There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field K, commonly denoted P1(K), as the set of one-dimensional subspaces of a two-dimensional K-vector space.
An arbitrary point in the projective line P1(K) may be represented by an equivalence class of homogeneous coordinates, which take the form of a pair of elements of K that are not both zero.
Adding a point at infinity to the complex plane results in a space that is topologically a sphere.
Any function field K(V) of an algebraic variety V over K, other than a single point, has a subfield isomorphic with K(T).
The rational maps play a role analogous to the meromorphic functions of complex analysis, and indeed in the case of compact Riemann surfaces the two concepts coincide.
Assuming C is non-singular (which is no loss of generality starting with K(C)), it can be shown that such a rational map from C to P1(K) will in fact be everywhere defined.
(That is not the case if there are singularities, since for example a double point where a curve crosses itself may give an indeterminate result after a rational map.)
A rational normal curve in projective space Pn is a rational curve that lies in no proper linear subspace; it is known that there is only one example (up to projective equivalence),[2] given parametrically in homogeneous coordinates as See Twisted cubic for the first interesting case.