Extract worldspace normal data from curve direction

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Hello dear magicians,

We are trying to extract a worldspace normal from curves.
The final idea is to use the curves direction to drive a vector motion in nuke.
So a vertical (y) curve would gives a full green value (0,1,0), horizontal (x) one a red value (1,0,0), and Z.

I am honestly struggling on the math parts so any help from you would be welcome

I will keep you updated if I end up with something,
Have a nice day,
H
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