Elliptic Paraboloids

Models III 10, 11, 12 are elliptic paraboloids. Model III 11 shows the intersection of the surface with horizontal planes, which are ellipses.

Models III 10; III 11; III 11 (from above)

Model III 10Model III 11 - Horizontal planar cuts meeting in ellipsesModel III 11 - Horizontal planar cuts meeting in ellipses

 

Model III 12

Recall that triaxial ellipsoids have four umbilic points. In the limiting process from an ellipsoid to an elliptic paraboloid two of the umbilic points go to infinity, so there are only two on an elliptic paraboloid. There are no umbilic points on circular paraboloids (hyperbolic paraboloids with rotational symmetry), and the same is true for circular hyperboloids. Model III 12 displays lines of curvature, and we can see an umbilic point near the top of the model - recall the configuration described in Principal Curvature, Umbilic Points, and Lines of Curvature.

Model III 12 - Lines of curvatureModel III 12 - Lines of curvatureModel III 12 - Lines of curvature