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RATIONAL SURFACE

Cartesian parametrization:
where P, Q, R and S are three polynomials with
real (setwise) coprime coefficients.
Homogeneous Cartesian parametrization: . Special case: surface with Cartesian equation: with rational f. |

A *rational* surface is a surface that has a parametrization
by rational functions.

It is an algebraic
surface of degree less than or equal to the square of the largest total
degree of the polynomials *P*,
*Q*, *R* and *S ?????*.

Attention: a planar section of a rational surface (all
the more so the intersection of two rational surfaces) is not necessarily
a rational curve.

Example: the section of the rational surface
by the plane *z* = 0 is the algebraic curve with equation:
which can be non-rational.

However, a cone or a cylinder is rational iif one of
its planar directrices is.

Examples:

- all the quadrics.

- cubics:
the skew ruled cubics, (of
which Plücker's conoid of order
2), the monkey saddle, the cubic
Dupin cyclides,
Costa's cubic surface.

- quartic: the Dupin
cyclides (including the torus).

- the Enneper
surface.

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© Robert FERRÉOL
2017