Lie geometry studies properties of curves and surfaces that are
invariant under inversions with respect to circles and spheres, and
that are also invariant under the offset operation, moving an object
away from itself to form a parallel curve or surface.  The key
examples in this theory are the circles and spheres, which are
preserved under the two operations, and the tori of revolution that
are sent into cyclides of Dupin under inversion and which are
preserved under the operation of taking parallel surfaces as long as
the parallel surface remains embedded.  In this presentation, we will
propose an extension of this geometry by studying properties that are
preserved even when the parallel object develops singularities.
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<IMG SRC="torus.gif">
<p>
One of the most basic properties of the torus of revolution is the
Spherical Two-Piece Property (STPP), which states that any sphere (or
plane) divides the object into at most two pieces.  Equivalently we
may say that the intersection of the surface with any ball, or the
complement of any ball, is connected. 
<p>
<IMG SRC="cyclide.gif">
<p>
T
his property is evidently
shared by the inverted image of any STPP surface, and for the torus of
revolution, this leads to the cyclides of Dupin, also characterized as
the envelopes of families of spheres tangent to three fixed spheres.
<p>
The parallel surfaces of a torus of revolution are also tori of
revolution as long as they remain embeddings, and each of these will
have the STPP.  However, as the parallel surface develops
singularities, the resulting objects will not have the ordinary STPP.
It is easy to find a height function that has two local maxima,
contradicting the usual STPP property that says in particular that no
height function has more than one critical point.
<p>
<IMG SRC="torus-horn.gif"> <IMG SRC="torus-spindle.gif">
<p>
Is there a way to salvage a geometric interpretation of the STPP even
for surfaces with conical singularities?  Yes, it turns out that there
is, using the concept of "intersection homology".  In this theory two
points will lie in the same connected component if there is a curve
from one to the other not passing through any conical singularity.
Under this definition the spindle torus will determine two components
for the 0th intersection homology group, so it still might be that any
point pair in a half space that bounds an arc (in the strict sence,
following standards of singles behavior) will already bound in the
half space.
<p>
We illustrate this phenomenon with some key examples.  The first is a
torus of revolution and the second a cyclide of Dupin formed by
inverting the torus in a sphere centered near the origin.  We then
show only the lower half of the torus to exhibit what happens when the
radius of the offset object is greater than the radii of circles of
curvature of a curve or surface.  We obtain horn cyclides and spindle
cyclides.
<p>
<IMG SRC="cyclide-horn.gif"> <IMG SRC="cyclide-spindle.gif">
<p>


To see a beautiful color film of the deformation from the torus to the
cyclide, select <A HREF = "torustocyclide.mpg">movie</A>.
<p>
To see a beautiful color film of the deformation from the torus to the
horn-torus to the horn-cyclide and back to the horn-torus select <A
HREF = "horn-torus.mpg">movie</A>.  Note that a portion of the
surface has been removed to exose the interior structure.
<p>
To see still another beautiful color film of the deformation from the horn-torus to the
spindle-torus to the spindle-cyclide and back to the spindle-torus and
finally back to the original torus, select <A
HREF = "spindle-torus.mpg">movie</A>.  <p>