How do I get distance between two curves that cannot be described with a polynomial?

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Hi guys so I have a unique problem. I'm comparing the distance between two path planning algorithms and these are the results I get in the MATLAB figure. I want to compare the distance between the two paths and output a single graph that has the offset of the distance between the curves, but as you can see, I cannot really use fit with smoothingspline ("fit(X,Y,"smoothingspline")") because both plots cannot be described in a single equation, trust me, I have tried.
I also tried using movmean to average the functions but that made the two graphs fall apart even further sicne the X and Y values don't exactly line up. I figure I will need to use a 3rd dimension like time to orient it but I would like to avoid doing that. My last resort could be running computer vision to get a white space between the two and grab the area between them but that is not a very exact analysis.
Jacinth Gudetti
Jacinth Gudetti on 17 Mar 2022
Edited: Jacinth Gudetti on 17 Mar 2022
@Sam Chak @Robert U @John D'Errico I have taken some more time to analyze my problem and I believe time should not be considered. rather I think it should be a function of perpendicular distance from the planned (red) path to the actual (blue) path. Time may in fact complicate matters even further as what I am looking for is for the algorithm's accuracy to stay close to the red line rather than anything with respect to time.

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Answers (1)

Sam Chak
Sam Chak on 15 Mar 2022
Edited: Sam Chak on 15 Mar 2022
Here is an example. Say, there are two paths: one is an onion-shaped path and the other is a circular path.
The onion-shaped path with a center at (, ) can be described with
, and ,
while the cicular path with a radius r and the same center is given by
, and .
t = 0:pi/50:2*pi;
r = 1;
xc = 2;
yc = 3;
% Onion-shaped path
x1 = r*(cos(t)).^3 + xc;
y1 = r*sin(t) + yc;
h1 = plot(x1, y1, xc, yc, 'o');
daspect([1, 1, 1])
f1 = @(t) sqrt((-3*((cos(t)).^2).*sin(t)).^2 + cos(t).^2);
len1 = integral(f1, 0, 2*pi)
hold on
% Circular path
x2 = r*cos(t) + xc;
y2 = r*sin(t) + yc;
h2 = plot(x2, y2, xc, yc, 'o');
daspect([1, 1, 1])
f2 = @(t) sqrt(sin(t).^2 + cos(t).^2);
len2 = integral(f2, 0, 2*pi)
hold off
len1 = 5.8290
len2 = 6.2832
The onion-shaped path is shorter than the circular path. You can also verify that circumference for the circle is .
Also check out @John D'Errico's amazing works on the File Exchange.

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