计算几何

Let $P$ be a convex polyhedron and $Q$ be a convex polygon with $n$ vertices in total in three-dimensional space. We present a deterministic algorithm that finds a translation vector $v \in \mathbb{R}^3$ maximizing the overlap area $|P \cap…

Geometric predicates are a basic ingredient to implement a vast range of algorithms in computational geometry. Modern implementations employ floating point filtering techniques to combine efficiency and robustness, and state-of-the-art…

The Fr\'echet distance is a computational mainstay for comparing polygonal curves. The Fr\'echet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in…

We study the problem of computing the Fr\'echet distance between two polygonal curves under transformations. First, we consider translations in the Euclidean plane. Given two curves $\pi$ and $\sigma$ of total complexity $n$ and a threshold…

A set of n segments in the plane may form a Euclidean TSP tour, a tree, or a matching, among others. Optimal TSP tours as well as minimum spanning trees and perfect matchings have no crossing segments, but several heuristics and…

Let $P$ be a set of $n$ points in the plane. We compute the value of $\theta\in [0,2\pi)$ for which the rectilinear convex hull of $P$, denoted by $\mathcal{RH}_\theta(P)$, has minimum (or maximum) area in optimal $O(n\log n)$ time and…

Simplicial partitions are a fundamental structure in computational geometry, as they form the basis of optimal data structures for range searching and several related problems. Current algorithms are built on very specific spatial…

We propose HyperSteiner -- an efficient heuristic algorithm for computing Steiner minimal trees in the hyperbolic space. HyperSteiner extends the Euclidean Smith-Lee-Liebman algorithm, which is grounded in a divide-and-conquer approach…

An orthogonal polygon is called an ortho-unit polygon if its vertices have integer coordinates, and all of its edges have length one. In this paper we prove that any ortho-unit polygon with $n \geq 12$ vertices can be guarded with at most…

We consider coresets for $k$-clustering problems, where the goal is to assign points to centers minimizing powers of distances. A popular example is the $k$-median objective $\sum_{p}\min_{c\in C}dist(p,C)$. Given a point set $P$, a coreset…

In this paper, we present a novel heuristic algorithm for the stable but NP-complete deformation-based edit distance on merge trees. Our key contribution is the introduction of a user-controlled look-ahead parameter that allows to trade off…

Merge trees are a common topological descriptor for data with a hierarchical component, such as terrains and scalar fields. The interleaving distance, in turn, is a common distance for comparing merge trees. However, the interleaving…

Given a set of $n\geq 1$ unit disk robots in the Euclidean plane, we consider the Pattern Formation problem, i.e., the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where…

We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and…

Given a set $ P $ of $n$ points and a set $ H $ of $n$ half-planes in the plane, we consider the problem of computing a smallest subset of points such that each half-plane contains at least one point of the subset. The previously best…

This paper introduces a new mathematical formulation and numerical approach for the computation of distances and geodesics between immersed planar curves. Our approach combines the general simplifying transform for first-order elastic…

Given a set $P$ of $n$ points in the plane, its unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ are connected by an edge if their (Euclidean) distance is at most $1$. We consider several classical…

For a set $P$ of $n$ points in the plane and a value $r > 0$, the unit-disk range reporting problem is to construct a data structure so that given any query disk of radius $r$, all points of $P$ in the disk can be reported efficiently. We…

Spectral analysis of open surfaces is gaining momentum for studying surface morphology in engineering, computer graphics, and medical domains. This analysis is enabled using proper parameterization approaches on the target analysis domain.…

We investigate approximation algorithms for several fundamental optimization problems on geometric packing. The geometric objects considered are very generic, namely $d$-dimensional convex fat objects. Our main contribution is a versatile…