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I have coordinates of a set of points that form a closed loop that lies in a 3D surface. I know the equation of the surface and I can calculate it's surface normal at any point. I found that for a unit speed curve $\alpha(t)$ I can calculate the geodesic curvature as $$ \kappa_g(t) = \alpha''(t)\cdot (n(t)\times\alpha'(t)) $$

where $n(t)$ is the surface normal. I am confused about how to calculate the unit-speed derivatives $\alpha'(t)$ and $\alpha''(t)$. I think I should use some finite difference scheme like $$ \alpha'(t) \approx \frac{\alpha(t + h) - \alpha(t - h)}{2h} $$

So my algorithm for calculating the geodesic curvature at any point $P$ of this curve should be as follows:

  1. Identify a reference point on the curve to measure arc-length from.
  2. For the point $P$ calculate the arc-length by adding up the Euclidean distances between successive points from the reference point to the point P. This arc length gives me $t$.
  3. Find the position vectors of points at $t + h$ and $t - h$ by interpolation and then use the above finite difference equation to calculate $\alpha'(t)$.

Is my algorithm correct? I am not clear about how to get arc-length parameterization from the points on the curve.

Are there any standard algorithms (or even Python libraries) for such a calculation?

3D Curve

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