  Traditional problems in computational geometry involve aspects that are both
discrete and continuous. One such example is nearest-neighbor searching, where
the input is discrete, but the result depends on distances, which vary
continuously. In many real-world applications of geometric data structures, it
is assumed that query results are continuous, free of jump discontinuities.
This is at odds with many modern data structures in computational geometry,
which employ approximations to achieve efficiency, but these approximations
often suffer from discontinuities.
  In this paper, we present a general method for transforming an approximate
but discontinuous data structure into one that produces a smooth approximation,
while matching the asymptotic space efficiencies of the original. We achieve
this by adapting an approach called the partition-of-unity method, which
smoothly blends multiple local approximations into a single smooth global
approximation.
  We illustrate the use of this technique in a specific application of
approximating the distance to the boundary of a convex polytope in
$\mathbb{R}^d$ from any point in its interior. We begin by developing a novel
data structure that efficiently computes an absolute
$\varepsilon$-approximation to this query in time $O(\log (1/\varepsilon))$
using $O(1/\varepsilon^{d/2})$ storage space. Then, we proceed to apply the
proposed partition-of-unity blending to guarantee the smoothness of the
approximate distance field, establishing optimal asymptotic bounds on the norms
of its gradient and Hessian.

Computational Geometry