Flat surfaces with singularities in Euclidean 3-space
By Satoko Murata, Masaaki Umehara. 2006
74 citations recorded by the source corpus.
It is classically known that complete flat surfaces in Euclidean 3-space are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface $f$ admits singularities but its Gauss map $\nu$ can be smoothly extended across the singular set, $f$ is called a frontal. In addition, if the pair $(f,\nu)$ gives an immersion, $f$ is called a front. A front $f$ is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the focal surface of a flat front $f$ are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fonts, and give a representation formula for complete flat fronts. As an application, we show that a complete flat front has properly embedded ends if and only if its Gauss image is a convex curve. Moreover, we show the existence of at least four singular points other than cuspidal edges on such a flat front with embedded ends, which is a variant of the classical four vertex theorem for convex plane curves.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/d07c61ca1cc11b568a0f3ee0c1ef6f3b70b26cc2
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Geometric Invariants of Cuspidal Edges (2014, background): “Recently, there have been numerous studies of wave fronts from the viewpoint of differential geometry, for example, [7, 11, 12, 15, 17].”