inflection: smooth point vs. singular point

From what I understand of the Cache text he seems to be using the term inflection somewhat loosely, or perhaps with reference to another philosopher's work. Cache uses the term inflection to define any curve that changes from being concave to convex. This is not the case in the mathematical definition of the term inflection.

Mathematically speaking the inflection point is a point on a curve at which the curvature changes sign via a smooth transition. The curve changes from being positive curvature to a negative curvature, or vice versa. For example when driving a vehicle along a curve, the point of inflection is a moment at which the steering-wheel is momentarily straight, being turned from left to right or right to left.

Cache's inflection rococo changes curvature from concave to convex but not at a smooth point. It changes at a singular point (one which is not smooth) so therefore it is not mathematical inflection.

Mathematical inflection occurs only through a smooth point. Cache's inflection occurs through a smooth point or a or a singular point.