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I am trying to get the curvature of a curve in 3D (I just have a bunch of data points). I wanted to test it on a simple known case of a helix.

s = 0:20/300:20;
x = 1.3*cos(s);
y = 1.3*sin(s);
z = 1*s;

and then get the tangent vectors at each point:

xtan = -1.3*sin(s);
ytan = 1.3*cos(s);
ztan = 1*ones(1,length(ytan));

Now with this information, I am trying to find out the angle it makes with the central Z axis.

Tangent = [xtan; ytan ;ztan];
CentralAxis = [0 0 1];

for i = 1:length(Tangent)
% Measure angle between two 3D vectors
    angleTan(i) = atan2d(norm(cross(Tangent(:,i),CentralAxis)),dot(Tangent(:,i),CentralAxis));
 end

    HelixCur = 1.3/(1.3^2+1^2);% curvature of helix which is  $\kappa_N$
    R_HelixCur = 1/HelixCur;

From Meusnier's theorem we know $\kappa_N = \kappa cos\alpha$. I am trying to verify this and I am not getting it right. Where am I going wrong?

I know the value of $\kappa$ which would be $1/1.3$ (in this example)

1 Answers 1

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There seem to be several suspicious things. First of all, to use your curvature formula, you need an arclength parametrization, and $s$ is not arclength for your helix. Second, Meusnier's formula applies to curves on surfaces, and $\kappa_N$ refers to the normal curvature of the surface in the direction of the curve (yet you comment that the curvature of the curve is $\kappa_N$). So, I think there is serious confusion here. What surface are you thinking about?

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    There was a typo in $s$ , i corrected it. Second here is the scenario which I am trying to solve. There is a cylinder with radius 1.3 units, centered at origin. Now there is a helix around this cylinder. Consider a point lying on the helix and take a tangent to it. Consider a plane with this tangent and normal to the surface at the point . This plane cuts the cylinder at angle $\theta$ . At this point I want to check and verify Meusnier's theorem. I need to solve this to use it in one of my other research problem. I am not very familiar with this material. – 2017-02-07
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    In order for $s$ to be arclength, the tangent vector needs to have length $1$. The normal curvature of the cylinder is more complicated, but in fact they angle between the surface normal and the principal normal oft he helix is either $0$ or $\pi$, so up to sign the two curvatures agree. Nothing to do with the axis of the cylinder. If you don't have a better reference, you might look at my text (free .pdf you can find linked in my profile. – 2017-02-07
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    Will go through the book. Was looking for a reference on Meusniers theorem. Do you have some suggestions which parts to focus on? – 2017-02-07