This figure shows a portion of a cubic ruled surface in 3-dimensional space
with two pinch points. This surface is
a generic projection of a nonsingular cubic ruled surface in 4-space. In fact, the projective closure
of this surface has exactly two pinch points, and the affine representative has been chosen so that
both pinch points are at finite distance. One of the pinch points is at the narrowest part of the yellow
region (at the front in the home position), and the other one is at the narrowest part of the blue region
(at the back in the home position). Along the line segment joining the two pinch points, the surface has
ordinary double points, i.e. it crosses itself transversally (except at the pinch points, of course).
This surface (or its set of real points anyway) can be described in cylindrical coordinates (r, θ, z)
by the following equation:
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I made the figure on this page by substituting my own data in a Geometry Center webpage.
Prof. Joel Roberts
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
Office: 351 Vincent Hall
Phone: (612) 625-1076
Dept. FAX: (612) 626-2017
e-mail: firstname.lastname@example.org TT>