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I've been looking for a proper way to make the creased vertices stay in line with the subdivided geometry but it ends up creating an unwanted corner stretching.

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I'm trying to get something like this, in this case I'm using a shrinkwrap modifier to get rid of the stretching but it's not a very reliable or practical solution:

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So, if anyone know of a proper way to do this I would really appreciate if you could share it, this has been giving me trouble for a long time.

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The way the subdivision modifier works, the final geometry is not going to touch the vertices.

What you want to use if you want that is a Nurbs surface

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This has a lot more limitations than subdivided geometry, like not being able to use booleans on it tho. And you still have control vertices that are not on the curve. They just need to be there to set the curvature one way or another.

I don't know why you need it to behave like that but if it's for retopology then the skinwrap modifier is that is commonly use. You can check this option tp show the vertices where they'd be if you apply the modifier but it's only visual, it does not work with snapping

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I'm not sure, but maybe you simply want to disable "boundary smooth" and resign from any creasing?

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