By Fulton W.
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Extra resources for Algebraic Curves. An Introduction to Algebraic Geometry
5 to recall the stratification properties of the medial axis. 6, and sufficient conditions involving the medial axis are also presented. 1 Generic surfaces Our focus is on generic phenomena on surfaces, and the statements presented are valid for generic surfaces only. Formally if one considers the set of all smooth surfaces S in E 3 as an infinite dimensional space with a well defined topology, a property is generic if the surfaces exhibiting this property form an open dense subset. Informally this notion means that (i) a generic property remains valid if one allows random perturbations, and (ii) that every surface is arbitrary close to another for which the property holds.
2) condition(s) on this family is expected to appear on lines (resp. isolated points) of the surface —a condition being an equation involving the Monge coefficients. A property requiring at least three conditions is not generic. e. the coefficient of the xy term vanishes and those of x2 and y2 are equal) are isolated points. A flat umbilic, requiring the additional condition k1 = 0, is not generic. CHAPTER 3. 3 Umbilics and lines of curvature, principal foliations This section is devoted to second order properties on a surface, and more precisely to umbilics and lines of curvature.
This means that the relevant contact to be considered is that of a plane with the surface at such a point. One can find a precise description of these parabolic points in [HGY+ 99]. 2 Generic contacts between a sphere and a surface Before presenting the generic contacts, let us illustrate the process of finding the first normal form using the Morse lemma. To ease the calculations, assume that the contact point is the origin, that the surface is given in Monge form, and that the center of the sphere has coordinates c(0, 0, r).