Co- authors Wanted for Journal Paper (related to "Jello-O... " thread)

That's possible. Then it would at least be a 3rd order polynomial in t instead of a 4th order polynomial. I will look into it tomorrow. Also, there is Aer's spherical-coordinates eggoid. But to use that I may have to figure out the parametric equation of a line in spherical coordinates.

-Dale

 

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Hehe. You got that right. I didn't even notice it with the spherical coordinates, but it is been a real pain for me with Mathematica's Fourier transform. Their default is such that the forward and inverse transform have the same multiplicative factor out front. It always messes up the communication when I share the results with coworkers, but the Wolfram people like it because it is more symmetrical.

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-Dale

 

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Ok, here is the situation from my perspective. The current eggoid equation winds up being a 4th order polynomial in t. There are "quadratic equations" for fourth-order polynomials, but the result is very messy, typically 2-3 pages long even after simplification. Of greater concern, however, is the fact that I get a division by zero for theta == 90º.

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As you can see, it is very well behaved until we get to 90º, this appears consistent regardless of the point under consideration. It is also gives the wrong answers for theta > 90º, but at least those are not infinite and they are not needed for the integration.

Basically, I cannot do an analytic integration of v around the unit sphere. I can probably do an approximate numeric integration by stopping at, say 89.5º, but there is no guarantee that the results would not miss some important feature.

As I see it we have two options at this point. 1) try a different function for the eggoid, hopefully one that is 3rd order or trigonometric or perhaps even a spherical harmonic. 2) try to take the Taylor series expansion as D->0 (and maybe r->R) and evaluate for purely radial v under the assumption that for small D all of the work will be done in the radial direction. Let me know what your preference is.

-Dale

 

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Why not try the "linear factor" (1 + Dz) eggoid. Do not bother initially to integrate for volume etc, only see if you can get the (t1 +t2) for all θ?

This is probablya crazy idea, but if you are calculating (t1 +t2) in one operation, try calculating each separately, then add. Does t1 alone blow up at 90 degrees? I hope to get to the problem this eve.

 

I used the factor (1 - Dz) because I like a substantial base. Anyway, the following is a plot of s² = (R² - z&sup2

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*(1 - Dz) where s² = x² + y² and D = 0.6

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The volume can be found by the following simple integration:

∫ A dz

where A is the area of a slice of the eggoid in the xy plane.

A = πs²

and s² = (R² - z&sup2

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*(1 - Dz) so,

∫ π(R² - z&sup2

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*(1 - Dz) dz with lower limit -R and upper limit R = 4πR³/3




Similiarly, we can get the eggoid that I initially thought we were analyzing which is the following:

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With the equation: s² + z²/(1 + Dz) = R² and D = 0.3

 

Yes, t1 and t2 are separate roots of the 4th order polynomial in t (there is also a t3 and t4). The first term for each root has a 1/Cos[θ] which is what becomes infinite at θ == 90º. One pair of roots works for θ [0º,90º) and the other works for θ (90º,180º], but neither work for θ == 90º.

Interestingly, the sum of the roots is a much more compact equation than the individual roots themselves (2-3 pages v. ~10 pages). There are a lot of terms that cancel out with the sum. Unfortunately, the 1/Cos[θ] term is not one that cancels out.

-Dale

 

DaleSpam, what are your thoughts on the following scaling equation?

s² + z²/(1 + Dz) = R²

Personally, I like the shape much better as relates to a change from a sphere to an eggoid.

 

It makes prettier eggoids too

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Much more similar-looking to chicken eggs. Of course, the real test will be when I solve for t.

-Dale

 

Well, I thought it might produce easier mathematical expressions which may turn out to be a farse. z as a function of s is this mess:

z=√((R² - s&sup2

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(D²R² - D²s²+4)) + D(R² - s&sup2

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/2

 

Does this just mean that in your plots (which are too small inmy display to read anything from, but I can see the shape.) that the plot has +z axis vertical above and the fat end is the "lower" end? You both are faster than me at all this, so I just say thanks. Looks like we now have around half a dozen eggs in our Easter baskets (if my original ones are included) Clearly we need to find one that we can get (t1 + t2) over the full θ range as top priority. I still plan my feeble efforts but a summary of the eggoid equations now in use would be useful.

If my guess about your meaning is correct, why not use the existing (1+Dz) and plug in D = -0.3 etc. ? - Then we can keep together on the equations and you can still have your "base down" egg plots.

 

Yeah, even fairly simple rational polynomial expressions can be surprisingly difficult to invert. So far it looks like three roots (as expected) but they are even more complicated than the original with (t1+t2) being about 10 pages long. Mathematica is currently working on a simplification.

-Dale

 

They are thumbnails. Click on them to get the full image.

-Dale

 

Yes.

It isn't that time of year yet is it?

You can do that too. Although, I don't know which shape we are using at the present.

 

I cannot even get an analytical solution for the volume. When I tried a numerical solution for R=1 and D=0.3, it did not match the volume of a sphere with radius R=1.

 

That works great. thanks. I note that Aer's second plot, (the 9K bit one) appears to me to have greater volume that the radius R sphere.

Are we all to gether on the evolved egg building plan? As I understand it, we distort the sphere with some thing like (1 + Dz) to compress one end and expand the other. Next the equation for its volume is found (by integration). Then we set volume to be the same as radius R sphere and use this equation to eliminate in the original eggoid equation something we can solve for. (For example, I originally did this to elimante B^2 of the semi-minor axis of an ellipsoid, and Dale thought that a good idea.) After that we normalize by setting R = 1 and hope for something we can get both (1)find (t1 + t2) for all θ and (2a)(in my stability study) still integrate dw or (2b)(in Aer's total force questions) still integrate dF over the volume.

If this is not reasonably our common scheme, please outline the altenatives that are in use. Sorry to be so slow.

 

Yes, but Dale and I have already (even from first post in my case) a lot with (1+DZ) in the equations so I ask as nicely as I can that you use the negative values of D to make your plots as you like and yet permit easy check of your equations by others.

 

I am going to try both, hopefully one will give a reasonably simple result when solving for t. After I find one that has a good form, then we can worry about all of the volume normalizations etc.

-Dale

 

That's funny. When I solved the volume numerically, I found that the volume was less than that of a sphere with Radius R=1.

Sphere: V = 4/3πR³ = 4.19

Eggoid: V = 4.07

Maybe there is something going on that I don't know about, but there is no B in any of these equations. Technically, there is a B, but it is dependent on our value for D. The way I see it, the best we can do is solve for an eggoid numerically with radius R>1 to get the same volume as a sphere of Radius R=1.

 

at least the first 1/4 of this thread (several pages) had two potential shapes under consideration. The ellipse with equation (z/A)^2 + (r/B)^2 = constant and the eggoid equation was derived then by modification of that ellipsoid equation. I made the B^2 disappear by procedure described in prior post. I don't want you to take your time to learn the history of this thread, but continue to help push it forward. Back then the A & B were lower case a & b as is the usual form, but now I at least try to use capital letters for constants.

 

Mathematica has been running for about 24 hours now on the analytical "field" integral and is showing no signs of progress. I think we are going to have to do these numerically. The good news is that by doing them numerically I think that all 3 eggoids are fine geometry. The first eggoid goes infinite near θ = 90º and the other two go complex near θ = 90º and θ = 0º.

So do you want to go with one of these three eggoids or do you want to continue the search for an eggoid that will give us an analytical expression for the field?

-Dale