The curve with Cartesian equation  is smooth in the real (resp. complex) affine plane if the system  has no real (resp. complex) solution. For an algebraic curve of degree n, with , the curve is smooth in the real (resp. complex) projective plane if the system  has no other real (resp. complex) solution than (0, 0, 0).

A curve is said to be smooth if it has no singular points, in other words if it has a (unique) tangent at all points.
For this, one has to clarify whether the curve is considered in the real affine plane, the real projective plane, or the complex projective plane, the conditions getting stronger and stronger. In the latter case and for an algebraic curve, this notion coincides with that of curve with maximal genus.

Examples
   - smooth curve of degree n in the complex projective plane: the Lamé curve with Cartesian equation .
    - smooth curve in the real affine plane, but not in the projective plane: the witch of Agnesi: , with homogeneous equation  (singular point (0, 1, 0)).
    - a curve with equation y = f(x), with f of class C¹, is always smooth in the affine plane.
    - a conic is smooth in the complex projective plane if and only if it is nondegenerate (iff it is projectively equivalent to the curve ).
    - a cubic is smooth iff????
 
 

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© Robert FERRÉOL   2017