This duality is also tangible from a practical perspective, since algorithms manipulating sampled smooth surfaces meshes or point clouds are more developed in the local than the global category. As a prerequisite for those interested in the development of algorithms for the manipulation of surfaces, we propose a concise overview of core concepts from differential topology applied to smooth embedded surfaces. We first recall the classification of umbilics, of curvature lines, and describe the corresponding stable foliations. Next, fundamentals of contact and singularity theory are recalled, together with the classification of points induced by the contact of the surface with a sphere.
This classification is further used to define ridges and their properties, and to recall the stratification properties of the medial axis.
Differential Topology M4P54
Finally, properties of the medial axis are used to present sufficient conditions ensuring that two embedded surfaces are ambient isotopic. From a theoretical perspective, we expect this survey to ease the access to intricate notions scattered over several sources. From a practical standpoint, we hope it will be useful for those interested in certified approximations of smooth surfaces.
Keywords : Smooth embedded surfaces differential geometry umbilics lines of curvature foliations ridges medial axis isotopy. Using the results of the first half, we show that the relative tangent bundle of a derived manifold over a derived stack has a Lie algebroid structure.
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We then provide a criterion for maps between Lie algebroids to integrate to maps between stacks, generalizing classical theorems of Lie and Van Est. This result is applied to show that finite-dimensional L-infinity algebras can be integrated to higher Lie groups.
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