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ALGEBRAIC CURVE

A curve is said to be *algebraic* when it's described by a
polynomial Cartesian equation with real coefficients; a not algebraic curve
is said to be *transcendent*.

The *degree* (or sometimes the *order*) of a
flat algebraic curve of equation *P*(*x, y*) = 0 is the degree
of the polynomial *P*, supposed square-free. It is also
the number of intersection points, counted with their multiplicities, of
the curve with an any straight line, in the complex projective plan (and
the maximum number of intersection points of the curve with a straight
line in the real affine plan is a lower bound of the degree and gives very
often the degree).

Therefore, a spiral curve, or having an asymptotic point is necessarily transcendent.

The notion of degree is a projective one, in other words,
that any homography would transform an algebraic curve into an algebraic
curve of the same degree.

The curve is said to be *decomposable* if it is
an union of algebraic curves of smaller degree and *undecomposable*
otherwise, in other words, when the polynomial *P* is unreducible.

Any bounded portion of an algebraic curve can be generated by an articulated mechanism (Kempe's theorem).

The curves of degree 1 are the straight
lines, of degree 2 the conics, of
degree 3 the cubics, of degree 4 the
quartics,
of degree 5

quintics, of degree
6 sextics, of degree 7 septics,
and of degree 8 biquartics.

Examples of families of algebraic curves of any degrees (for rational values of the parameters): Lissajous curves, Lamé curves, epi- and hypotrochoids, roses, conchoids of roses, folioids, knots curves, epispirals, tear drop curves, cassinian curves, Clairaut's curves, sinusoidal spirals.

See also :

- genus
of an algebraic curve

- 3D
algebraic curve

- algebraic
surface

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© Robert FERRÉOL 2016