Related papers: Slime mould computes planar shapes
The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…
The identification of the interfacial molecules in fluid-fluid equilibrium is a long-standing problem in the area of simulation. We here propose a new point of view, making use of concepts taken from the field of computational geometry,…
Convex hulls are useful as tight bounding proxies for a variety of tasks including collision detection, ray intersection, and distance computation. Unfortunately, the complexity of polyhedral convex hulls grows linearly with their input. We…
The slime mould Physarum polycephalum displays adaptive transport dynamics and network formation that have inspired its use as a model of biological computation. We develop a Lagrangian formulation of Physarum's adaptive dynamics on…
Writing an uncomplicated, robust, and scalable three-dimensional convex hull algorithm is challenging and problematic. This includes, coplanar and collinear issues, numerical accuracy, performance, and complexity trade-offs. While there are…
Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation…
A fully implementable filtered polynomial approximation on spherical shells is considered. The method proposed is a quadrature-based version of a filtered polynomial approximation. The radial direction and the angular direction of the…
A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and…
Let C be a real nonsingular affine curve of genus one, embedded in affine n-space, whose set of real points is compact. For any polynomial f which is nonnegative on C(R), we prove that there exist polynomials f_i with f \equiv \sum_i f_i^2…
We study the convex hulls of trajectories of polynomial dynamical systems. Such trajectories include real algebraic curves. The boundaries of the resulting convex bodies are stratified into families of faces. We present numerical algorithms…
We propose a novel 3D shape correspondence method based on the iterative alignment of so-called smooth shells. Smooth shells define a series of coarse-to-fine shape approximations designed to work well with multiscale algorithms. The main…
We describe the implementation of a subfield of the field of formal Puiseux series in polymake. This is employed for solving linear programs and computing convex hulls depending on a real parameter. Moreover, this approach is also useful…
This paper presents an alternate choice of computing the convex hulls (CHs) for planar point sets. We firstly discard the interior points and then sort the remaining vertices by x- / y- coordinates separately, and later create a group…
Any subset of the plane can be approximated by a set of square pixels. This transition from a shape to its pixelation is rather brutal since it destroys geometric and topological information about the shape. Using a technique inspired by…
Many studies have shown that Physarum polycephalum slime mold is able to find the shortest path in a maze. In this paper we study this behavior in a network, using a hyperbolic model of chemotaxis. Suitable transmission and boundary…
The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for…
The convex hull of N independent random points chosen on the boundary of a simple polytope in R^n is investigated. Asymptotic formulas for the expected number of vertices and facets, and for the expectation of the volume difference are…
The tropical convex hull of a finite set of points in tropical projective space has a natural structure of a cellular free resolution. Therefore, methods from computational commutative algebra can be used to compute tropical convex hulls.…
We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some…
A central problem of geometry is the tiling of space with simple structures. The classical solutions, such as triangles, squares, and hexagons in the plane and cubes and other polyhedra in three-dimensional space are built with sharp…