The image of a smooth surface, under generic projection to
P³, is a surface with a 1-dimensional singular locus.
Most of the singular points are ordinary double points (where two smooth
sheets of the surface cross transversally). There are finitely
many pinch points and finitely many triple points.
The Java GeomView software was developed at the Geometry Center.
You don't need this software for viewing my pages. If, however, you'd
like to obtain it for your own webpage, you can download it from the
When you view one of these figures, your computer downloads the Java
applet (a small application) from the U of MN math department
web server, along with the surface coordinates and related data from my
web page. When you rotate, translate, or scale the figure by dragging it
with your computer's mouse, the computations are done on your
computer in real time. Thus, you don't have to wait for the server
to re-draw the figure.
Polyhedral approximation vs. ray tracing.
A triangulation of a surface is a literally correct form of
polyhedral approximation. Generally, we are more likely to use
approximations of a surface by collections of other polygonal figures.
Quadrilaterals are very common; for example that is
what you'd have with a parametric sketch
created in Matlab, Maple, or Mathematica. And we're often not looking
at actual planar polygons. But this turns out not to matter!!
Indeed, the edges of our polygon approximate curve segments on the
surface. These are projected onto the plane of the computer screen,
finally giving an actual polygon. This is finally filled in with whatever
color is given in my webpage data.
If we use a grid with relatively few vertices to plot our surface,
we may get a picture that's not as smooth as what can be done by tools such
as surf. But, as mentioned above,
we are able to take advantage of the cross-platform capabilities of the
Java language, thereby obtaining figures that can be rotated much more