The real projective line consists of one-dimensional vector subspaces of , written with and for every nonzero real . The chart gives the affine coordinate , while is one added point at infinity.
A unit vector and its negative determine the same line, so the angular coordinate is taken modulo . This quotient is itself a circle: the map sending that line to is well defined and gives a homeomorphism with .
Hence the real projective line is compact and can be viewed as the one-point compactification of the real line. Both and approach the same added projective point, unlike the two distinct endpoints of an extended closed real interval.
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The real projective line, denoted as \(\mathbb{RP}^1\), is a fundamental concept in projective geometry. It can be understood as the space of all lines that pass through the origin in \(\mathbb{R}^2\). Each line corresponds to a unique direction in the plane, and projective geometry allows for a more compact representation of these directions.
