In modern terms, the defect at a vertex is a discrete version of the curvature of the polyhedral surface concentrated at that point.
Negative defect indicates that the vertex resembles a saddle point (negative curvature), whereas positive defect indicates that the vertex resembles a local maximum or minimum (positive curvature).
If the sum of the angles exceeds a full turn, as occurs in some vertices of many non-convex polyhedra, then the defect is negative.
The concept of defect extends to higher dimensions as the amount by which the sum of the dihedral angles of the cells at a peak falls short of a full circle.
[1] A generalization says the number of circles in the total defect equals the Euler characteristic of the polyhedron.
This is a special case of the Gauss–Bonnet theorem which relates the integral of the Gaussian curvature to the Euler characteristic.
A converse to Descartes' theorem is given by Alexandrov's uniqueness theorem, according to which a metric space that is locally Euclidean (hence zero curvature) except for a finite number of points of positive angular defect, adding to
It is tempting to think that every non-convex polyhedron must have some vertices whose defect is negative, but this need not be the case if the Euler characteristic is positive (a topological sphere).