## A surface with a pinch point

This figure shows a surface with a standard pinch point.
It is given parametrically by:

(*s,t*) ---> (*x, y, z*) =
(*s, t*^{2}, *st*).

The range of values shown in the figure is -1 __<__
*s *__<__ 1 and
-1 __<__ *t* __<__ 1. Accordingly,
the images of points

(*s,t*) with *s* close to -1
are shown in orange, while the images of points (*s,t*)
with *s* close to +1 are shown

in pale green. Finally, the
images of points (*s,t*) with *s* close to 0 are
shown in purple.

In the "home position" of this sketch we are looking at the surface from
the direction of the negative y-axis.

In that same position, the x-axis
runs from left to right -- with a slight tilt. There are parabolic
cross sections

in the planes perpendicular to the x-axis; you can see
those by rotating the surface with the mouse. (If you

want to
return to the home position after doing that, just type "h".) So, this
surface is somewhat like what we

would get if we started with a
parabolic cylinder and then pinched the cross-section in the yz-plane
down

to a line, namely the y-axis. The tightest pinching
occurs at the origin. (In the home position, the origin is the

closest point of the surface the y-axis. In any position,
the pinch point is at the middle of the purple region.)

The origin is the one point on this surface to which the term
"pinch point" is usually applied.

This surface has the following implicit equation:

x^{2}y = z^{2}.

Using this equation, we can check that the surface is singular along
the y-axis.

`Click here` to see
another view of this surface, focused along the singular locus.

*The projective closure* of our surface has a second
pinch point: it is the point at infinity

on the y-axis. If w is the homogeneous coordinate
at infinity, then the projective closure

satisfies the following implicit equation:

x^{2}y = z^{2}w.

Incidentally, this equation exhibits the surface as a cubic ruled
surface.

We can make a projective change of coordinates so that both pinch points
will be

at finite distance. One way of doing this is to replace w
and y by p := w + y

and q := w - y. Then, we set p = 1 to get affine
coordinates x,z,q and the following

implicit equation:

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

`Click here` to see the
transformed surface.

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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`