8 research outputs found
Computing largest circles separating two sets of segments
A circle separates two planar sets if it encloses one of the sets and its
open interior disk does not meet the other set. A separating circle is a
largest one if it cannot be locally increased while still separating the two
given sets. An Theta(n log n) optimal algorithm is proposed to find all largest
circles separating two given sets of line segments when line segments are
allowed to meet only at their endpoints. In the general case, when line
segments may intersect times, our algorithm can be adapted to
work in O(n alpha(n) log n) time and O(n \alpha(n)) space, where alpha(n)
represents the extremely slowly growing inverse of the Ackermann function.Comment: 14 pages, 3 figures, abstract presented at 8th Canadian Conference on
Computational Geometry, 199
Constrained Geodesic Centers of a Simple Polygon
For any two points in a simple polygon P, the geodesic distance between them is the length of the shortest path contained in P that connects them. A geodesic center of a set S of sites (points) with respect to P is a point in P that minimizes the geodesic distance to its farthest site. In many realistic facility location problems, however, the facilities are constrained to lie in feasible regions. In this paper, we show how to compute the geodesic centers constrained to a set of line segments or simple polygonal regions contained in P. Our results provide substantial improvements over previous algorithms
An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams
Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O(k(n-k) log^2 n +n log^3 n) steps, where O(k(n-k)) is the number of faces in the worst case. Due to those axioms, this result applies to disjoint line segments in the L_p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, this kind of run time with a polylog factor to the number of faces was only achieved for point sites in the L_1 or Euclidean metric before
Differential operators on sketches via alpha contours
A vector sketch is a popular and natural geometry representation depicting
a 2D shape. When viewed from afar, the disconnected vector strokes of a
sketch and the empty space around them visually merge into positive space
and negative space, respectively. Positive and negative spaces are the key
elements in the composition of a sketch and define what we perceive as the
shape. Nevertheless, the notion of positive or negative space is mathematically ambiguous: While the strokes unambiguously indicate the interior
or boundary of a 2D shape, the empty space may or may not belong to the
shape’s exterior.
For standard discrete geometry representations, such as meshes or point
clouds, some of the most robust pipelines rely on discretizations of differential operators, such as Laplace-Beltrami. Such discretizations are not
available for vector sketches; defining them may enable numerous applications of classical methods on vector sketches. However, to do so, one needs
to define the positive space of a vector sketch, or the sketch shape.
Even though extracting this 2D sketch shape is mathematically ambiguous,
we propose a robust algorithm, Alpha Contours, constructing its conservative
estimate: a 2D shape containing all the input strokes, which lie in its interior
or on its boundary, and aligning tightly to a sketch. This allows us to define
popular differential operators on vector sketches, such as Laplacian and
Steklov operators.
We demonstrate that our construction enables robust tools for vector
sketches, such as As-Rigid-As-Possible sketch deformation and functional
maps between sketches, as well as solving partial differential equations on a
vector sketch
An Introduction to Randomization in Computational Geometry.
International audienceThis paper is not a complete survey on randomized algorithms in computational geometry, but an introduction to this subject providing intuitions and references. In a first time, some basic ideas are illustrated by the sorting problem, and in a second time few results on computational geometry are briefly explained