Difference between revisions of "Joachimsthal surface"
From Encyclopedia of Mathematics
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| − | The surface formed by the orthogonal trajectories of a one-parameter family of spheres with centres on a straight line. If the | + | {{TEX|done}} |
| + | The surface formed by the orthogonal trajectories of a one-parameter family of spheres with centres on a straight line. If the $z$-axis is taken for this straight line, if the $z$-coordinates of the centres of the spheres are denoted by $u$, and if the radius of the sphere is denoted by $R=R(u)$, then the position vector of the Joachimsthal surface is: | ||
| − | + | $$r=\left\lbrace\frac{R\cos u}{\cosh\tau},\frac{R\sin u}{\cosh\tau},u+R\tanh\tau\right\rbrace,$$ | |
where | where | ||
| − | + | $$\tau=\int\frac{du}{R}+V.$$ | |
| − | One of the families of curvature lines | + | One of the families of curvature lines $(\nu=\text{const})$ of the Joachimsthal surface is located in the planes of a pencil. The surface was studied by F. Joachimsthal [[#References|[1]]]. |
====References==== | ====References==== | ||
Latest revision as of 04:56, 16 September 2014
The surface formed by the orthogonal trajectories of a one-parameter family of spheres with centres on a straight line. If the $z$-axis is taken for this straight line, if the $z$-coordinates of the centres of the spheres are denoted by $u$, and if the radius of the sphere is denoted by $R=R(u)$, then the position vector of the Joachimsthal surface is:
$$r=\left\lbrace\frac{R\cos u}{\cosh\tau},\frac{R\sin u}{\cosh\tau},u+R\tanh\tau\right\rbrace,$$
where
$$\tau=\int\frac{du}{R}+V.$$
One of the families of curvature lines $(\nu=\text{const})$ of the Joachimsthal surface is located in the planes of a pencil. The surface was studied by F. Joachimsthal [1].
References
| [1] | F. Joachimsthal, "Demonstratio theorematum ad superficies curvas spectantium" J. Reine Angew. Math. , 30 (1846) pp. 347–350 |
Comments
References
| [a1] | G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887) pp. 1–18 |
How to Cite This Entry:
Joachimsthal surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Joachimsthal_surface&oldid=18490
Joachimsthal surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Joachimsthal_surface&oldid=18490
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article