If by "just work" you mean testing each constituent vertex, it'll fail. Consider the case when the needle is at the midpoint of a line between two points far away.
A point is represented by a z-value at the resolution of space.
A non-point is approximated by multiple z-values, at varying resolutions. You can trade the accuracy of the approximation against the number of z-values.
I explained the NN algorithm in another comment on this thread. It shouldn't matter if some of the adjacent z-values are actually from the approximation of the same object.
