Related papers: Linear Complexity Hexahedral Mesh Generation
We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts. By contrast, the best previous such result used exponentially many cuts. More precisely,…
Given a non-empty genus in $n$ dimensions with determinant $d$, we give a randomized algorithm that outputs a quadratic form from this genus. The time complexity of the algorithm is poly$(n,\log d)$; assuming Generalized Riemann Hypothesis…
In this work, we show the geometric properties of a family of polyhedra obtained by folding a regular tetrahedron along regular triangular grids. Each polyhedron is identified by a pair of nonnegative integers. The polyhedron can be cut…
Adaptive mesh refinement is a key component of efficient unstructured space-time finite element methods. Underlying any adaptive mesh refinement scheme is, of course, a method for local refinement of simplices. However, simplex bisection…
Contemporary large models often exhibit behaviors suggesting the presence of low-level primitives that compose into modules with richer functionality, but these fundamental building blocks remain poorly understood. We investigate this…
The bilinear equations of the $N$-component KP and BKP hierarchies and a corresponding extended Miwa transformation allow us to generate quadrilateral and circular lattices from conjugate and orthogonal nets, respectively. The main…
Finite elements of higher continuity, say conforming in $H^2$ instead of $H^1$, require a mapping from reference cells to mesh cells which is continuously differentiable across cell interfaces. In this article, we propose an algorithm to…
The synthesis of a metasurface exhibiting a specific set of desired scattering properties is a time-consuming and resource-demanding process, which conventionally relies on many cycles of full-wave simulations. It requires an experienced…
Let $M$ be a volume finite non-compact complete hyperbolic $n$-manifold with totally geodesic boundary. We show that there exists a polyhedral decomposition of $M$ such that each cell is either an ideal polyhedron or a partially truncated…
We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges of a 4x5x1…
A polyiamond is a polygon composed of unit equilateral triangles, and a generalized deltahedron is a convex polyhedron whose every face is a convex polyiamond. We study a variant where one face may be an exception. For a convex polygon P,…
We show that no minimal vertex triangulation of a closed, connected, orientable 2-manifold of genus 6 admits a polyhedral embedding in R^3. We also provide examples of minimal vertex triangulations of closed, connected, orientable…
For a polyhedron $P$ in $\mathbb{R}^d$, denote by $|P|$ its combinatorial complexity, i.e., the number of faces of all dimensions of the polyhedra. In this paper, we revisit the classic problem of preprocessing polyhedra independently so…
A classic theorem by Steinitz states that a graph G is realizable by a convex polyhedron if and only if G is 3-connected planar. Zonohedra are an important subclass of convex polyhedra having the property that the faces of a zonohedron are…
We consider the combinatorial question of how many convex polygons can be made by using the edges taken from a fixed triangulation of n vertices. For general triangulations, there can be exponentially many: we show a construction that has…
We construct an explicit bijection between bipartite pointed maps of an arbitrary surface $\mathbb{S}$, and specific unicellular blossoming maps of the same surface. Our bijection gives access to the degrees of all the faces, and distances…
An interesting description of a collinear configuration of four particles is found in terms of two spherical coordinates. An algorithm to compute the four coordinates of particles of a collinear Four-Body central configuration is presented…
Mesh generation remains a key technology in many areas where numerical simulations are required. As numerical algorithms become more efficient and computers become more powerful, the percentage of time devoted to mesh generation becomes…
Compacting orthogonal drawings is a challenging task. Usually algorithms try to compute drawings with small area or edge length while preserving the underlying orthogonal shape. We present a one-dimensional compaction algorithm that alters…
We consider a subclass of tilings, the tilings obtained by cut and projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent \alpha in terms of the ranks of…