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Produce one smooth curve on one triangle mesh

I hope to get one smooth curve on one triangle mesh. I get one path on the mesh at first. The path consists of vertices of the mesh. I can see the path from the image below. … To project the smooth curve, I find the closest point on the mesh for each vertex of the smooth curve. However the curve is not on the mesh because it intersects with the mesh. …
Jogging Song's user avatar