"complex projective line" meaning in English

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  "lang_code": "en",
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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.",
          "type": "quotation"
        },
        {
          "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).",
          "type": "quotation"
        },
        {
          "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.",
          "type": "quotation"
        },
        {
          "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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      "glosses": [
        "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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          "point at infinity",
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          "projective line",
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          "equivalence class",
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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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          "word": "complex projective plane"
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          "word": "complex projective space"
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        {
          "english": "= homogeneous coordinates",
          "word": "projective coordinates"
        },
        {
          "word": "projective line"
        },
        {
          "word": "real projective line"
        }
      ],
      "synonyms": [
        {
          "sense": "complex numbers plus point at infinity",
          "word": "extended complex numbers"
        },
        {
          "sense": "complex numbers plus point at infinity",
          "word": "Riemann sphere"
        }
      ],
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        "analytic"
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  "lang_code": "en",
  "pos": "noun",
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      "english": "= homogeneous coordinates",
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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.",
          "type": "quotation"
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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).",
          "type": "quotation"
        },
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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.",
          "type": "quotation"
        },
        {
          "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.",
          "type": "quotation"
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      "glosses": [
        "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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      "qualifier": "projective geometry",
      "raw_glosses": [
        "(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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  "synonyms": [
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      "sense": "complex numbers plus point at infinity",
      "word": "extended complex numbers"
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      "sense": "complex numbers plus point at infinity",
      "word": "Riemann sphere"
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  "word": "complex projective line"
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