BERTRAND CURVE

 Curve studied by Bertrand in 1850, Serret in 1851, Bioche in 1889 and then Darboux. Joseph Bertrand (1822-1900): French mathematician.  [Gomes t2] p 447 , [Mir] p58 , [Berger] p 351, [Lelong- Ferrand] p 695, [Valiron] p 421, [Loria] p 90.

 Intrinsic equation: . Cartesian parametrization:  where  is the parametrization of a curve with constant curvature (a skew circle), and  is the parametrization of a curve with constant torsion.

Bertrand curves are the 3D curves the curvature and torsion of which are linked by an affine non linear relation (hence the above intrinsic equation) - the linear case yields the helices.

A curve  is a Bertrand curve if and only if there exists a curve  different from  with the same principal normal line as .

Except for the case of a circular helix, the curve  is unique; the distance between two corresponding points along the common normal line is constant, and the angle formed by the corresponding tangents is constant.

Examples: the circular helix and, more generally, the skew circles (case where b = 0; the angle formed by the tangents is then a right angle and each curve is the locus of the centers of curvature of the other).