Find the radius ( or curvature ) at a point on a bspline using geomdl / nurbs

Question

I would like to find the curvature at a given point on a 3D b-spline. I believe that I want to use the derivatives of the spline at that point to calculate the curvature at that point but I do not understand how.

I have defined a 3D bspline (taken from SolidWorks (where it came from is not relevant )) in geomdl. I can evaluate any point on the spline using Curve.evaluate_single(). This appears to work correctly. I have verified the returned points against the SolidWorks model so I assume I have implemented the bspline in nurbs correctly.

I need to find the radius at a various points along this curve. From my Google searching I think that I want to use Curve.derivatives() to calculate the instantaneous curvature at that point. But I do not understand how to get the results of Curve.derivatives() into a radius.

So below is the result of Curve.derivatives(SomePointOnPath,4):

[   [74.66019681782404, 131.77035668055586, 19.88498274391211],
    [-2719.7097781710354, -598.8099790539873, -711.5032638750225],
    [-5384.519486543373, 1273.8662545231637, 19431.220551950217],
    [93757.48746982217, -22247.397114396095, 31343.52746776864],
    [0.0, 0.0, 0.0]]

By taking points a small step either side of this point I have calculated the radius at this point to be about 409 ( Solving for radius of a circle given 3 points )

I do not understand what the results from Curve.derivatives() are telling me. (the first tuple is the coordinates of the point, beyond that I am lost)

I expect the radius at this particular point to be about 409.

My fundamental question is two parts:

What are the results telling me. What do they mean.

How do I use these result to calculate the radius at this point.

fang · Accepted Answer · 2019-06-15T03:15:51.543

The radius of curvature of a curve can be computed as |C'|^3/|C' X C"| where C' and C" are the first and the 2nd derivative vectors, X is the cross product operator and |.| is the vector's magnitude. So, you will need the first and the 2nd derivatives at that point to compute the radius of curvature.

Curve.derivatives(SomePointOnPath,4) returns the first 4 derivatives of the curve and the 0-th derivative is the position of the point on the curve. Therefore, the first derivative is [-2719.7097781710354, -598.8099790539873, -711.5032638750225] and the 2nd derivative is [-5384.519486543373, 1273.8662545231637, 19431.220551950217]. So, we can use these two vectors to do our computation and obtain a radius of curvature 408.9176414, which is close to your estimated value 409.

On a side note, you can simply pass '3' as the 2nd argument to Curve.derivatives() as the 3rd derivatvie vector has no use in the computation of radius of curvature.

Find the radius ( or curvature ) at a point on a bspline using geomdl / nurbs

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