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:

Accordingly, for any point (

On the other hand, if (*x,y*) is not the origin, then
*x*^{2} + *y*^{2} 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.

`Go back to`
the JGV homepage.

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the Math 5385 homepage

`Back to` my homepage

*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
`