I am interesting in seeing and counting (enumerating) the most basic
unit volumes and their associated surfaces for inclusion in a computer software library of "atomic" features from which "everything" (to second order) may be designed, in a attempt to provide a reasonable and
nontrivial basis for para-universal constructors.
It seems to me that including advanced math in CAD representations of manufacturable objects would reduce file sizes and eliminate
digitization and tiling errors which are becoming a problem as the
resolution of additive and subtractive manufacturing machinery
increases, which is why I am writing about this here.
This post would go to sci.math were it not for the ubiquitous use of
change of coordinate system in solving the most advanced physics
problems. Briefly, when an initial, constraining, or terminal condition
of a physics problem is representable most effectively in a coordinate
system other than Cartesian, translating the entire problem into that
system can provide a solution where no other method will work. The
solution to the Navier-Stokes equations with viscosity for the case of
flow over a sphere is a famous example--after the change of system the problem is thereby reduced from 3 dimensions to only 1, and is readily solved.
Douglas Goncz
Replikon Research FCN 783774974
The Change of Variable Theorem is of interest as it generalizes the transformations between coordinate systems.
There are 16 orthogonal systems listed at Math World under Orthogonal Coordinates and it is asserted on that page they are all degenerate
cases of elliptical coordinates, a mistake; Giankoplis gives the
derivation of the degenerate cases from the matrix equation.
x=[-1,1]; y=[=1,1]; z=[-1,1].
I am open to suggestions today and have not started programming yet.
Cheers,
Douglas Goncz
Replikon Research FCN 7837774974
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