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, t2, 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:
x2y = z2.

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:
x2y = z2w.

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(x2 + z2) = x2 - z2.

Click here to see the transformed surface.

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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