Projective line

English

Noun

1. Template:Lb 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, [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:

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

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

   • 2008, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique] Springer, page 37,

     Consider the projective line ${𝐏}^1$, with homogeneous coordinates $x$ and $t$ and open sets $U_0(x\ne 0)$ and $U_1(t\ne 0)$.

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Further reading

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• Projective straight line on Template:W
• Point at Infinity on Template:W
• Projective lines on Template:W