Grasshopper

algorithmic modeling for Rhino

I want to "sweep"(?) a small rectangle - the pen - along a curve - we cannot see here - to finally get the light grey surface. Can this easily be done without using a script?

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Sort of. You can use [Sum Surface] to sweep one curve along another, but the resulting surface will be weirdly self-intersecting in places. You may have to post-process the edge curves to get a nice result out of it, but since I don't know what exactly you need it for, I can't give you specifics.

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Here's a way to get the outlines, but because it involves going via meshes you'll get a polyline result instead of smooth curves...

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