Dictionary

complex projective line en · NOUN

Meanings

1. (analytic, qualifier: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".
   • By way of an example, we consider in detail the complex projective line #92;C#92;mathbb#123;P#125;¹.[…]Thus via the function w#95;0 the complex projective line #92;C#92;mathbb#123;P#125;¹ becomes identified with the "extended complex plane" (i.e. the ordinary complex plane with an additional "point at infinity"). 2..2.1 Theorem The complex projective line #92;C#92;mathbb#123;P#125;¹ is diffeomorphic to the 2-dimensional sphere S². 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:
   • As what is commonly called the Riemann sphere (after Bernhard Riemann, 1826–1866), the parabolic sphere #92;dot#123;S#125;² ̇provides a conformal model for the complex projective line #92;C#92;mathbb#123;P#125;¹. 2018, Norman W. Johnson, Geometries and Transformations, Cambridge University Press, page 194:
   • Hence, C(#92;Gamma) is an uncountable union of complex projective lines. 2013, Angel Cano, Juan Pablo Navarrete, José Seade, Complex Kleinian Groups, Springer (Birkhäuser), page 142:
   • Finally, C is interested in the geometry of the complex projective line P¹(#92;C). 2009, Jean-Pierre Marquis, From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory, Springer, page 14:

Forms

Relateds

real projective line · projective coordinates · projective line · complex projective plane · complex projective space

Synonyms

extended complex numbers (complex numbers plus point at infinity) · Riemann sphere (complex numbers plus point at infinity)

Translations (1)

pl sfera Riemanna (complex numbers plus point at infinity — see also Riemann sphere)