Joachimsthal surface

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

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