Linear Regression in Polar Space

I have posted this question before but I don't think I was clear on what i was trying to do exactly. I am trying to simulate a set of muon detecting drift tubes in 2d space. I have 2 sets of detector tubes (shown as black circles in the image), a particle trajectory goes through all tubes instantaneously (represented as a blue line 2 shown), if a particle passes through the tube you can tell how far from the center of the tube the particle was but not in what direction this closest point was (The closest point in the trajectory is represented omnidirectionaly as a colored circle). Using the information about how close the particle was to the center of the respective tubes it passed through, i need to recreate the trajectory of the particle. Fundamentally I am given one set of colored circles and need to find the best fit line that is tangent to all the circles. Since this data is represented as the distance from the center of a tube to the trajectory without any directional data (i.e. no θ) I believe a linear regression must be preformed in polar coordinates.

If anyone could point me in the right direction it would be greatly appreciated. I am trying to implement this in MATLAB so i cannot use any external programs.

Let me know if i need to clarify anything else about the problem.

 PhysOrg.com science news on PhysOrg.com >> Front-row seats to climate change>> Attacking MRSA with metals from antibacterial clays>> New formula invented for microscope viewing, substitutes for federally controlled drug
 Recognitions: Homework Help Science Advisor This looks very similar to the outer tracker at LHCb. A simple solution would be to test all different options with a regression in cartesian coordinates - with a fit quality determined by the difference ((simulated radius)-(measured radius))/(estimated error) or something like that. If your problem has a real application, I could try to contact someone at LHCb who worked with that.
 I am developing this algorithm using matlab simulations so that I can test how accurate it is, but I will then map it into 3 dimensions and implement it on hardware. Once implemented on real hardware there will be no way to reliably test the accuracy. There are also some other considerations that I need to account for in the algorithm but this is a starting point. The algorithm I am trying to develop will probably be much simpler that those implemented on other similar drift tube muon detectors such as those installed at particle colliders and the like. I cant convert the radi into Cartesian coordinates because they are represented as a circles. I only know how far from the center of a tube the trajectory was not in what direction. Thanks

Recognitions:
Homework Help

Linear Regression in Polar Space

Your radius measurement gives something like "##r=300\pm 50 {\mu}m##". For a given track in cartesian coordinates, you can calculate the radius, and get the deviation from the measured radius. As there is an analytic expression for this radius (based on coordinates for track and straw), you can even calculate its derivative in your parameter space.

Each double layer gives 4 possible track layouts, for most tracks 2 or 3 of them should have a very bad fit, so you can quickly reduce the number of possible orientations and perform a regular regression for them afterwards.

Recognitions: