On the Convex Hulls of Parametric Plane Curves

Linear-time convex hull algorithms are known to exist for polygons and planar objects bounded by piecewise algebraic curves. Most of the existing algorithms laboriously trace out the so-called “pockets” formed by the concave portions of an object’s boundary. Objects bounded by general parametric curves have been excluded from the previous study partly because the nature of computation is seemingly more numerical than combinatorial so asymptotic analysis cannot be performed as for algebraic curves. This paper deals with two-dimensional objects with parametric curve boundaries. It introduces a design scheme that considerably simplifies the maintenance of a “partial hull” which bounds the traversed portion of an object’s boundary. This design scheme spawns three convex hull algorithms respectively for objects bounded by smooth curves, by polygons, and by piecewise smooth curves. All three algorithms run in time linear in the number of monotone segments (or vertices) and under a fixed precision. Curve preprocessing, nevertheless, is still subject to the limitations of nonlinear root finding (and thus not included in the running time analysis). Extensions to open curves and shapes with holes are straightforward, just as to objects bounded by non-parametric curves (e.g., algebraic curves). The algorithms have been implemented with presentation of experimental results.