# projective line

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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 $P2(\mathbb {R} )$ , the projective lines are defined by two projective points, that is, by two linearly independent directions of $\mathbb {R} ^{3}$ ; if we take one vector for each direction, the two vectors generate a plane through the origin in $\mathbb {R} ^{3}$ , that is, a subspace of dimension 2, and a projective line can be defined as follows:
A projective line in $P^{2}(\mathbb {R} )$ , consists of the projective points defined by coplanar directions in $\mathbb {R} ^{3}$ .
In other words, just as the points in $P^{2}(\mathbb {R} )$ correspond to one-dimensional subspaces in $\mathbb {R} ^{3}$ , the projective lines correspond to two-dimensional subspaces in $\mathbb {R} ^{3}$ .
• 2008, Catriona Maclean (translator), Daniel Perrin, Algebraic Geometry: An Introduction, [1995, D. Perrin, Géométrie algébrique] Springer, page 37,
Consider the projective line $\mathbf {P} ^{1}$ , with homogeneous coordinates $x$ and $t$ and open sets $U_{0}\ (x\neq 0)$ and $U_{1}\ (t\neq 0)$ .
• 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 ).