Case n = 5/16

Case n = 5/7

 
 

Curve studied by E. Salkowski in 1909, and by J. Monterde in 2009 and in 2024.

 

Cartesian paramétrization : ,  ;  where  is the constant angle between the normal vector  and . Curve traced on the revolution quadric : . Abscisse curviligne : . Constant curvature radius : . Torsion radius: .

Salkowski curves are curves of constant curvature whose principal normal makes a constant angle with a fixed direction (these latter curves being designated by "slant helices", compare with helices).

For , i.e. , the curve is plotted on an revolution ellipsoid, and for , i.e. , it is plotted on a revolution hyperboloid.

In the first case, the projection on xOy is the same as that of the spherical epicycloid with parameters  and  (see the article by J. Monterde). The Salkowski curve is therefore formed of arches whose number is equal to the numerator of 2n if n is rational.

The cuspidal points being obtained for , so , we get an arch for , and the complete curve for .
 
 

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© Robert FERRÉOL 2024