The Stability of the Apparent Contour of an Orientable 2-Manifold

Herbert Edelsbrunner, Dmitriy Morozov, Amit Patel.
Proceedings of the Workshop on Topological Methods in Data Analysis and Visualization, 2009.
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Abstract
The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f: M → R2 , is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming M is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable.
Torus contour
References
[11]
Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. Quantifying transversality by measuring the robustness of intersections. Manuscript, Deptartment of Computer Science, Duke University, Durham, North Carolina, 2009.