RATIONAL CURVE

 From the Latin ratio "quotient" (quotient of two polynomials...). Synonym: unicursal curve (from the Latin unus "one", and cursus "course").

 Cartesian parametrization:  where P, Q and R are three coprime polynomials with real coefficients such that the polynomials and are coprime (proper representation). Homogeneous Cartesian parametrization: . Replacing t by , we get the parametrization .

A rational curve is a curve that can be parametrized with rational function in R(x).
The parametrization is said to be proper if all points on the curve (except a finite number of multiple points) can be obtained for only one value of the parameter: see the practical condition in the boxed paragraph. A NSC is that the parameter can be written as a rational function of the coordinates.
The proper representations can be derived from one another by a homographic change of parameter (Lüroth theorem).

All rational curves are algebraic curves of degree equal to the highest degree of the above polynomials P, Q, R of a proper representation.

An algebraic curve is rational if and only if the number of its singularities, counted with multiplicity, is maximal, in other words, if its genus is zero.
Example: all algebraic curves of degree n with a multiple point of order n - 1 are rational;
In particular, the conics are all rational.
For the cubics, quartics, and sextics, see rational cubic, rational quartic, and rational sextic.

Attention: a planar section of a rational surface isn't necessarily a rational curve.

The rational curves are also called unicursal curves, which means, etymologically, that they can be traced with the pen never leaving the plane of the paper; which is perfectly true in the affine plane only when the polynomial R has no real zero. Otherwise, we have to place ourselves in the projective plane to imagine that the pen does not leave the plane (to trace a hyperbola or a mixed cubic for example).
By contraposition, this proves that a curve that has several connected components (called multipartite curve), one of which is a closed curve, is not rational. Examples: the divergent parabola or the Cartesian ovals.
The converse is false: the divergent parabola, which is probably the non-rational curve with the simplest equation, can be traced with one stroke of the pen; such a curve is called unipartite curve.

When R = 1, we get the polynomial curves.

© Robert FERRÉOL, Jacques MANDONNET 2017