"projective line" meaning in All languages combined

{
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      "form": "projective lines",
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        "plural"
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  "lang_code": "en",
  "pos": "noun",
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      "derived": [
        {
          "word": "complex projective line"
        }
      ],
      "examples": [
        {
          "ref": "2007, Unnamed translator, Ana Irene Ramírez Galarza, José Seade, Introduction to Classical Geometries, [2002, Introducciòn a la Geometria Avanzada], Springer (Birkhäuser), page 97, In P2( R ), the projective lines are defined by two projective points, that is, by two linearly independent directions of R ³; if we take one vector for each direction, the two vectors generate a plane through the origin in R ³, that is, a subspace of dimension 2, and a projective line can be defined as follows",
          "text": "A projective line in P²( R ), consists of the projective points defined by coplanar directions in R ³.\nIn other words, just as the points in P²( R ) correspond to one-dimensional subspaces in R ³, the projective lines correspond to two-dimensional subspaces in R ³."
        },
        {
          "text": "2008, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique] Springer, page 37,\nConsider the projective line mathbf P¹, with homogeneous coordinates x and t and open sets U_0(x ne 0) and U_1(t ne 0)."
        },
        {
          "ref": "2009, Dirk Kussin, Noncommutative Curves of Genus Zero: Related to Finite Dimensional Algebras, American Mathematical Society, page 13:",
          "text": "The projective line is related to the Kronecker algebra,[…]In general one has to deal with the so-called weighted case which leads to the study of the canonical algebras and to the weighted projective lines (as Ringel pointed out in his survey [93]).",
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        "A line that includes a point at infinity; a line in a projective space; a projective space of dimension 1."
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          "line",
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          "point at infinity",
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          "projective space",
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      "qualifier": "projective geometry",
      "raw_glosses": [
        "(projective geometry) A line that includes a point at infinity; a line in a projective space; a projective space of dimension 1."
      ],
      "related": [
        {
          "word": "projective completion"
        },
        {
          "word": "projective plane"
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        {
          "word": "projective space"
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      "synonyms": [
        {
          "sense": "line with a point at infinity",
          "word": "projective straight line"
        }
      ],
      "translations": [
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          "code": "it",
          "lang": "Italian",
          "sense": "line with point at infinity",
          "tags": [
            "feminine"
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          "word": "retta proiettiva"
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        "projective line"
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  "word": "projective line"
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      "word": "projective completion"
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      "word": "projective plane"
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      "word": "projective space"
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          "ref": "2007, Unnamed translator, Ana Irene Ramírez Galarza, José Seade, Introduction to Classical Geometries, [2002, Introducciòn a la Geometria Avanzada], Springer (Birkhäuser), page 97, In P2( R ), the projective lines are defined by two projective points, that is, by two linearly independent directions of R ³; if we take one vector for each direction, the two vectors generate a plane through the origin in R ³, that is, a subspace of dimension 2, and a projective line can be defined as follows",
          "text": "A projective line in P²( R ), consists of the projective points defined by coplanar directions in R ³.\nIn other words, just as the points in P²( R ) correspond to one-dimensional subspaces in R ³, the projective lines correspond to two-dimensional subspaces in R ³."
        },
        {
          "text": "2008, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique] Springer, page 37,\nConsider the projective line mathbf P¹, with homogeneous coordinates x and t and open sets U_0(x ne 0) and U_1(t ne 0)."
        },
        {
          "ref": "2009, Dirk Kussin, Noncommutative Curves of Genus Zero: Related to Finite Dimensional Algebras, American Mathematical Society, page 13:",
          "text": "The projective line is related to the Kronecker algebra,[…]In general one has to deal with the so-called weighted case which leads to the study of the canonical algebras and to the weighted projective lines (as Ringel pointed out in his survey [93]).",
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      ],
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        "(projective geometry) A line that includes a point at infinity; a line in a projective space; a projective space of dimension 1."
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  "synonyms": [
    {
      "sense": "line with a point at infinity",
      "word": "projective straight line"
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  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "line with point at infinity",
      "tags": [
        "feminine"
      ],
      "word": "retta proiettiva"
    }
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  "word": "projective line"
}