Frenet

FRENET - Frenet-Serret Space Curve Invarients

Description

FRENET - Frenet-Serret Space Curve Invarients
  
  [T,N,B,k,t] = frenet(x,y);
  [T,N,B,k,t] = frenet(x,y,z);

  Returns the 3 vector and 2 scaler
  invarients of a space curve defined
  by vectors x,y and z. If z is omitted
  then the curve is only a 2D,
  but the equations are still valid.

   _ r'
   T = ---- (Tangent)
       |r'|

   _ T'
   N = ---- (Normal)
       |T'|
   _ _ _
   B = T x N (Binormal)

   k = |T'| (Curvature)

   t = dot(-B',N) (Torsion)

   Example:
   theta = 2*pi*linspace(0,2,100);
   x = cos(theta);
   y = sin(theta);
   z = theta/(2*pi);
   [T,N,B,k,t] = frenet(x,y,z);
   line(x,y,z), hold on
    
quiver3(x,y,z,T(:,1),T(:,2),T(:,3),'color','r')
quiver3(x,y,z,N(:,1),N(:,2),N(:,3),'color','g')
quiver3(x,y,z,B(:,1),B(:,2),B(:,3),'color','b')
legend('Curve','Tangent','Normal','Binormal')

See also: GRADIENT

• MATLAB release: MATLAB 7 (R14)

[J. Li]

It is excellent for a curve that is really smoothly curved.

Do you have any idea how to handle the case that the curve is reduced to a straight line and/or the curved is reduced to a polygon as in the case of a nurb curve of order 2 in the Nurbs toolbox?

Thanks!

[G. Z.]

Hi all,
It is a very nice and usefull code, however:
I am no expert on this kind of math, but there may be an error in the torsion calculation in the code:
t = dot(-B,N), instead t = dot(-B',N)
I know that the code as it is now does not give a correct answer for a synthetic helix..
worth checking.
Thanks!
  

[david Fernandez]

Thanks it works perfect! but the example is wrong write, you have to transpose the T to use quiver3:
 quiver3(x,y,z,T(:,1)',T(:,2)',T(:,3)','color','r')
   quiver3(x,y,z,N(:,1)',N(:,2)',N(:,3)','color','g')
   quiver3(x,y,z,B(:,1)',B(:,2)',B(:,3)','color','y')

[Peter Vatter]

Thanks for the very nice code. It's the only piece I know which implements the way back - from a point cloud to the curvature and torsion values. Only the calculation of the torsion is actually incorrect. For an generic helix the calculated torsion is not constant. The fix by G.Z. unfortunately didn't work for me. I corrected the code as following (altough it certainly can be done shorter and easier):

dddx = gradient(ddx);
dddy = gradient(ddy);
dddz = gradient(ddz);
dddr = [dddx dddy dddz];

t = vdot(cross(dr, ddr), dddr) ./ mag(cross(dr, ddr),1).^2;

function N = vdot(A, B)
%row-wise dot-product of A and B
N=zeros(size(A,1),1);
for i=1:size(A,1)
    N(i) = dot(A(i,:), B(i,:));
end