Find outside/inside curve of two curves

Question

in geometry nodes i want to determine which curves lies on the inside (our outside) of a turn.

I found out that it should, in general, be possible to do so by calculating the curvature of of the curves and then comparing those values. The larger one should usually be the outside curve. My goal is to get and use these values as "mask" for further GN Stuff.

I base these two curves on one single bezier curve:

Here i take my input curve and offset it by a factor in the normal direction, however i'm unsure about calculating the curvature and then comparing it. Does anyone have an idea how i could calculate the curvature for each point in the curve and then compare them? Thanks in advance.

Answer 1

Curvature of a Curve

This solution should work for curves with variable z-position, as long as the general shape is somewhat planar.

By using the Cross Product on the tangent of the current point and the next one, we get a perpendicular vector, pointing up when curve goes to the left, and pointing down when the curve goes to the right. We can then extract the z position and test its signature.

In orange is the current tangent $\vec{t_n}$, in red is the next tangent $\vec{t_{n+1}}$ and in blue the cross product. I've drawn another example a little further along the curve, where curvature is opposite.

Let's store the boolean isRightTurn as attribute. From there it's straight forward. The outter part is only define by the left/right curve and if we are in a left/right turn.

For the right curve, we reverse the boolean with Not. For the left curve, we take the boolean as is. Here is for the right curve :

If we plug the isOutter attribute, we get the expected result :

Oversight

The last point of the curve will not get a correct computation of the curvature. I would suggest manually giving its correct value if it generates some issue.

Blend file