Vertices and inflexions of plane sections of surfaces in R^3
Abstract
We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. More precisely we will trace the patterns of inflexions and vertices (maxima and minima of curvature) on the sections of a surface as the section passes through a tangential point. We also keep track on the evolution of the curvature of the curves at vertices and control its limit when the vertices collapse at singular points. In particular, we cover all the generic cases, namely when the tangential points are elliptic (A1), umbilic (A1), hyperbolic(A1), parabolic (A2) and cusp of Gauss (A3) points. This has some applications in Computer vision and is also related to interesting mathematical problems such as Legendrian collapse, foliations of surfaces, the 4-vertex Theorem or in general the behaviour of vertices and inflexions in parameter families of curves etc.
Keywords
symmetry sets
medial axes
vertex
affine invariant of the surface
inflexion
curves on surfaces
evolute
elliptic point
umbilic point
hyperbolic point
parabolic point
cusp of Gauss
Computer vision
plane section of a surface
Legendrian collapse
foliations of surfaces
the 4-vertex Theorem
singularity theory
plane section of a surface.