Difference between revisions of "Indicatrix of tangents"
From Encyclopedia of Mathematics
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| − | ''of a curve | + | {{TEX|done}} |
| + | ''of a curve $\Gamma$ in a Euclidean space $E^n$'' | ||
| − | The curve | + | The curve $\Gamma^*$ on the sphere $S^{n-1}\subset E^n$ which assigns to the parameter value $\tau$ the point whose position vector is parallel to the tangent vectors at $\tau$ to $\Gamma$. In order that a spherical curve $L$ be the indicatrix of tangents of some closed curve in $E^n$ it is necessary and sufficient that $L$ is not confined to some open half-sphere (Krein's theorem). |
====References==== | ====References==== | ||
Latest revision as of 09:29, 27 June 2014
of a curve $\Gamma$ in a Euclidean space $E^n$
The curve $\Gamma^*$ on the sphere $S^{n-1}\subset E^n$ which assigns to the parameter value $\tau$ the point whose position vector is parallel to the tangent vectors at $\tau$ to $\Gamma$. In order that a spherical curve $L$ be the indicatrix of tangents of some closed curve in $E^n$ it is necessary and sufficient that $L$ is not confined to some open half-sphere (Krein's theorem).
References
| [1] | M.Ya. Vygodskii, "Differential geometry" , Moscow-Leningrad (1949) (In Russian) |
Comments
The indicatrix of tangents is also called the spherical tangent image. See also Spherical indicatrix.
How to Cite This Entry:
Indicatrix of tangents. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Indicatrix_of_tangents&oldid=17066
Indicatrix of tangents. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Indicatrix_of_tangents&oldid=17066
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article