Everyone is familiar with the roundcone, a convex object, and the sphere sweep
derived from it. A while ago I thought about connecting three spheres smoothly
with cones. When you join two roundcones, there are a pair of points where the
normals to the sphere and the two cones are identical.

Adding a pair of smooth triangles with vertices at these points gives a "solid"
object. This doesn't have C_1 since the line joining one normal point
to another usually penetrates the cone. However, the interpolation of the normal
on the smooth triangle is not a terrible match for the normal on the curve of
the intersection of the smooth triangle and the cone. The discontinuity appears
no more prominent than the false discontinuity observed when there is a
curvature discontinuity.

The image gives a fat triangle, a fat quadrilateral, some spheres with smooth
holes and a hollow roundcone (surprisingly tricky).

The fat quadrilateral is best not done with a pair of fat triangles - a
prominent crease is almost unavoidable on one side for non trivial
quadrilaterals. This one uses four fat triangles with a central sphere. The
default is to use the average of the four vertex spheres, but I've had to
multiply that by 1.5 to minimise any creases.

I've not seen such objects used, though it's hard to believe I've produced these
for the first time.

The next step is to produce Bezier patches using fat triangles. See the next
post in this thread

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