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:
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(2).
This leads to the conclusion that all remaining parts of the surface
are given parametrically as follows:
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 < < 2.
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: email@example.com TT>