# On the bivariate function minimization problem and its applications to motion planning

## Abstract

An important technical problem which arises in many geometric contexts including robot motion planning is to analyze the combinatorial complexity κ(**F**) of the minimum *M* (*x*,*y*) of a collection **F** of *n* continuous bivariate functions *f*_{1}(*x*,*y*), ..., *f*_{n}(*x*,*y*), such that each triple of function graphs intersect in at most *s* points, and each pair of functions intersect in a curve having at most *t* singular points. We have obtained the following results for this problem: (1) If the intersection curve of each pair of functions intersects each plane *x* = *const* in exactly one point and *s*=1 (but not if *s*=2) then κ(**F**) is at most *O* (*n*), and can be calculated in time *O* (*n* log *n*) by a method extending Shamos' algorithm for the calculation of planar Voronoi diagrams. (2) If *s*=2 and the intersection of each pair of functions is connected then κ(**F**)=*O* (*n*^{2}). (3) If the intersection curve of each pair of functions intersects every plane *x* = *const* in at most two points, then κ(**F**) is at most *O* (*n*λ_{s}+2(*n*)), where the constant of proportionality depends on *s* and *t*, and where λ_{r}(*q*) is the (almost linear) maximum length of a (*q*,*r*) Davenport-Schinzel sequence. We also present an algorithm for calculating *M* in this case, running in time *O* (*n*λ_{s}+2(*n*) log *n*). (4) Various new geometric applications of these results have also been derived.

## Keywords

Voronoi Diagram Piecewise Linear Function Intersection Curve Lower Envelope Courant Institute## Preview

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