How to find the arclength and radius of curvature for a set of 3D data points?

I have a set of 3-D points that represent the coordinates of a curve. I need to find the arclength and radius of curvature of this curve. How might I go about doing this? I've tried fitting the data to a polynomial but it doesn't seem to be working. Please Help!

[Matt J]

What do you mean, it's not working? If the polynomial provides a good fit to your curve, the calculation should be straightforward.

[John D'Errico]

My guess is the OP does not know how to fit a polynomial to a curve embedded in a 3-d domain. I'm also willing to bet the data itself is not amenable to such a curve fit, since traditional surface fitting tools are not appropriate for that task, if this is truly a curvilinear path in R^3.

So, if you want help, you need to show your data. You need to show what you tried, how you tried to model it. And finally, you need to show what you tried to compute the radius of curvature. Because in the end, we cannot guess what you did wrong if we are not given any indication of what you did.

[Matt J]

@Matthew Wong,

If the problem is with the fitting, as John surmises, and not with the actual arclength and curvature calculation, then please go back to your original thread on that topic and pick up the discussion there, adding data and other information as proposed by John.

[Matthew Wong]

Thanks for the feedback guys! I've attached my data this time. Basically right now I try to break up the data into smaller sections that encompass only one section of the curve so that it better fits a polynomial. Then I use the 'fit' equation to go through and try to fit a polynomial to each section. Then I try to find the arclength of each section (Right now I'm only trying to calculate the arclength, after I get a correct fit for the data I'll move on to radius of curvature).

if true
  %Initialize empty arrays and variables to prevent memory overload
syms s x y;
Pts = zeros(100,3);
magnitudes = zeros(99,1);
dx = zeros(99,1);
dy = zeros(99,1);
dz = zeros(99,1);
dscore = zeros(99,1);
magdiff = zeros(99,1);
Sigpts = zeros(100,3);
lowbound = 1;
fitdata=zeros(100,3);
arclength=zeros(100,1);

 %Load points into a 100 x 3 array
for j=1:100
    Pts(j,1) = xpts(j);
    Pts(j,2) = ypts(j);
    Pts(j,3) = zpts(j);
end

 %Plot the curvature
%plot3(Pts(:,1),Pts(:,2),Pts(:,3));
% grid on;
hold off;
  end

[Matthew Wong]

Update: I had some trouble posting my entire code, but I've attached the MATLAB file above.

Answer 1

[Are Mjaavatten]

I was inspired by your question to write a functon that calculates the arc length and curvature of a 1D curve in 2D or 3D space. I believe that does what you want. I submitted it to The Mathworks File Exchange today. You will find it here.

[Matthew Wong]

Exactly what I was looking for, thanks!