  We show that the Chern-Schwartz-MacPherson class of a hypersurface X in a
nonsingular variety M `interpolates' between two other notions of
characteristic classes for singular varieties, provided that the singular locus
of X is smooth and that certain numerical invariants of X are constant along
this locus. This allows us to define a lift of the Chern-Schwartz-MacPherson
class of such `nice' hypersurfaces to intersection homology. As another
application, the interpolation result leads to an explicit formula for the
Chern-Schwartz-MacPherson class of X in terms of its polar classes.
