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Is there any statistical method to visually compare two curves?

What is the best and correct way to compare two similar curves and calculate the error/difference in percentage?

I have created a program that generates a curve of a column base using Bezier curve. Now, I want to find out how accurate my generation is. So I have a function for the first curve I defined, but I dont have a function for the second one, which is only on the picture.

enter image description here

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Are these curves functions, $y = f(x)$, or general curves in the plane like spirals, ellipses, etc.? –  Rahul Apr 19 at 19:16
    
general 2D curves, I am trying to compare curves I created using Bezier curve with the curve on the image (so I don't have function definitions for both of them) –  Riko Apr 19 at 19:20
    
but evaluation with certain number of vertices would be good enough, but how to calculate the error of the whole curve? I was thinking about (average deviation) / (curve length), but I'm not sure if this is statistically correct –  Riko Apr 19 at 19:23
    
or if there is any standard for such a evaluation... –  Riko Apr 19 at 19:24
    
Is the original curve parameterized? Is the Bezier curve parameterized? PS it might help if you put the actual equations into the question so we have some more context. –  NotNotLogical Apr 19 at 19:33

1 Answer 1

up vote 3 down vote accepted

A standard way to compare two (sufficiently nice) functions $f(x)$ and $g(x)$ over the interval $[a,b]$ is to use the inner product $$\left<f(x),g(x)\right>:=\int_a^b{f(x)g(x)\,\mathrm{d}x}$$ from which we get $$||f(x)-g(x)||=\sqrt{\int_a^b{\left(f(x)-g(x)\right)^2\,\mathrm{d}x}}$$ where you can think of $||f(x)-g(x)||$ as being the "distance" between the functions $f$ and $g$.

If you are dealing with parametric curves you could use $$\text{dist}\,\left(x(t),y(t)\right):=\sqrt{\int_{t_0}^{t_1}{||x(t)-y(t)||^2\,\mathrm{d}t}}$$ to get a reasonable measure, but you would have to ensure that both curves are parameterized in the "same way".

EDIT: If you want a measure of "percent error" I suppose you could do something like $$\text{% error}=\frac{\text{magnitude of error}}{\text{original magnitude}}=\frac{\int{||x(t)-y(t)||\,\mathrm{d}t}}{\int{||x(t)||\,\mathrm{d}t}}$$ which is the integral of the difference divided by the arclength of the original path. Since you only have points, you would have to approximate by computing $$\frac{{\Delta t\over 10}\sum{||P_i-B_i||}}{\sum{\sqrt{(x_{i+1}-x_i)^2+(y_{i+1}-y_i)^2}}}$$ where $P_i=(x_i,y_i)$ is the $i$'th point on the path and $B_i$ is the corresponding point on the Bezier curve. So if the Bezier approximation is parameterized with $0\le t\le 1$ then $$B_i=y\left(i{1\over 10}\right)$$ where $y(t)$ is the curve.

Keep in mind that I'm making this up as I go ;) But hopefully you can work with some of these ideas and see if anything fits what you're wanting to get...

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I am not sure, if this is what I need because I don't know the definition of g(x), I elaborated my question plz have a look –  Riko Apr 19 at 20:59
    
@Riko See the edit I've tried to change it accordingly –  NotNotLogical Apr 19 at 21:39
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In discussing curve fitting, one uses this metric. However, there is a drawback of the error percentages: "Slim" column bases will get higher error percentages. I would suggest to standardise it somehow by the error of taking one solid block as the column base. But that's just a suggestion to review other possibilities. Some kind of total variation would also be reasonable. –  Horst Grünbusch Apr 19 at 22:16
    
@NotNotLogical Thanks for your answer, thinking of it: and please correct me if I am wrong, but since integral is the area between two curve segments, and taking into account that a segment between Pi and Pi+1 will be treated as a straight line segment, it would give me some error even if there was none in case that the corresponding B segment was not straight –  Riko Apr 19 at 22:49
    
If the points all match up, the sum in the numerator will be zero so you will get no error. –  NotNotLogical Apr 19 at 22:55

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