## A cubic ruled surface with two pinch points

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:

z = cos(2θ).

This equation can be used to produce a reasonably nice plot of the surface in Matlab, Maple, or Mathematica.
The portion shown in the sketch corresponds to the parameter values  0 ≤ r ≤ 1, and  0 ≤ θ ≤ 2π.
Multiplying the equation by  r2  we can convert it to the following implicit equation in rectangular coordinates:

z(x2 + y2) = x2 - y2.

As usual, you can rotate the surface by grabbing it with the mouse. For instance, the blue pinch point can
be brought to the front by a rotation of 180° in either of the axes perpendicular to the line that joins the
two pinch points. You can do that by dragging either vertically or horizontally with the mouse -- for a total
distance equal to about 11/2 times the height [or width] of the figure. To return to the home position
at any time, just type "h".

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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
USA

Office: 351 Vincent Hall
Phone: (612) 625-1076
Dept. FAX: (612) 626-2017
e-mail: roberts@math.umn.edu
http://www.math.umn.edu/~roberts