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Projective line
open-in-new
<p>In <a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a> and <a href="/facts/Mathematics/pxTouaz4">mathematics</a> more generally, a projective line is, roughly speaking, the extension of a usual <a href="/facts/Line_(geometry)/gJqsr4Gj">line</a> by a point called a <i><a href="/facts/Point_at_infinity/GAuS0wHQ">point at infinity</a></i>. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a <a href="/facts/Projective_plane/p6ot8oNv">projective plane</a> meet in exactly one point (there is no "parallel" case). </p><p>There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> <i>K</i>, commonly denoted P1(<i>K</i>), as the set of one-dimensional <a href="/facts/Linear_subspace/2wtKTgHL">subspaces</a> of a two-dimensional <i>K</i>-<a href="/facts/Vector_space/M1pxMLx2">vector space</a>. This definition is a special instance of the general definition of a <a href="/facts/Projective_space/7MSpA9cM">projective space</a>. </p><p>The projective line over the <a href="/facts/Real_number/R02gw5Pb">reals</a> is a <a href="/facts/Manifold/rXPERJtu">manifold</a>; see <i><a href="/facts/Real_projective_line/1g8z8v3U">Real projective line</a></i> for details. </p>