Related papers: Plane-filling trails
A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
We study the version of the C-Planarity problem in which edges connecting the same pair of clusters must be grouped into pipes, which generalizes the Strip Planarity problem. We give algorithms to decide several families of instances for…
It is an attractive hypothesis that the spatial structure of visual cortical architecture can be explained by the coordinated optimization of multiple visual cortical maps representing orientation preference (OP), ocular dominance (OD),…
We consider applications involving a large set of instances of projecting points to polytopes. We develop an intuition guided by theoretical and empirical analysis to show that when these instances follow certain structures, a large…
We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…
Space-filling designs are commonly used in computer experiments to fill the space of inputs so that the input-output relationship can be accurately estimated. However, in certain applications such as inverse design or feature-based…
A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this paper is to introduce a new tool for sampling diagrams via sampling of plane…
It is common to encounter situations where one must solve a sequence of similar computational problems. Running a standard algorithm with worst-case runtime guarantees on each instance will fail to take advantage of valuable structure…
Using the Semple bundle construction, we derive an intersection-theoretic formula for the number of simultaneous contacts of specified orders between members of a generic family of degree $d$ plane curves and finitely many fixed curves. The…
Cutting plane methods, particularly outer approximation, are a well-established approach for solving nonlinear discrete optimization problems without relaxing the integrality of decision variables. While powerful in theory, their…
In this paper we show how, under surprisingly weak assumptions, one can split a planar curve into three arcs and rearrange them (matching tangent directions) to obtain a closed curve. We also generalize this construction to curves split…
Modern methods of graph theory describe a graph up to isomorphism, which makes it difficult to create mathematical models for visualizing graph drawings on a plane. The topological drawing of the planar part of a graph allows representing…
We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let $P$ be a set of $n$ points in the plane and let $S$ be a set of…
We introduce space-efficient plane-sweep algorithms for basic planar geometric problems. It is assumed that the input is in a read-only array of $n$ items and that the available workspace is $\Theta(s)$ bits, where $\lg n \leq s \leq n…
Alignment algorithms usually rely on simplified models of gaps for computational efficiency. Based on an isomorphism between alignments and physical helix-coil models, we show in statistical mechanics that alignments with realistic laws for…
We present a characterisation of blenders based on mapping properties of certain sets of curves that can be rigorously verified by computer-assisted methods. We develop an algorithm to construct these sets of curves that requires only a…
Parallel coordinates plot (PCP) is an excellent tool for multivariate visualization and analysis, but it may fail to reveal inherent structures for datasets with a large number of items. In this paper, we propose a suite of novel…
For each integer n, an n-folding curve is obtained by folding n times a strip of paper in two, possibly up or down, and unfolding it with right angles. Generalizing the usual notion of infinite folding curve, we define complete folding…
Intersections are the bottlenecks of the urban road system because an intersection's capacity is only a fraction of the flows that the roads connecting to the intersection can carry. This capacity can be increased if vehicles can cross the…