JGV Example: a cubic ruled surface with two pinch points

  The line segment that seems to be skewering our cubic ruled surface in this picture has been intentionally added to the figure because it is part of the real locus of this real algebraic variety.

  Indeed, we have the following implicit equation:

z(x2 + y2) = x2 - y2.

  Accordingly, for any point (x,y,z) with x = y = 0, the equation is trivially satisfied. It follows that the entire z-axis is contained in the real algebraic variety.

  On the other hand, if (x,y) is not the origin, then x2 + y2 is nonzero, then we can solve the implicit equation to obtain the equation z = cos(2theta). This leads to the conclusion that all remaining parts of the surface are given parametrically as follows:

x = r cos(theta),
y = r sin(theta),
z = cos(2theta).

  This is the parametrization that can be used to produce a plot of the surface in Matlab, Maple, or Mathematica.
The purely 2-dimensional portion shown in the sketch corresponds to the parameter values  0 < r < 1, and  0 < theta< 2pi.

  This variety is very similar to the Whitney umbrella that is fairly well known to differential geometers.

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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: roberts@math.umn.edu