We had our first meeting today. Joe explained the construction of the ruled racetrack-surfaces, which are examples of D-forms, and of
their predecessors, the pita-forms.
He explained how discontinuous curvature and torsion can integrate to give a smooth curve.
We suppose that the curvature of sections where the straightaway meets the half-circle is 1/r where r is the radius of the circle,
and that sections where two straightaways meet have zero curvature. It was formerly thought that when half-circle met half-circle the
curvature was still 1/r. There doesn't seem to be much justification for this, and perhaps even some experimental data against the hypothesis.
Is this curvature even a constant value? I have no intuition wrt this question. If, however, the curvature and torsion were simple step functions,
we would be able to calculate the curve quiet easily since we would simply measure the arc-length (in the plane, since we do not stretch the
surfaces when we glue them, and we have parameterization for the racetracks in the plane) of the sections where one of the above three combinations
of curve to curve, curve to straightaway, and straightaway to straightaway met to determine the curvature and torsion functions, and then apply the
integration algorithm from Gray's differential geometry/Mathematica book to obtain the curve. We have been focusing on thinking of methods
to prove our first two conjectures about the curvature, have been ignoring the curve to curve case, and haven't even touched the seemingly
more complicated subject of torsion.
It seems that there are two different approaches to calculating the curvature and torsion of the racetrack D-forms: one,
use the fact that the surface is an intersection of two ruled surfaces with Gaussian curvature zero at all interior points, or
two, somehow use distances and arc lengths in the plane to calculate the curvature and torsion of the intersection curve itself and
then form the surface by taking the convex hull of the intersection curve. The first approach seems to be a bit of a dead end
since Gaussian curvature of zero tells us nothing about the principle curvatures, only that, since we're looking at a ruled surface
the minimal curvature and, therefore, one of the principle curvatures is zero and so the product is zero. The other principle direction
is in the direction of the tangent line to the intersection curve where the "ruled reflection line" hits the intersection curve. This is
just a result of the surface being a ruled surface and is not particularly useful.
The second approach is difficult because the curvature of the space curve, which I've been calling the intersection curve, will be different
from the curvature of the plane curve before the two racetracks are glued together. The curvatures should change in a predictable way since we
know the shapes of the two surfaces being glued together, but it does not seem like a simple task to find the way. Joe has already proven that the d-form is the convex hull of the curve so if we could find the intersection curve we would have the entire object.
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