Enumerating the Unit Volumes and Their Surfaces in the Confocal Ellipsoidal Coordianate Systems

[Douglas Dana Edward^2 Parker-Goncz (fully)]

There is a matrix equation including a 3x3 matrix which, when solved for
its degenerate cases, gives orthonormal surfaces in the various confocal
ellipsoidal coordinate systems. There are 11 solutions if memory
serves. Adding the toroidal and helical systems, we have 13 systems and
so, at least 13 unit volumes. Since in Cartesian coordinates, the axes
are fully interchangeable, the unit volume (the unit cube) is fully
defined by its volume of 1 L^3. However, in all of the other systems,
the axes are not fully interchangeable.

Let's examine the system I consider next most challenging, the
cylindrical coordinate system. I believe it to be a degenerate case of
projected ellipsoidal coordinates. (That is not the most popular name
for that system)

With a surface of constant r=1, and surfaces at l (ell) = 0 and l = 1
and including a (alpha) from [-pi, ... pi] the unit "slug" has a volume
of pi.

We could normalize this volume to 1 by scaling each axis by pi. But that
won't work.

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.

K. Eric Drexler leads the field in attempts to implement an atomic scale
additive universal constructor. Adrian Bowyer leads in attempts to
implement a human-scale universal constructor. Julian Leland Bell has
made significant progress in implementing a subtractive universal
constructor--his Swarthmore project was a self-reproducing externally
framed milling machine.

I built a four-axis mill with some self-reproducing features in 1997
and sold one of two copies to a hobbyist, advertising it in The Want AD
as a "self-reproducing milling machine" for $300. I wrote that up at
ESG at MIT. The writeup was mentioned in Kinetic Self-Replicating
Machines (KSRM) by Frietas and Merkle in 2004. I recovered the web site
mentioned in KSRM using the Wayback Machine maintain the page
first.replikon.net to this day, documenting that machine build.

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

[Douglas Dana Edward^2 Parker-Goncz (fully)]

The Change of Variable Theorem is of interest as it generalizes the
transformations between coordinate systems.

I have found pictures of something like unit volumes for the Cartesian,
Cylindrical, and Spherical 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.

I do not have a chart and may have to program in Mathcad. That's pretty
easy using the transformation equations to Cartesian coordinates tracing
along each edge of the coordinate system specific unit volume (not the
differential volume, but a substantial "chunk" of spaces, near the
origin, in each system). I think I can articulate some of those limits
here today before trying it:

For each coordinate axis with range from 0 to oo, apply limits of 1/2 to 1.
For each coordinate axis with range from 0 to 2pi, apply limits of pi/2 to 3pi/2.
For each coordinate axis with range from 0 to pi, apply limits of pi/4 to 3pi/4.
There are others, however, and there are inequalities.

Ideally the volume of each "chunk" would be 8 since the obvious chunk of
Cartesian space is:
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

On Sunday, June 6, 2021 at 10:22:08 AM UTC-4, Douglas Dana Edward^2 Parker-Goncz (fully) wrote (I wrote):
...

>
> 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.
>

...

[Douglas Dana Edward^2 Parker-Goncz (fully)]

Well aside from 8 where 1 is better, it's what I meant.

On Saturday, September 25, 2021 at 3:12:20 AM UTC-4, Douglas Dana Edward^2 Parker-Goncz (fully) wrote:
> 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.
>