Information Processing Letters 36 (1990) 95-102 North-Holland 15 October 1990 ORACLE COMPLEXITIES FOR COMPUTATIONAL GEOMETRY OF SEMI-ALGEBRAIC SETS AND VORONOI DIAGRAMS Chandan HALDAR and L.M. PATNAIK Department of Computer Science and Automation, Indian Institute of Science, Bangalore 540012. India Communicated by L. Boasson Received 17 April 1989 Revised 28 March 1990 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA KeYWOrb: Computationd complexity, computational geometry, convex hull, semi-algebraic set, symbolic algebraic computa- tion, Voronoi diagram 1. Introduction Computational geometry has traditionally been concerned with computing geometric properties of sets of objects bounded by piecewise linear boundaries, for example, polygons and polyhedra. For accurately representing the geometry of most real-life objects, however, the polygonal and poly- hedral models are inadequate and the need for algorithmic methods to deal with objects whose boundaries are piecewise algebraic curves or surfaces has recently been recognized. On the other hand, piecewise algebraic curves and surfaces have been widely used in geometric modeling where these have been dealt with solely by curve-tracing type of purely numerical iterative methods. The new algebraic methods, drawing techniques from developments in symbolic and algebraic computa- tion, need to minimize their dependence on numerical iterative procedures, so as to permit elegant symbolic solutions of geometric problems concerning semi-algebraic objects. Almost all of the few published works on geo- metric algorithms for objects with curved boundaries (with the exception of Bajaj and Kim [2]) deal with the problem of accounting for the algebraic complexities of primitive computational operations on algebraic curves either by assuming a bound on the degrees of the curves (in which case each primitive operation can be assumed to take constant time) or by assuming the existence of a set of oracles which return the desired results in specific times irrespective of the algebraic com- plexity of the input curves. The complexities of the various algorithms on semi-algebraic sets are then analyzable in terms of the number (as a function of number of vertices of the input object) of calls to the relevant oracles. For example, Schaffer and Van Wyk [6] present an O(n) time algorithm for computing the convex hulI of a planar object the boundary of which consists of n edges each of which is a degree-2 (at most) curve segment (a conic section) without any inflection or singular points. They assume unit-time oracles for the computation of curve-line segment intersec- tions, tangent at a point on a curve, and curve- curve supporting line. While both procedures (as- suming degree bounds on curves, and assuming existence of oracles) have basically the same effect and facilitate extension of standard computational geometry algorithms for point sets and objects with piecewise linear boundaries to objects bounded by piecewise algebraic curves (or even transcendental curves, but we do not consider them here), they obfuscate the underlying alge- braic complexities of the primitive operations on 0020-0190/90/% 03.50 0 1990 - Ekevier Science Publishers B.V. (North-Holland) 95