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The differently treated vertices

I applied subdivision surface modifier to my model and afterwards, I found some annoying burrs on the surface. Taking a closer look revealed that these burrs are caused by vertices - the surface is kind of snapped to these vertices instead of going smoothly.

I have pretty just begun learning Blender so can neither explain or solve this problem. Please help.

P.S. Most vertices have surface goes smoothly under them.

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  • $\begingroup$ Selected vertices are slightly offset, but when you deselect them they're back to normal. $\endgroup$ – Lukasz-40sth Aug 18 '19 at 8:38
  • $\begingroup$ Doubled checked for duplicates, result is negative. Only one vertex there. As for selecting, unfortunately it is not the case. $\endgroup$ – user288208 Aug 18 '19 at 8:41
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To keep it simple:

If the faces normals are in the same orientation, they can be smoothly interpolated at the subdivision: their mean stays positive.

enter image description here

If they are not, their mean becomes null and so their is no interpolation possible and their mean (so the orientation) is null.

enter image description here

This is like to say the function is not derivable at the face junction.

So the junction stays a singularity and stays angular.

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Though totally have no idea why, after selecting all the vertices (thus all the faces) and recalculating their normal (ctrl + shift + N), the problem is solved.

However, answers explaining reasons will still be appreciated.

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    $\begingroup$ Normals give orientation to the calculation. If normals are opposite on each side of an edge, that lead to angular result, if in the same orientation, smooth result. $\endgroup$ – lemon Aug 18 '19 at 8:48

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