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Examples ofcollineation
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Such a mapping permutes the lines of the incidence structure, and the notion ofcollineationpersists.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Some authors restrict the definition ofcollineationto the case where it is an automorphism.
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This example is from Wikipedia and may be reused under a CC BY-SA license.A duality is acollineationfrom a projective space onto its dual space, taking points to hyperplanes (and vice versa) and preserving incidence.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Every affine vector field is a curvaturecollineation.
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This example is from Wikipedia and may be reused under a CC BY-SA license.One may also definecollineationgroups for axiomatically defined projective spaces, where there is no natural notion of a projective "linear" transform.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thus, the fixed point and fixed line structure for anycollineationeither form a projective plane by themselves, or a degenerate plane.
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This example is from Wikipedia and may be reused under a CC BY-SA license.A mapping of a geometry to itself which sends lines to lines is called acollineation, it preserves the collinearity property.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Synonyms include projectivity, projective transformation, and projectivecollineation.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The composition of two central collineations, while still a homography, will in general not be a centralcollineation.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Not that a centralcollineationis uniquely defined by its center, its axis, a point and its image.
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This example is from Wikipedia and may be reused under a CC BY-SA license.More precisely, thecollineationgroup is the projective semilinear group, which is the semidirect product of homographies by automorphic collineations.
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Wikipedia
This example is from Wikipedia and may be reused under a CC BY-SA license.Acollineationis an invertible (or more generally one-to-one) map which sends collinear points to collinear points.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Indeed, cross-ratio is invariant under anycollineation, and the stable absolute enables the metric comparison, which will be equality.
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This example is from Wikipedia and may be reused under a CC BY-SA license.It is obvious now that is acollineation.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Acollineationis thus an "isomorphism" between projective spaces, or an automorphism from a projective space to itself.
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This example is from Wikipedia and may be reused under a CC BY-SA license.It is a bijection that maps lines to lines, and thus acollineation.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The geometric view of a centralcollineationis easiest to see in a projective plane.
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Wikipedia
This example is from Wikipedia and may be reused under a CC BY-SA license.A permutation of the seven points that carries collinear points (points on the same line) to collinear points is called a "collineation" or "symmetry" of the plane.
FromThis example is from Wikipedia and may be reused under a CC BY-SA license.
Wikipedia
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.
- downward price
- downward revision
- downwards
- downward slide
- downward slope
- downward stroke
- downward trend
- downward turn
- down-way
- downwind
- down with...!
- down-with
- downy
- downy feathers
- downy mildew
- downy-mildew
- downzoning
- do-wonders-for
- do wonders for something/someone
- do-worst
- dowry
- dowse
- dowsed
- dowsing
- dox