## 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 r^{2} we can convert
it to the following implicit equation in rectangular coordinates:

z(x^{2} + y^{2}) = x^{2} - y^{2}.

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 1^{1}/_{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`