PSEUDOGEODESIC of a surface

 Notion studied by W. Wunderlich in 1951. See also Erwin Kruppa, Analytische und konstruktive Differentialgeometrie, p. 159.

The pseudogeodesic lines of a surface are the curves traced on the surface the osculating planes of which form a fixed angle  with the tangent plane of the surface; when this angle is a right angle, we get the proper geodesic and when it is equal to zero, the asymptotic lines.
According to the formulas  (see the notations), they are the curves for which the normal curvature, or the geodesic curvature, is proportional to the curvature, or for which the geodesic torsion is equal to the torsion.

 Computation of the pseudogeodesics in the case of a vertical cylinder (see the notations). The first principal curvature  is that of the horizontal section; the other one being equal to zero, the Euler formula yields , where  is the angle between the curve and the vertical; hence the geodesic curvature  (1). But  and  where  is the tangential angle of the horizontal section; using (1), we get , hence, taking  for , , ; but , hence the general equation of the pseudogeodesics: . For a cylinder of revolution, , hence , see below.

Example: the proper pseudogeodesic of the cylinder of revolution are the curves that develop onto catenaries with axis parallel to the axis of the cylinder (therefore, special case of generalized catenaries). Equations:
 Parametrization where  is the angle between the osculating plane and the tangent plane: . Opposite, animation for  describing . See here the surface folded along this curve.