See complex projective line on Wiktionary
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"ref": "1985, B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, translated by R. G. Burns, Modern Geometry — Methods and Applications: Part II, Springer, page 18:",
"text": "By way of an example, we consider in detail the complex projective line #x5C;C#x5C;mathbb#x7B;P#x7D;¹.[…]Thus via the function w#x5F;0 the complex projective line #x5C;C#x5C;mathbb#x7B;P#x7D;¹ becomes identified with the \"extended complex plane\" (i.e. the ordinary complex plane with an additional \"point at infinity\").\n2..2.1 Theorem The complex projective line #x5C;C#x5C;mathbb#x7B;P#x7D;¹ is diffeomorphic to the 2-dimensional sphere S².",
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"ref": "2009, Jean-Pierre Marquis, From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory, Springer, page 14:",
"text": "Finally, C is interested in the geometry of the complex projective line P¹(#x5C;C).",
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"ref": "2013, Angel Cano, Juan Pablo Navarrete, José Seade, Complex Kleinian Groups, Springer (Birkhäuser), page 142:",
"text": "Hence, C(#x5C;Gamma) is an uncountable union of complex projective lines.",
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"ref": "2018, Norman W. Johnson, Geometries and Transformations, Cambridge University Press, page 194:",
"text": "As what is commonly called the Riemann sphere (after Bernhard Riemann, 1826–1866), the parabolic sphere #x5C;dot#x7B;S#x7D;² ̇provides a conformal model for the complex projective line #x5C;C#x5C;mathbb#x7B;P#x7D;¹.",
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"A complex line (especially, the set of complex numbers regarded as such) endowed with a point at infinity (thus becoming a projective line); (more formally) the set of equivalence classes of ordered pairs (α, β) of complex numbers, not both zero, with respect to the equivalence relation \"(α, β) ≡ (λα, λβ) for all nonzero complex λ\"."
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"(analytic geometry, projective geometry) A complex line (especially, the set of complex numbers regarded as such) endowed with a point at infinity (thus becoming a projective line); (more formally) the set of equivalence classes of ordered pairs (α, β) of complex numbers, not both zero, with respect to the equivalence relation \"(α, β) ≡ (λα, λβ) for all nonzero complex λ\"."
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"(analytic geometry, projective geometry) A complex line (especially, the set of complex numbers regarded as such) endowed with a point at infinity (thus becoming a projective line); (more formally) the set of equivalence classes of ordered pairs (α, β) of complex numbers, not both zero, with respect to the equivalence relation \"(α, β) ≡ (λα, λβ) for all nonzero complex λ\"."
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This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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