Edge of regression

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A type of singularity of differentiable mappings (cf. Singularities of differentiable mappings) of a manifold into a Euclidean space. In the simplest case of a mapping $f$ of a surface $M$ into the three-dimensional Euclidean space $E^3$ an edge of regression represents a smooth curve $L$ in $M$, with smooth image $f(L)\subset E^3$, such that for any $p\in f(L)$ the intersection of $M$ with the plane $\pi$ through $p$ and perpendicular to $f(L)$ is a cusp. It occurs in the pseudo-sphere.


Cf. also the edge of regression of a developable surface.

An edge of regression is also called a cuspidal edge. It is the stable caustic of type $A_3$, see [a1], p. 331.


[a1] V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , 1 , Birkhäuser (1985) (Translated from Russian) MR777682 Zbl 0554.58001
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Edge of regression. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article