 Solution

A subdivision curve should support subdivision curve interrogation rather than merely supply successively larger control polygons. Its interface should provide the parameter interval, orientation and open/closed status; endpoint and limit-point evaluation to a prescribed accuracy; and tangent vector or derivative evaluation where defined. It should be possible to split a parameter interval, commonly at its midpoint, into equivalent restricted curve definitions while retaining all neighboring controls needed by the refinement rule.

For geometric algorithms, request a certified bounding volume for each restricted curve, such as a convex hull or an axis-aligned bounding box, and the minimum and maximum of a plane's linear function over that bound. Request also a certified flatness bound: the maximum distance of the restricted limit curve from its endpoint chord, together with its spatial extent. These allow safe rejection, adaptive drawing and accuracy control. A positive, constant-reproducing subdivision mask often makes a control-point convex hull a valid bound. With negative mask coefficients this must be proved separately or replaced by a padded bound; it cannot simply be assumed.

Further useful enquiries include roots of a scalar function along the curve, closest-point searches, arc length, and inversion of an arc-length parametrization. They can be implemented through evaluation, differentiation, certified bounds and recursive subdivision. Evaluation, restriction, enclosure and error bounds are the essential primitives needed for reliable clipping and rendering; an unqualified control-polygon approximation is insufficient.