Difference between revisions of "Edge of regression"
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| − | A type of singularity of differentiable mappings (cf. [[Singularities of differentiable mappings|Singularities of differentiable mappings]]) of a manifold into a Euclidean space. In the simplest case of a mapping | + | {{TEX|done}} |
| + | A type of singularity of differentiable mappings (cf. [[Singularities of differentiable mappings|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(2)|cusp]]. It occurs in the pseudo-sphere. | ||
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Cf. also the edge of regression of a [[Developable surface|developable surface]]. | Cf. also the edge of regression of a [[Developable surface|developable surface]]. | ||
| − | An edge of regression is also called a cuspidal edge. It is the stable [[Caustic|caustic]] of type | + | An edge of regression is also called a cuspidal edge. It is the stable [[Caustic|caustic]] of type $A_3$, see [[#References|[a1]]], p. 331. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> 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) {{MR|777682}} {{ZBL|0554.58001}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> 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) {{MR|777682}} {{ZBL|0554.58001}} </TD></TR></table> | ||
Latest revision as of 16:30, 15 April 2014
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.
Comments
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.
References
| [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 |
How to Cite This Entry:
Edge of regression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Edge_of_regression&oldid=24430
Edge of regression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Edge_of_regression&oldid=24430
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article