Dupin cyclides osculating surfaces (Q2015858)

The osculating planes and osculating spheres of a surface are defined for its special degree of tangency with a surface. In this work, the authors use the same idea for a special type of canal surfaces, namely Dupin cyclides, and they analyze their highest possible degree of tangency with generic surfaces in 3-dimensional space. They show that generically, at any point of a surface, one has a one-parameter family of cyclides tangent to a surface curve of a order three and among them just one is tangent to this curve of order four (osculating cyclide). Directions of this higher-order tangency form a line filed on the surface and its integral curves are called Dupin lines. Each point of a generic surface meets two orthogonal lines of curvature but only one Dupin line. The authors provide some examples which show that foliations by curves making a constant angle with the principal foliation can be realized as Dupin ones.