On the right end, you have an infinite slope. On the left end, not so much of a problem. But it will be better to do a couple of things. First, these look almost like arcs of an ellipse. So lets pretend they are that, or at least something close.
What you need to understand is an ellipse is just a circle that has been stretched. So we can undo that.
First, I'll rescale the axes.
whos
Name Size Bytes Class Attributes
R 1x9 433320 cell
Z 1x9 433320 cell
ans 1x29 58 char
I'll set the center of the coordinate system at (Z,R) = (30,-15), and then rescale R and Z, so they will now be roughly circular. Then I could transform to polar coordinates. At least, if I needed to do so. But do I?
Rtrans = @(r) (r - R0)/(max(r) - R0);
Ztrans = @(z) (z - Z0)/(max(z) - Z0);
plot(Ztrans(Z{i}),Rtrans(R{i}))
Not too bad. Do you see that by simply rescaling the axes by the endpoints of the curves, we now have all the curves virtually on top of each other?
My gut tells me that this may be sufficient for your purposes. Essentially, all you need to know are those endpoints, and you can reconstruct any curve, from this almost circular common arc.
Now, how, you might ask, can you use that curve, as a common arc, and to then reconstruct the original curves from it? Simpler (maybe) than you think.
Rmax = cellfun(@max,R)
Rmax =
13.8180 16.2501 15.5263 15.4948 15.5853 15.5561 15.5919 15.7650 15.6824
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Zmax = cellfun(@max,Z)
Zmax =
54.9765 62.7786 60.7609 59.5278 58.4727 57.2374 56.0746 55.0867 53.5833
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Next, once all of the curves are in this globally consistent form, convert to polar and model rhe result.
Rcommon = [Rcommon;Rtrans(R{i})];
Zcommon = [Zcommon;Ztrans(Z{i})];
[thet,rad] = cart2pol(Zcommon,Rcommon);
There we see that one of your curves is a little unlike the others. But the difference is not too immense, if you look at the scale on the y-axis in that last plot.
mdl = fit(thet,rad,'poly4')
mdl =
Linear model Poly4:
mdl(x) = p1*x^4 + p2*x^3 + p3*x^2 + p4*x + p5
Coefficients (with 95% confidence bounds):
p1 = -0.05061 (-0.05154, -0.04967)
p2 = 0.2918 (0.2889, 0.2948)
p3 = -0.3361 (-0.3392, -0.3331)
p4 = 0.004989 (0.003827, 0.006151)
p5 = 1.001 (1, 1.001)
If you look carefully, the blue curve falls neatly in the middle of that band. Now, we can reconstruct each of the curves, as well as plot a common curve on top, with an average set of parameters.
The model is a function of polar angle theta, varying from 0 to pi/2 radians. So the global curve will be:
thetglob = linspace(0,pi/2,20)';
[Zglob,Rglob] = pol2cart(thetglob,radglob);
plot(Zglob*(Zmaxbar - Z0)+ Z0,Rglob*(Rmaxbar - R0) + R0)
I've used average values to undo the transformations.
