# projective line

## English

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### 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, [2002, Introducciòn a la Geometria Avanzada], 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, [1995, D. Perrin, Géométrie algébrique] 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]).