Chasing Puppies: Mobile Beacon Routing on Closed Curves

We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others' work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time.Comment: Full version of a SOCG 2021 paper, 28 pages, 27 figure

Chromatic kk-Nearest Neighbor Queries

Let P be a set of n colored points. We develop efficient data structures that store P and can answer chromatic k-nearest neighbor (k-NN) queries. Such a query consists of a query point q and a number k, and asks for the color that appears most frequently among the k points in P closest to q. Answering such queries efficiently is the key to obtain fast k-NN classifiers. Our main aim is to obtain query times that are independent of k while using near-linear space. We show that this is possible using a combination of two data structures. The first data structure allow us to compute a region containing exactly the k-nearest neighbors of a query point q, and the second data structure can then report the most frequent color in such a region. This leads to linear space data structures with query times of O(n1/2 log n) for points in R1, and with query times varying between O(n2/3 log2/3 n) and O(n5/6 polylog n), depending on the distance measure used, for points in R2. These results can be extended to work in higher dimensions as well. Since the query times are still fairly large we also consider approximations. If we are allowed to report a color that appears at least (1 - ϵ)f*times, where f*is the frequency of the most frequent color, we obtain a query time of O(log n + log log 1 1-ϵ n) in R1 and expected query times ranging between O (n1/2ϵ-3/2) and O(n1/2ϵ-5/2) in R2 using near-linear space (ignoring polylogarithmic factors)

Terrain Prickliness: Theoretical Grounds for High Complexity Viewsheds

An important task in terrain analysis is computing viewsheds. A viewshed is the union of all the parts of the terrain that are visible from a given viewpoint or set of viewpoints. The complexity of a viewshed can vary significantly depending on the terrain topography and the viewpoint position. In this work we study a new topographic attribute, the prickliness, that measures the number of local maxima in a terrain from all possible angles of view. We show that the prickliness effectively captures the potential of terrains to have high complexity viewsheds. We present near-optimal algorithms to compute it for TIN terrains, and efficient approximate algorithms for raster DEMs. We validate the usefulness of the prickliness attribute with experiments in a large set of real terrains

Efficient Fréchet distance queries for segments

We study the problem of constructing a data structure that can store a two-dimensional polygonal curve P, such that for any query segment ab one can efficiently compute the Fréchet distance between P and ab. First we present a data structure of size O(n log n) that can compute the Fréchet distance between P and a horizontal query segment ab in O(log n) time, where n is the number of vertices of P. In comparison to prior work, this significantly reduces the required space. We extend the type of queries allowed, as we allow a query to be a horizontal segment ab together with two points s, t ∈ P (not necessarily vertices), and ask for the Fréchet distance between ab and the curve of P in between s and t. Using O(n log2 n) storage, such queries take O(log3 n) time, simplifying and significantly improving previous results. We then generalize our results to query segments of arbitrary orientation. We present an O(nk3+ϵ + n2) size data structure, where k ∈ [1, n] is a parameter the user can choose, and ϵ > 0 is an arbitrarily small constant, such that given any segment ab and two points s, t ∈ P we can compute the Fréchet distance between ab and the curve of P in between s and t in O((n/k) log2 n + log4 n) time. This is the first result that allows efficient exact Fréchet distance queries for arbitrarily oriented segments. We also present two applications of our data structure. First, we show that our data structure allows us to compute a local δ-simplification (with respect to the Fréchet distance) of a polygonal curve in O(n5/2+ϵ) time, improving a previous O(n3) time algorithm. Second, we show that we can efficiently find a translation of an arbitrary query segment ab that minimizes the Fréchet distance with respect to a subcurve of P

Mapping Multiple Regions to the Grid with Bounded Hausdorff Distance

We study a problem motivated by digital geometry: given a set of disjoint geometric regions, assign each region Ri a set of grid cells Pi, so that Pi is connected, similar to Ri, and does not touch any grid cell assigned to another region. Similarity is measured using the Hausdorff distance. We analyze the achievable Hausdorff distance in terms of the number of input regions, and prove asymptotically tight bounds for several classes of input regions

Discrete curvature and torsion from cross-ratios

Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a Möbius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature

Saturated K-Plane Drawings with Few Edges

A drawing of a graph is k-plane if no edge is crossed more than k times. In this paper we study saturated k-plane drawings with few edges. This are k-plane drawings in which no edge can be added without violating k-planarity. For every number of vertices n>k+1, we present a tight construction with n−1k+1 edges for the case in which the edges can self-intersect. If we restrict the drawings to be ℓ-simple we show that the number of edges in saturated k-plane drawings must be higher. We present constructions with few edges for different values of k and ℓ. Finally, we investigate saturated straight-line k-plane drawings

Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the mixed finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure preserving: it reproduces curl-free fields precisely and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation

Optimal transport-based polar interpolation of directional fields

We propose an algorithm that interpolates between vector and frame fields on triangulated surfaces, designed to complement field design methods in geometry processing and simulation. Our algorithm is based on a polar construction, leveraging a conservation law from the Hopf-Poincaré theorem to match singular points using ideas from optimal transport; the remaining detail of the field is interpolated using straightforward machinery. Our model is designed with topology in mind, sliding singular points along the surface rather than having them appear and disappear, and it caters to all surface topologies, including boundary and generator loops