complex projective line

complex projective line

English

Noun

complex projective line (plural complex projective lines)

1. (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".
   • 1985, R. G. Burns (translator), B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry — Methods and Applications: Part II, Springer, page 18,

     By way of an example, we consider in detail the complex projective line ${\\displaystyle \\mathbb {C} \\mathbb {P} ^{1}}$. […] Thus via the function ${\\displaystyle w_{0}}$ the complex projective line ${\\displaystyle \\mathbb {C} \\mathbb {P} ^{1}}$ 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 ${\\displaystyle \\mathbb {C} \\mathbb {P} ^{1}}$ is diffeomorphic to the 2-dimensional sphere ${\\displaystyle S^{2}}$.

   • 2009, Jean-Pierre Marquis, From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory, Springer, page 14,

     Finally, C is interested in the geometry of the complex projective line ${\\displaystyle P^{1}(\\mathbb {C} )}$.

   • 2013, Angel Cano, Juan Pablo Navarrete, José Seade, Complex Kleinian Groups, Springer (Birkhäuser), page 142,

     Hence, ${\\displaystyle C(\\Gamma )}$ is an uncountable union of complex projective lines.

   • 2018, Norman W. Johnson, Geometries and Transformations, Cambridge University Press, page 194,

     As what is commonly called the Riemann sphere (after Bernhard Riemann, 1826–1866), the parabolic sphere ${\\displaystyle {\\dot {S}}^{2}}$ ̇provides a conformal model for the complex projective line ${\\displaystyle \\mathbb {C} \\mathbb {P} ^{1}}$.

Usage notes

(complex numbers plus point at infinity):

• The formal definition above is a version of the definition of a projective space by homogeneous coordinates. (See also Projective space#Introduction on Wikipedia.Wikipedia )
• Notations include: ${\\displaystyle \\mathbb {P} (\\mathbb {C} ^{2}),\\ \\mathbb {P} _{1}(\\mathbb {C} ){\\text{ and }}\\mathbb {C} \\mathbb {P} ^{1}}$.

Synonyms

• (complex numbers plus point at infinity): extended complex numbers, Riemann sphere

Translations

complex numbers plus point at infinity — See also translations at Riemann sphere

See also

• complex projective plane
• complex projective space
• projective coordinates (= homogeneous coordinates)
• projective line
• real projective line

Further reading

• Riemann sphere#As the complex projective line on Wikipedia.Wikipedia
• Complex projective plane on Wikipedia.Wikipedia
• Complex projective space on Wikipedia.Wikipedia
• Real projective line on Wikipedia.Wikipedia
• Projective line on Wikipedia.Wikipedia
• Complex line on Wikipedia.Wikipedia
• Projective space on Wikipedia.Wikipedia
• Homogeneous coordinates on Wikipedia.Wikipedia
• Projective straight line on Encyclopedia of Mathematics