Dr. Zarebski,

> My work with POV-Ray has gone smoothly; it seems almost tailor-made for such
> illustrations due to its precise mathematical approach to description,
> calculation, and rendering of cones and cylinders as lathe objects. It is a
> wonderful tool, and I am grateful to all its creators.

Yes, it's pretty difficult to find something that's so suited to such geometric
tasks yet so accessible to anyone, free, and open-source as well.

"rendering of cones and cylinders as lathe objects"
I'm not sure if going that route gains you anything - cones and cylinders are
basic primitive shapes.
https://wiki.povray.org/content/Reference:Cone
But perhaps the shapes you're using don't have simple flat endcaps.

> I viewed the animation of the hypoid
> gear assembly you sent, and it looks very good, so from that perspective, the
> matter is likely settled.

Well that got modeled by
https://www.otvinta.com/download06.html
and I just wanted something that would clearly depict how the gears work, as
supplementary material to your article.

> For flanks such as those in hypoid gears, a
> unique geometry description method is the use of a point cloud with
> corresponding normal vectors. So, it would be necessary to create an object that
> uses such data as input and generates the surface accordingly.

That sounds to me like numerical methods are used to calculate the geometry, and
a mesh of triangles (perhaps the result of FEM) would be the preferred
embodiment of such data.

> Another known method is simulation of machining by calculating successive tool
> positions in relation to the teeth and subtracting solids from each other.
> Surfaces obtained this way tend to be stepped, not perfectly smooth, require
> extensive computation, and very small angular differences at intersections can
> cause unexpected effects.

Yes, I read a lot about that method when modeling very small gears, as it's
quite commonly used to provide an undercut for clearance of the larger gear's
teeth.

I used the same approach to "machine" the toolpath for the flutes of a twist
drill bit, and encountered everything that you mentioned above.

I'm tempted to suggest that a parametric curve ought to be able to describe the
tooth shape - perhaps a Coons patch for the tooth flank.  Presumably the stepped
tool path could be interpolated and thus provide a smoother result.


> Regarding the transition curve at the tooth root, this involves another set of
> phenomena, which have been highlighted, for example, in
>
https://gearsolutions.com/features/transition-curve-much-more-than-a-radius-at-the-root-fillet-of-a-tooth/

Yeah - I have dealt with stress risers and the consequences of inadequate tooth
strength.  I opened up a manual transmission, and a pile of sheared off teeth
just poured out.  (oof.)

> If the modeling goal was solely animation, it would probably suffice to replace
> it simply with a rounding.

Well, yes.  Graphics and engineering are vastly different fields.
I recall reading a quote attributed to Archimedes about art vs math (which I
have since been unable to find again)
It was basically the observation that in art, one can do whatever one wants, but
with math, there are rigorously enforced rules.
(And having known many artists, they are prone to just do whatever and then
rationalize the result after the fact)


> I also want to say that I have found here a group of very interesting people,
> many apparently passionate about gearing. I admire your achievements and believe
> humanity can be proud of you.

Gears are what lie between accomplishing our useful tasks, and Ayn Rand's engine
that drives the world.
And hypoid gears are what drive most heavy vehicles in the rear differential.

Are you familiar with Brandon Sander of Inheritance Machining?
He recently tried to machine some gears, and could probably use some expert
advice from someone of your caliber.
https://www.youtube.com/watch?v=sQ_blQAkC4Y

> I've noticed that collaboration among diverse communities can yield surprising
> and beautiful results. A relative of mine, uncle Zdzisiek, an industrial
> conveyor designer, used to say: "If something can't be done, give it to a
> student. The student doesn't know it can't be done and will just do it."

I agree, and have made the same point.  And I will point out that we both know
the real meaning of "just".  ;)

> For example, I had an industrial project in mind related to electromagnetic wave
> simulation in an industrial setting. If of any interest, anyone wishing to
> contact me directly can find me at the email starting with spigor and ending
> with op.pl.

I can't say that I could help, but I'm interested in hearing about it.  :)

Also, are you familiar with Paul Nylander?
http://www.bugman123.com/Physics/index.html

> I wish you all the best!
> Kind regards,
> Igor

Thank you,
for the kind words, and for sharing the results of your work.

- BW

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