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@Fabian,

Thanks for your responce.

To me it looks like eq. 40 is all about k, kappa (curvature).

Then x~``traj(s0) for example :
- This is second derivate = curvature, given the ``
- Is already transfomed, given the ~ above the x
- Is for startpoint s0.
- Is for x coordinate.

- It is the curvature kappa value for the transformed startpos?

- I now get it, that a line has no curvature, so value is : 0
- So for a arc, we just put in the curvature value like : "1/Radius" normally.  But they use eq.30 :   -1/r*cos(s/r).....etc.... Why so difficult?



I did a little play around. Nothing seriously.
We can use this little program also to validate other curves.

Setup:
2 red lines, lenght 100mm, are connected by a clothoid at xy ground plane.
The lines have a offset 100mm.
After drawing the clothoid, we know the clothoid total length.

Test:
For the first line we lift the startpoint off the plane, given the z value.
For the second line we lift the endpoint off the plane, given the -z value.
Then both lines are out off plane, they don't share any plane anymore.

Still we are able to project a clothoid on the groundplane.
In the second picture the z value of one end of the line is already z219 and z-219.

On the ground plane i projected the red 2 lines, given the yz plane. (side view)
between the 2 lines is the clothoid lenght for xy. So clothoid is streched out to straight line.
Then another algo growns the xy lenght until the 3d clothoid ends are colinear to the red lines in 3d.

 
 
 
 

I took a stab at making a hal component but was having issues getting it to compile with comp. Seems like comp isnt installed ("which comp" returns nothing) running the 2.9.3 iso

Sounds like you might want to read the docs:
linuxcnc.org/docs/html/hal/comp.html