View source for Joachimsthal surface

The surface formed by the orthogonal trajectories of a one-parameter family of spheres with centres on a straight line. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054240/j0542401.png" />-axis is taken for this straight line, if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054240/j0542402.png" />-coordinates of the centres of the spheres are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054240/j0542403.png" />, and if the radius of the sphere is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054240/j0542404.png" />, then the position vector of the Joachimsthal surface is:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054240/j0542405.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054240/j0542406.png" /></td> </tr></table>

One of the families of curvature lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054240/j0542407.png" /> of the Joachimsthal surface is located in the planes of a pencil. The surface was studied by F. Joachimsthal [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Joachimsthal,   "Demonstratio theorematum ad superficies curvas spectantium"  ''J. Reine Angew. Math.'' , '''30'''  (1846)  pp. 347–350</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Darboux,   "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1''' , Gauthier-Villars  (1887)  pp. 1–18</TD></TR></table>