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Examples ofcross ratio
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In this paper, by thecrossratiodistortion we mean the distortion of the secondcrossratio.
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We will rather use the straightforward definition of thecrossratiofor the product of rank-one symmetric spaces.
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Then anycrossratiopreserving homeomorphism on the boundary can be extended isometrically into the space.
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But in our situation, it is not obvious to relate the marked length spectrum to this generalizedcrossratio.
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Then by the definition of thecrossratio, the claim follows.
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Therefore, in every realization, there are four points having the samecross-ratioas thecross-ratio of the four collinear points in the realization derived from the regular pentagon.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thecross-ratiois preserved by the fractional linear transformations and it is essentially the only projective invariant of a quadruple of points, which underlies its importance for projective geometry.
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This example is from Wikipedia and may be reused under a CC BY-SA license.It is also related to thecross-ratio.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The power of the method ofcross-ratio, as an instrument of analysis, it is not easy to over-rate.
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Therefore, thecross-ratiois invariant under the affine transformations.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The computational aspect of this is thecross-ratio.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This can not be extended to 4-tuples of points, because thecross-ratiois invariant.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Using these symmetries, there can then be 6 possible values of thecross-ratio, depending on the order in which the points are given.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The interpretation in terms of the infinitesimalcross-ratiowas given in [21].
From theCambridge English Corpus
The advantage of this representation is that the landmarks are more robust than the singlecross-ratio.
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A harmonic quadruple of lines are four lines whosecross-ratioequals -1.
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This causes an infinite distortion ofcross-ratioat the critical point and hence thecross-ratiocannot be restored in a finite number of iterations.
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Our functionals are expressed in terms of a complex-valued 2-form, which can be considered as thecross-ratioof a pair of infinitesimal segments of the knot.
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Unlike local invariants defined in [4], the infinitesimalcross-ratiocan be defined by local data up to the first derivatives at a pair of points on a curve.
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Lengths are not invariant under inversion transformations but there is across-ratioon four points that is invariant.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.