I wonder if it is okey on this site to offer payment for solving a specific math problem?

Or what other sites are recommended for this? Are there mathematicians for hire on a problem by problem basis, somewhere?

Cheers!

Tutors are always available for hire, but what type of problem is this? Is this something that is incredibly hard? Here at MHF, we are here to help you without charge. You will not get any courses taught to you, but you will be given help as you need it!

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Mathematical contributions of all kinds are welcome - IT IS LIVE AND READY!

Do you have a legitimate reason for wanting math solutions ? If you don't I doubt anyone on this site is going to help you.

I don't mean to be rude and forgive me if I have made a false assumption, but usually when people want solution to specific problems it means they somehow obtained their examination papers before they should have. Cheating is not approved of by anyone on this site and if you are going to cheat make and effort to be discrete about it.

Bobak

We will help you understand math, but we will not help you cheat.

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Mathematical contributions of all kinds are welcome - IT IS LIVE AND READY!

Oh, I don't want to cheat! I don't study or anything. But I now realize that this suspicion and misunderstanding is natural given my brief post.

I'd need some code that solves a specific kind of problem. It includes curve fitting and probably requires numerical methods. Also, it needs to be a bit efficient, nothing extreme but it shouldn't waste computational power. Matlab M-code would be fine if it compiles well. I've looked at it and just might work my way through it in M-code, but it'd take me quite some time and effort. While I like math, I'm not very good at it...
:P

The code could be completely open source if someone else ever has some use for it. I leave that to the author. I just want to be able to use it myself, w/o commercial interests.

The problem is probably not very hard or time consuming at all for someone with more than rudimetnary math education. But it's a bit more than I'd ask someone to do for free, and the purpose is to get a piece of working code, rather than learning math.

I'll take a look at the problem again before I try to specify it.

Try posting it first without offering payment and see if anyone isn't capable of doing it.

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That was very honorable of you. It amazes me how many people on the net or in person demand programming/application help and when it comes time to pay the piper, they want to offer mere pesos and shekels for services rendered. Seeing as though you determined that this was more than a simple piece of code to be put together, and determined that it had financial value, shows you have respect for yourself and the market, and I commend that.

Because you spoke like a gentleman and respect the principles of capitalism, I too will try to provide you a solution at no-cost as well. Except I will avoid MatLab like the plague, and most likely provide you a user friendly and quick solution on Excel/VBA.

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Well, I simply know to value the time and effort I could save by having the right person do the right thing. I thank you for your offer, but I will insist on compensating any work put into this after the fact at least.

The problem is...:

I have a set of points to which I want to fit ellipses (all five parameters: ellipse center coordinates, axis lengths, tilt angel). Fitting one ellipse to a set of points, is a standard problem with ready-made solutions. But I want to do this in plural. Two, or three, ellipses need to be fitted simultaneously. Each ellipse has its own set of points. The trick is that there is a certain relationship between the ellipses which must be respected.

Now I will try to describe this relationship, or "restriction", between the ellipses. All points and all ellipses are in the same plane. But consider each ellipse as the projection of a circle in 3D. These two or three circles are concentric and lie in parallell planes, but on different heights on their common central axis. It is as if any two of the circles are on the surface of a cone, but all three of them are not necessarily on the same cone. The radii of and height distances between the circles are known.

Fitting the ellipses one by one and then average things out between them, isn't ideal. The main problem with this is that normally I will only have points from the upper half of one ellipse, and only points from the lower half of another ellipse. This makes rather small "errors" in the points near the end of each ellipse-half, cause quite large distortions in the parameters in the ellipses fitted to each of them. Imagine, the wide half of a horisontally oriented ellipse, is not far off from being the narrow end of a much larger vertically oriented ellipse. Such fitting errors in the individual ellipses don't average out well if the restrictions are imposed afterwards.

If I could fit points to both the upper and the lower half of the same ellipse simultaneously, the results would be much more stable. However, this cannot be done. But if the relationship between the ellipses could be respected during the fitting process, then the "multi-ellipse object" would be surrounded by points and this would reduce the fitting errors considerably.

Does this make sense?

Quote:

Except I will avoid MatLab like the plague, and most likely provide you a user friendly and quick solution on Excel/VBA.

I'd need something which executes fairly swiftly. But if it can be compiled, or if I can understand the solution method well enough to implement it myself in for example Matlab and then compile it, it'd do it for me. And any step on the way is of course helpful.

Cheers!

Do you want graphics? Or do you just want the points for the ellipses?

I have an idea what you want to do I think, but a picture may help me understand more.

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Here are some attempts to illustrations:


Cricles.


From a perspective



Three ellipses (or halves thereof) for which points are available are coloured for illustration. There will be three distinct sets of points to fit. Red points for the red ellipse and so on. For example, no red point is to be used in fitting the green ellipse.

Generally, the fragment of each ellipse available, is a bit too small for reliable parameter estimation. But if they could all be put together and fitted at once, the estimates should become much better.

If it is of any help, here is some of my reasoning about the relationship between circles and ellispes. It is difficult for me to describe this, and I'm not 100% it is correct, but I make a try:

Introduce an imaginary circle at right angle to a circle observed. The two circles intersect in two points. The diameter between these two points forms the vertical axis of the ellipse for this circle. I introduce two triangels. The angle at the center of the imaginary circle is the same as the angle at the "viewing point" on the periphery of the same imaginary circle, no matter where on the imaginary circle the viewing point is located. And this angle is proportional to the length of the vertical axis of the ellipse seen from the viewing point.


A new line is drawn from the viewing point to the "bottom" point of the imaginary circle, right under its center. Now consider the point where this line intersects the diameter of the circle. This is where we find the center of the ellipse, so it is how the length of the horisontal axis of the ellipse is determined. The length of the horisontal axis is proportonal to the viewing angle in the corresponding triangle.

So, the ratio between the lengths of the two axes of an ellipse, is (I think) the same as the ratio between those two angles at the viewing point which I have tried to describe.

Each circle has its own "imaginary circle". But all these imaginary circles intersect in the same "viewing point". I think that this is the geometry one has to use in order to translate the relationship between the circles in 3D space, to a relationship between the parameters of the ellipses in the 2D plane where they are "projected".

I write "vertical" and "horisontal" here, but the axes are tilted at some angle.

Ok, I cannot assist you with shapes/drawing/geometric figures. I'm horrible with art and not much of a web/artistic designer.

If you did want an engine built that figures distances, or loops through possible combinations of points, I can do that.

If that doesn't help your cause, I understand. Reading what you just described, a java based calculator with a program that can build shapes and 3D-animation would be the way to go.

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Oh, I forgot to answer that question:

No, I am not at all interested in any graphics stuff! I just want the parameters for the ellipses. 5 numbers for each one of them.

I just drew those figures now in order to help explain the mathematical problem. That red-green-blue thing maybe was confusing. I did it just to explain that for each of the ellipse halves (the colored ones) there belongs a certain set of points. The points and ellipses and circles actually have no colors. They are just mathematical abstractions. The points are randomly distributed around each of their ellipse (half).
This part I don't understand. But it doesn't sound like anything I am interested in!

Please, before you put any work into this, please clear things out with me. It is so easy to misunderstand, especially since I am not used to strict mathematical language.

I want to put in 2D coordinates for two or three sets of points, together with radii and distances defining the circles' relations to each other, and get back the ellipse parameters that fits it best according to least square or something like that.

I guess one makes an initial guess for parameters of one of the ellipses, then uses the geometry of the circles to calculate which parameter this implies for the other ellispes. Then one calculates the fit (least square or something like that). Then iterates and refits until the fit is good enough. From the geometry of it, one needs to formulate some optimization function which hopefully can have its derivatives solved so that an efficient numerical method can be applied.

There is a duality between the 2D projection, where there exists ellipses and points; and the 3D physical world, where there exists circles. A certain angle in the 3D world, corresponds to the length of an ellipse axis in the 2D projection.

I'll think it through and make a better attempt at explaining things tomorrow.

I attached a shell program for you. It has 2 points you enter, radius lengths, and a filler loop program when you press the button.

It sounds like what you want to do is guess through scenarios until you meet a fit with the ellipses/circles.

If this is what you want, please supply me the equations for what you want to check.

Attached Files

Dirlewanger.xls (23.0 KB, 27 views)

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Well, I want a numerical method to efficiently find a good enough fit of ellispes to points. Much like this code does in Matlab:
MATLAB Central File Exchange - fit_ellipse
But for multiple related ellipses at once.

I'll solve the geometry and provide the equations, yes!