projective line

English

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projective line

Wikipedia

Noun

projective line (plural projective lines)

1. (projective geometry) A line that includes a point at infinity; a line in a projective space; a projective space of dimension 1.
   • 2007, Unnamed translator, Ana Irene Ramírez Galarza, José Seade, Introduction to Classical Geometries, , Springer (Birkhäuser), page 97,

     In ${\displaystyle P2(\mathbb {R} )}$, the projective lines are defined by two projective points, that is, by two linearly independent directions of ${\displaystyle \mathbb {R} ^{3}}$; if we take one vector for each direction, the two vectors generate a plane through the origin in ${\displaystyle \mathbb {R} ^{3}}$, that is, a subspace of dimension 2, and a projective line can be defined as follows:

     A projective line in ${\displaystyle P^{2}(\mathbb {R} )}$, consists of the projective points defined by coplanar directions in ${\displaystyle \mathbb {R} ^{3}}$.

     In other words, just as the points in ${\displaystyle P^{2}(\mathbb {R} )}$ correspond to one-dimensional subspaces in ${\displaystyle \mathbb {R} ^{3}}$, the projective lines correspond to two-dimensional subspaces in ${\displaystyle \mathbb {R} ^{3}}$.

   • 2008, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, Springer, page 37,

     Consider the projective line ${\displaystyle \mathbf {P} ^{1}}$, with homogeneous coordinates ${\displaystyle x}$ and ${\displaystyle t}$ and open sets ${\displaystyle U_{0}\ (x\neq 0)}$ and ${\displaystyle U_{1}\ (t\neq 0)}$.

   • 2009, Dirk Kussin, Noncommutative Curves of Genus Zero: Related to Finite Dimensional Algebras, American Mathematical Society, page 13:

     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]).

Synonyms

• (line with a point at infinity): projective straight line

Derived terms

• complex projective line

Translations

See also

• projective completion
• projective plane
• projective space

Further reading

• Point at infinity on Wikipedia.Wikipedia
• Projective geometry on Wikipedia.Wikipedia
• Projective plane on Wikipedia.Wikipedia
• Projective space on Wikipedia.Wikipedia
• Projection (mathematics) on Wikipedia.Wikipedia
  • Graphical projection on Wikipedia.Wikipedia
• Perspective (graphical) on Wikipedia.Wikipedia
  • Vanishing point on Wikipedia.Wikipedia
• Projective straight line on Encyclopedia of Mathematics
• Point at Infinity on Wolfram MathWorld
• Projective lines on nLab