- Authors:
- K P Agarwal,
- Otfried Schwarzkopf,
- Micha Sharir
Let $F$ and $G$ be two collections of a total of $n$ bivariate algebraic functions of constant maximum degree. The minimization diagrams of $F$, $G$ are the planar maps obtained by the $xy$-projections of the lower envelopes of $F$, $G$, respectively. We show that the combinatorial complexity of the overlay of the minimization diagrams of $F$ and of $G$ is $O(n^{2+\varepsilon})$, for any $\varepsilon<0$. This result has several applications: (i) a near-quadratic upper bound on the complexity of the region in 3-space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an efficient and simple divide-and-conquer algorithm for constructing lower envelopesein three dimensions; and (iii) a near-quadratic upper bound on the complexity of the space of all plane transversals of a collection of simply-shaped convex sets in three dimensions.
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