Curvature of Catmullrom spline

[gingaz]

Hello guys!

I'm stuck with this for a 4th day now..

I have a set of data and for every data point I want to calculate a curvature. In order to do that I use Catmullrom spline to interpolate points and get derivatives f' and f". Curvature is defined as y"/ (1+y'^2)^3/2.

However, at some points calculated curvature is incorrect.It is known, that Catmullrom is C1 continuous, so f" is NOT continuous.
I have read somewhere, that f' means slope and f" - curvature.

My question would be: for curvature calculations, can I rely on Catmullrom spline if it is only C1 continuous (not C2)?
Or should i use NURBS? Any easier approach?

Thank you very much!

Ginga

 

[haruspex]

Does http://tom.cs.byu.edu/~455/bs.pdf help?

 

[HallsofIvy]

Probably not. B-spline are piecewise cubic and the second derivative is always continuous at knots, unlike Catmullrom splines.

 

[haruspex]

Probably not. B-spline are piecewise cubic and the second derivative is always continuous at knots, unlike Catmullrom splines.

The OP appeared to be open to the possibility of using different splines, so I was suggesting B-splines.

 

[CellCoree]

Hello Ginga,

First of all, I understand your frustration and I'm sorry to hear that you have been stuck on this issue for several days. Let me try to provide some insight and potential solutions to your problem.

From what you have described, it seems like you are trying to calculate the curvature of a Catmullrom spline using the formula y"/ (1+y'^2)^3/2. This formula is commonly used to calculate curvature for any type of curve, not just Catmullrom splines. However, as you have mentioned, the curvature is incorrect at some points. This is most likely due to the fact that Catmullrom splines are only C1 continuous, meaning that the first derivative (slope) is continuous, but the second derivative (curvature) may not be. This can result in incorrect curvature values at certain points.

To answer your question, yes, you can still rely on Catmullrom splines for curvature calculations even if they are only C1 continuous. However, there are a few things you can try to improve the accuracy of your calculations. One approach is to use a higher degree of interpolation, such as using a Catmullrom spline with a higher degree (e.g. cubic instead of quadratic). This can help improve the continuity of the second derivative and may result in more accurate curvature values.

Another option is to use NURBS (Non-Uniform Rational B-Splines) instead of Catmullrom splines. NURBS are C2 continuous, meaning that both the first and second derivatives are continuous. This can result in more accurate curvature calculations. However, NURBS can be more complex to implement and may not be necessary depending on your specific application.

In terms of an easier approach, you may want to consider using a library or software that has built-in functions for calculating curvature of curves. This can save you time and effort in implementing the calculations yourself and may also provide more accurate results.

I hope this helps and good luck with your project!