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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, and p ?????.

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, 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