How to cut a geometry with an alpha texture?

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I want to use an alpha texture to cut a geometry. Basically want a trace node but can work with an input geometry with any shape.

My idea is to have uved geometry. Then I map a black and white texture to that geometry according to the uv. At the edges of the black and white geometry, a bunch of points are generated to draw a line to cut the geometry.

However, I do not know how to get the position of the black and white texture in world space of the geometry. Any help is very much appreciated!

I tried scattering many points on the geometry and transfer the texture color to point color, then removing the black colored point to cut the geometry. But I think this method is not efficient and elegant enough. The contour of generated geometry is usually too rough and not precise enough.
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Attribute transfer by UV, or the similar VEX functions could be used to deform the traced geometry onto a surface with uv's.

A boolean can be used in uv space to ensure the trace geometry has the same topology as the target.

Attachments:
uv_transfer_trace.png (974.2 KB)
trace_uv_texture.hip (134.1 KB)

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Thanks man! Your suggestions really helped me a lot.
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