52 research outputs found
Spindle configurations of skew lines
We prove a conjecture of Crapo and Penne which characterizes isotopy classes
of skew configurations with spindle-structure. We use this result in order to
define an invariant, spindle-genus, for spindle-configurations.
We also slightly simplify the exposition of some known invariants for
configurations of skew lines and use them to define a natural partition of the
lines in a skew configuration.
Finally, we describe an algorithm which constructs a spindle in a given
switching class, or proves non-existence of such a spindle.Comment: 42 pages, many figures. A new corrected proof of a conjecture of
Crapo and Penne is added. More new material is also adde
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
Maximally inflected real rational curves
We introduce and begin the topological study of real rational plane curves,
all of whose inflection points are real. The existence of such curves is a
corollary of results in the real Schubert calculus, and their study has
consequences for the important Shapiro and Shapiro conjecture in the real
Schubert calculus. We establish restrictions on the number of real nodes of
such curves and construct curves realizing the extreme numbers of real nodes.
These constructions imply the existence of real solutions to some problems in
the Schubert calculus. We conclude with a discussion of maximally inflected
curves of low degree.Comment: Revised with minor corrections. 37 pages with 106 .eps figures. Over
250 additional pictures on accompanying web page (See
http://www.math.umass.edu/~sottile/pages/inflected/index.html
Searching edges in the overlap of two plane graphs
Consider a pair of plane straight-line graphs, whose edges are colored red
and blue, respectively, and let n be the total complexity of both graphs. We
present a O(n log n)-time O(n)-space technique to preprocess such pair of
graphs, that enables efficient searches among the red-blue intersections along
edges of one of the graphs. Our technique has a number of applications to
geometric problems. This includes: (1) a solution to the batched red-blue
search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an
algorithm to compute the maximum vertical distance between a pair of 3D
polyhedral terrains one of which is convex in O(n log n) time, where n is the
total complexity of both terrains; (3) an algorithm to construct the Hausdorff
Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n)
time and O(n+m) space, where n is the total number of points in all clusters
and m is the number of crossings between all clusters; (4) an algorithm to
construct the farthest-color Voronoi diagram of the corners of n axis-aligned
rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle
problem for n parallel line segments in the plane in optimal O(n log n) time.
All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure
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