Hyperboloids

Models III 5 and III 7 are one-sheet hyperboloids, with III 5 showing intersections with coordinate planes (the $x=0$, $y=0$, $z=0$ planes), and III 7 displaying lines of curvature. Models III 8 and III 9 are two-sheet hyperboloids, with III 8 showing intersections with coordinate planes, and III 9 displaying lines of curvature. On III 9, the approximate location of umbilic points can be seen near the centre - recall the configuration described in Principal Curvature, Umbilic Points, and Lines of Curvature. Note also the doubly ruled property for the one-sheet (but not the two-sheet) hyperboloids. We had a model of a 1-sheet hyperboloid exhibiting the doubly ruled property, but only a very small piece now remains.

Elliptic hyperboloids of 1 sheet have no umbilic points, whereas 2-sheet hyperboloids have 4 umbilic points. Cones (which are degenerate hyperbolic paraboloids) have no umbilic points: in the limit from a 2-sheet hyperboloid to a cone (i.e. as $a, b, c$ tend to infinity), all 4 "vanish" into the singularity.

 

Models: III 5; III 7; III 7

Model III 5Model III 7 - Lines of curvatureModel III 7 - Lines of curvature

 

Models: III 8; III 9; III 9

Model III 8Model III 9 - Lines of curvatureModel III 9 - Umbilic Point