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I just stumbled across this while trying to regain my sanity,
and if even Ramanujan couldn't come up with anything but an approximation, than
I think the rest of us are Fuuuuuuuuuuuuuuuuuuuuuumbling around in the dark.
:)
https://www.youtube.com/watch?v=5nW3nJhBHL0
But don't anyone think that that will stop us from making another attempt, using
new inspiration and mathematical tools!
I think the above implies that due to the interesting properties of ellipses,
that a workable solution will use numerical methods.
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On 2020-09-20 6:24 PM (-4), Bald Eagle wrote:
> I just stumbled across this while trying to regain my sanity,
> and if even Ramanujan couldn't come up with anything but an approximation, than
> I think the rest of us are Fuuuuuuuuuuuuuuuuuuuuuumbling around in the dark.
>
> :)
>
> https://www.youtube.com/watch?v=5nW3nJhBHL0
You watch Matt Parker to regain your sanity? I have some of his videos
languishing in my "Watch later" list because I'm afraid I'll *lose* my
sanity if I watch them.
> But don't anyone think that that will stop us from making another attempt, using
> new inspiration and mathematical tools!
>
> I think the above implies that due to the interesting properties of ellipses,
> that a workable solution will use numerical methods.
I am not convinced that this property is directly relevant to the
problem I am trying to solve. I still think the key is in that quartic
equation I came up with, which should not require numerical methods
(beyond what's implicit in POV-Ray's built-in functions). The problem
is that I am not practiced in solving quartic equations.
P.S. I have known since 2007 that there is no exact formula for an
ellipse's circumference. I discovered that while trying to weld this
'O' to the rectangular plate using an ellipse made out of blobs.
Post a reply to this message
Attachments:
Download 'povanized-b.png' (100 KB)
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Cousin Ricky <ric### [at] yahoo com> wrote:
> You watch Matt Parker to regain your sanity?
After spending most of the weekend trying to work out partial differential
equations of 2D Bernstein polynomials, and Gaussian curvature?
Sure.
After that, lifting boxes at work on Monday was relaxing :D
Watching Daniel Shiffman would throw me over the edge, because it's
comedy/tragedy of errors, but I'm learning "important" things at the same
time.... It's like elective surgery with insufficient anaesthetic.
And, take a pre-emptive Advil for this one...
https://www.youtube.com/watch?v=YueAtA_YnSY
> I am not convinced that this property is directly relevant to the
> problem I am trying to solve. I still think the key is in that quartic
> equation I came up with, which should not require numerical methods
> (beyond what's implicit in POV-Ray's built-in functions). The problem
> is that I am not practiced in solving quartic equations.
Dumb question, but is there an online solver?
A language with the proper symbolic logic?
Cheat and get the right answer first, then use that knowledge as a light to
guide you along the path of working out the quartic.
> P.S. I have known since 2007 that there is no exact formula for an
> ellipse's circumference. I discovered that while trying to weld this
> 'O' to the rectangular plate using an ellipse made out of blobs.
It's always the "simple" things, isn't it?
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Le 22/09/2020 à 12:28, Bald Eagle a écrit :
> Cousin Ricky <ric### [at] yahoocom> wrote:
>> I am not convinced that this property is directly relevant to the
>> problem I am trying to solve. I still think the key is in that quartic
>> equation I came up with, which should not require numerical methods
>> (beyond what's implicit in POV-Ray's built-in functions). The problem
>> is that I am not practiced in solving quartic equations.
>
> Dumb question, but is there an online solver?
Sir, Yes Sir !
mathworld wolfram (alpha) has such beast.
It take a bit of learning to get it do what is desired, but there is
some examples along the pages.
You can ask it to refactor, solve (even based on chosen parameters), and
far more.
We are more in algebra than geometry.
https://www.wolframalpha.com/examples/mathematics/algebra/
> A language with the proper symbolic logic?
>
> Cheat and get the right answer first, then use that knowledge as a light to
> guide you along the path of working out the quartic.
>
I think the elliptical torus with a minor radius of 0 should match the
equation of the ellipse.
That's the basic check of any solution: minor radius of 0 would simplify
the equation to an ellipse.
The difficulty is going backward: reintroduce a non-0 minor radius.
Post a reply to this message
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Le 21/09/2020 à 00:24, Bald Eagle a écrit :
> I just stumbled across this while trying to regain my sanity,
> and if even Ramanujan couldn't come up with anything but an approximation, than
> I think the rest of us are Fuuuuuuuuuuuuuuuuuuuuuumbling around in the dark.
>
> :)
>
> https://www.youtube.com/watch?v=5nW3nJhBHL0
>
> But don't anyone think that that will stop us from making another attempt, using
> new inspiration and mathematical tools!
>
> I think the above implies that due to the interesting properties of ellipses,
> that a workable solution will use numerical methods.
>
I tried to play a bit with the hope to transform the quartic equation of
circular torus into elliptical torus.
From (sqrt(x²+y²)-R)²+z²=r², replacing x²+y²=R² with (x/a)²+(y/b)²=1
And it is deception.
x^4/a^4
+ (2 x^2 y^2)/(a^2 b²)
- (2 r^2 x^2)/a²- (2 x^2)/a²
+ (2 x^2 z^2)/a²
+ y^4/b⁴
- (2 r^2 y^2)/b²- (2 y^2)/b²
+ (2 y^2 z^2)/b²
+ r⁴- 2 r²+ 1
- 2 r^2 z²+ 2 z²
+ z⁴
Because the scaling of x & y also influence the radius of the minor circle.
But it must be possible to have it with a 4th degree poly.
If F( Point ) is such polynomial, we have the following requirement
F( 0, +/- B , +/- Minor ) = 0 (point B)
F( +/- A, 0, +/- Minor ) = 0 (point A)
F( 0, +/- B +/- Minor, 0 ) = 0 (points M & N)
F( +/- A +/- Minor, 0, 0 ) = 0 (points K & L)
I guess, but cannot yet assert, that to avoid compressing the minor
circle, the z⁴, x⁴ and y⁴ must be 1.
And there might be some non-traditional power of x,y and z (aka odd) in
the non-null coefficient, which would disappear when A=B to have a
continuity with the circular torus.
Also, any point W On the central ellipse with z=+/- Minor is also F(W)=0
And if any more point is needed, there is the two minor circle at A & B
to explore more deeply than K, L, M & N.
Is that enough to try some spreadsheet solver, and how to do it ?
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Attachments:
Download 'torusellipse.png' (91 KB)
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So....
After the whole "surround a function with a tube" got explored (and still not
completely sorted out in my head), my brain decided to alert me that therein
might lay the possible solution to this problem.
I copy-pasted some code, fought with Phi again, "won", and got an elliptical
shaped torus. For a=2, b=1.5 it looked good, but the problem had always been
the eccentric tori. a=4 and b=0.75 revealed the same old problem.
I can't find the thread where you list the required specifications, but I don't
think that it's possible using a cross-sectional circle of a constant radius.
Because that's what we're doing with the tube method. And this doesn't give
inner and outer curves that are themselves ellipses.
https://www.frassek.org/3d-mathe/funktionsgraph-als-rohr-r%C3%B6hre/
https://image.jimcdn.com/app/cms/image/transf/dimension=1920x400:format=gif/path/scee86bccd27a6ab2/image/i16edd265f402b
f7c/version/1580469937/image.gif
We can get signed distance functions,
https://iquilezles.org/articles/distfunctions2d/
https://www.shadertoy.com/view/4sS3zz
and we know that we can calculate the tangent and normal vectors to the central
ellipse.
What I'm thinking right now is that we can ditch trying to create the whole
surface and focus on the 3 ellipses - the central, and the inner and outer. If
the inner and outer aren't the same distance away from the central curve along
the normal vector, then I'd say that this demonstrates that a solution isn't
possible, given the parameters.
I think I ran into this issue with Mike Horvath wanting ellipsoid shells -
simply scaling an ellipsoid doesn't give a constant-distance shell from the
original ellipsoid. But those ARE ellipsoids themselves. And calculating a
constant-distance shell would give objects that WEREN'T ellipsoids.
I'mm betting that those level sets of the signed distance function of an ellipse
- aren't ellipses.
So I think that some of the constraints are mutually exclusive.
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Attachments:
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"Bald Eagle" <cre### [at] netscapenet> wrote:
> https://www.shadertoy.com/view/4sS3zz
Referenced in that code is this article with some comprehensive treatment of the
quartic equation.
https://iquilezles.org/articles/ellipsedist/
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