation of real numbers they will be " signed " and form a double proportion known as the cross ratio ( sometimes " double ratio ")
13.
Laguerre formula can be useful in computer vision, since the absolute conic has an image on the retinal plane which is invariant under camera displacements, and the cross ratio of four collinear points is the same for their images on the retinal plane.
14.
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15.
This therefore means that " R " ( " QB "; " CY " ) } }, where one of the points in the two tetrads overlap, hence meaning that other lines connecting the other three pairs must coincide to preserve cross ratio.
16.
The Zassenhaus groups notably include the projective general linear group of a projective line over a finite field, PGL ( 2, "'F " "'q " ), which is sharply 3-transitive ( see cross ratio ) on q + 1 elements.
17.
The explanation is " since the cross ratio of z _ 1, z _ 2, z _ 3, z _ 4 is the image of the ( unique ) projective transformation carrying z _ 1 to 1, z _ 2 to infinity and z _ 3 to 0 ".
18.
Note that not all points in general position are projectively equivalent, which is a much stronger condition; for example, any " k " distinct points in the line are in general position, but projective transformations are only 3-transitive, with the invariant of 4 points being the cross ratio.
19.
But for a harmonic range in particular there are just three values of cross ratio : since & minus; 1 is self-inverse so exchanging the last two points merely reciprocates each of these values but produces no new value, and is known classically as the "'harmonic cross-ratio " '.
20.
In fact the homeomorphisms of "'S "'1 induced by quasiconformal self-maps of "'D "'are precisely the quasisymmetric homeomorphisms of "'S "'1; these are exactly homeomorphisms which do not send four points with cross ratio 1 / 2 to points with cross ratio near 1 or 0.
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