University of Liverpool
This paper was presented at the Sixth IMA confrence on the MAthematics of Surfaces at Brunel University, 1994 and appears in
Mathematics of Surfaces VI, Ed. Glen Mullineux, IMA new series 58, Clarendon Press, Oxford, 1996. pp 79 - 102
One important feature of the ridges and sub-parabolic lines is that, like the parabolic curves, they are robust, i.e. if we slightly deform the surface then the curve will deform. This contrasts with the geodesics and lines of curvature of the surface which reform as the surface is deformed. Hence it makes sense to observe how a sub-parabolic line changes in a family of surfaces but it does not make sense to study how a line of curvature changes.
The ridges can be thought of as the set of points where one of the principal curvatures has an extremal value (max or min), when moving along a line of curvature of the same colour. They can also be thought of as the pre-image of cuspidal edges on the focal surface. I. Porteous first gave the curves a thorough mathematical treatment in [Port1], which was expanded on in [Port2], by studying the singularities of the distance squared map. An alternative approach used in [Fold1], [Fold2] and [Fold3] is to consider folding maps of the surface, which captures the local reflectional symmetry. Two papers on ridges have previously been presented in this series of conferences, [MoS1] [MoS2].
In [Koenderink], J. Koenderink recognised that ridges are significant features of a surface and they are beginning to find several applications. They have been used by Thirion and Gourdon in the study of medical Magnetic Resonance Image data [INRIA1]. G. Gordon, D. Mumford has used them for face recognition from range data [INRIA2]. In geology the ridges appear as the hinge lines of folds, [Ramsay].
Whilst not being as visually obvious as ridges, the significance of sub-parabolic lines is increasing. They were first observed as the pre images of parabolic lines on the focal surface, hence the name. They also appear as the locus of geodesic inflections, of the lines of curvature. For this reason Porteous has proposed the shorter name flexcords for these curves. We shall see later that the a recent result of Thirion suggests an alternative characterisation: as the counter-ridges, points where the principal curvature of one colour has an extremal value when moving along a line of curvature of the other colour. Sub-parabolic lines can be found by examining the profiles of surfaces. Sub-parabolic lines have a fairly short history, first being studied in terms of folding maps in [Fold1] and [Fold2]. In [Thesis] they are studied geometrically in terms of the focal surface and the distance squared function and experimentally where examples of the transitions were found by computer. In [Fold3] and [Fold4] the transitions which occur on both curves are discussed.
One early reference is Eisenhart [Eisen] where an explicit equation for the Gaussian Curvature of the focal surface is presented. In his recent book [GeomDiff] Porteous gives both ridges and sub-parabolic lines a thorough treatment.
In section § 1 we present some basic facts about surfaces and focal surfaces. In section § 2 we present simple proofs for the characterisations of sub-parabolic lines and ridges and also discuss the Gaussian curvature of the focal surface. Section § 3 deals with the special points which can occur on these curves, including the highly spherical umbilic points. Some practical formulae for calculating the curves for implicit and explicitly defined surfaces are discussed in section § 4 as well as some of the problems which may occur in a computer implementation. Finally, in section § 5 we discuss some of the possible applications of these curves.
Many thanks to Ian Porteous for help in the preparation of this paper.