Make vertices act as nodes when subdividing.

When you use the pen tool in adobe illustrator, you get the affect of the right drawing. In blender when you add a subdivision it shrinks the bends. Can I make the subdivision act like the pen tool in illustrator?

I need the mesh to make a 3d scanned surface workable. Issue is when a build a surface out using the surface it clips everywhere due to the above issue.

When you use the pen tool in adobe illustrator, you get the affect of the right drawing. In blender when you add a subdivision it shrinks the bends. Can I make the subdivision act like the pen tool in illustrator?

I need the mesh to make a 3d scanned surface workable. Issue is when a build a surface out using the surface it clips everywhere due to the above issue.