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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: .

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 nonrational 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.
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© Robert FERRÉOL 2017