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SMOOTH SURFACE
The surface with Cartesian equation For an algebraic surface of degree n, with |
A surface is said to be smooth if it does not have singular points, in other words, if it has a (unique) tangent plane at every point.
For this we need to clarify if the surface is considered from a real affine, a real projective, or a complex projective point of view, the conditions becoming stronger and stronger.
Examples:
- smooth surface of degree n
(in the latter sense): the Lamé surface with Cartesian equation
.
- smooth surface from an affine point of view, but not a projective one: a cylinder based on a curve having the same property.
- a surface with equation z
= f(x,
y) (with f of class C1) is always smooth in the affine point of view.
- a cone
(different from a plane), or Whitney's umbrella are not smooth.
- a quadric is smooth in the complex projective space if and only if it is non degenerate (iff it is projectively equivalent to the surface ).
- a cubic surface is smooth iff????
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© Robert FERRÉOL 2017