In this video we highlight the application of Computational Geometry to our understanding of the formation and dynamics of the Cosmic Web. The emergence of this intricate and pervasive weblike structure of the Universe on Megaparsec scales can be ...

We present a Java implementation of Lang's Universal Molecule algorithm, alongside with a visualization of its interconnected structures: the input metric tree and compatible convex polygon, whose 2D crease pattern and 3D uniaxial base are computed by ...

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their ...

Approximating convex bodies is a fundamental question in geometry and has applications to a wide variety of optimization problems. Given a convex body K in RE^d for fixed d, the objective is to minimize the number of vertices or facets of an ...

Flexibility studies of macromolecules modeled as mechanical frameworks rely on computationally expensive, yet numerically imprecise simulations. Much faster approaches for degree-of-freedom counting and rigid component calculations are known for finite ...

We investigate one of the fundamental areas in computational geometry: lower bounds for range reporting problems in the pointer machine and the external memory models. We develop new techniques that lead to new and improved lower bounds for simplex ...

The colorful Carathéodory theorem [B82] states that given d+1 sets of points in R^d, the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the ...

In this paper, we consider two fundamental problems in the pointer machine model of computation, namely orthogonal range reporting and rectangle stabbing. Orthogonal range reporting is the problem of storing a set of n points in d-dimensional space in a ...

We consider maintaining a dynamic set S of N horizontal segments in R² such that, given a vertical ray Q in R², the segments in S intersecting Q can be reported efficiently. In the external memory model, we give a structure that consumes O(N/B) space, ...

Let P be a d-dimensional n-point set. A Tverberg partition of P is a partition of P into r sets P1, ..., Pr such that the convex hulls ch(P1), ..., ch(Pr) have non-empty intersection. A point in the intersection of the convex hulls is called a Tverberg ...

In the conflict-free coloring problem, for a given range space, we want to bound the minimum value F(n) such that every set P of n points can be colored with F(n) colors with the property that every nonempty range contains a unique color. We prove a new ...

Let P be a set of n points in the plane. We present an efficient algorithm for preprocessing P, so that, for a given query point q, we can quickly report the largest disk that contains q but its interior is disjoint from P. The storage required by the ...

Computing the coordinate-wise maxima of a planar point set is a classic and well-studied problem in computational geometry. We give an algorithm for this problem in the self-improving setting. We have n (unknown) independent distributions cD₁, cD₂, ..., ...

We present a deterministic algorithm to compute the Reeb graph of a PL real-valued function on a simplicial complex in O(m log m) time, where m is the size of the 2-skeleton. The problem reduces to dynamic graph connectivity. We obtain the running time ...

The Morse-Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse-Smale complexes have been presented in case of piecewise linear manifolds. However, previous ...

This note contributes to the point calculus of persistent homology by extending Alexander duality from spaces to real-valued functions. Given a perfect Morse function f: Sspaceⁿ⁺¹ -> [0,1] and a decomposition Sspaceⁿ⁺¹ = Uspace ∪ Vspace into two (n+1)-...

The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes ...

We introduce a parametrized notion of genericity for Delaunay triangulations which, in particular, implies that the Delaunay simplices of δ-generic point sets are thick. Equipped with this notion, we study the stability of Delaunay triangulations under ...

Given an anisotropic Voronoi diagram, we address the fundamental question of when its dual is embedded. We show that, by requiring only that the primal be orphan-free (have connected Voronoi regions), its dual is always guaranteed to be an embedded ...

Let P be a set of n ≥ 4 points in the plane that is in general position and such that n is even. We investigate the problem whether there is a cubic plane straight-line graph on P. No polynomial-time algorithm is known for this problem. Based on a ...