Geodesic
Bifurcations of Affine Equidistants
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Singularities and applications, Tome 267 (2009), pp. 65-81.

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The bifurcations of so-called affine equidistants for a surface in three-space are classified and described geometrically. An affine equidistant is formed by the points dividing in a given ratio the segment with the endpoints lying on a given surface provided that the tangent planes to the surface at these endpoints are parallel. The most interesting case corresponds to segments near parabolic lines. All singularities turn out to be stable and simple. 
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P. J. Giblin; J. P. Warder; V. M. Zakalyukin. Bifurcations of Affine Equidistants. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Singularities and applications, Tome 267 (2009), pp. 65-81. http://geodesic.mathdoc.fr/item/TM_2009_267_a4/

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