The perpendicular bisector construction

1995 ◽  
Vol 56 (1) ◽  
pp. 75-84 ◽  
Author(s):  
G. C. Shephard
Keyword(s):  
Perpendicular Bisector

scholarly journals Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

2018 ◽  
Vol 28 (3) ◽  
pp. 365-387
Author(s):  
S. CANNON ◽  
D. A. LEVIN ◽  
A. STAUFFER
Keyword(s):  
Markov Chain ◽  
Relaxation Time ◽  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Mixing Time ◽  
Perpendicular Bisector ◽  
Unit Square

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

scholarly journals Generalized Perpendicular Bisector and exhaustive discrete circle recognition

2011 ◽  
Vol 73 (6) ◽  
pp. 354-364 ◽  
Author(s):  
Eric Andres ◽  
Gaëlle Largeteau-Skapin ◽  
Marc Rodríguez
Keyword(s):  
Perpendicular Bisector

Vorticity-Based Perpendicular-Bisector Grids for Improved Upscaling of Two-Phase Flow

SPE Journal ◽  
2010 ◽  
Vol 15 (04) ◽  
pp. 989-1002 ◽  
Author(s):  
Hassan Mahani ◽  
Mohammad Evazi
Keyword(s):  
Two Phase Flow ◽  
Phase Flow ◽  
Perpendicular Bisector ◽  
Two Phase

Mutuality

1939 ◽  
Vol 23 (256) ◽  
pp. 342-348
Author(s):  
N. M. Gibbins
Keyword(s):  
Subject Matter ◽  
The Other ◽  
Perpendicular Bisector ◽  
The Subject ◽  
The Common

1. This lecture grew out of an attempt to trace the consequences of putting together two examination questions. The subject-matter of both is the reflections X, Y, Z of a point P in the sides of a triangle ABC- In the first question we have to show that the perpendiculars from A on YZ, from B on ZX, from C on XY meet in P′, the centre of the circle XYZ, and that the relation between P and P′ is mutual. Since AY = AP =AZ, the perpendicular from A to YZ bisects it. Hence this perpendicular passes through the centre of the circle XYZ, as do similarly the other two perpendiculars. Let X′, Y′, Z′ be the reflections of P′ in the sides of ABC. Since BC is the common perpendicular bisector of PX and P′X′, PX′ =P′X) and similarly PY′ = P′Y, PZ′ =P′Z. Hence P is the centre of the circle X′Y′Z′, and the two radii are equal.

scholarly journals On the centre of spherical curvature of a curve

1937 ◽  
Vol 30 ◽  
pp. xi-xii
Author(s):  
C. E. Weatherburn
Keyword(s):  
Position Vector ◽  
Perpendicular Bisector ◽  
Arc Length ◽  
Normal Plane ◽  
Adjacent Point ◽  
Current Point ◽  
Image Position ◽  
Spherical Curvature ◽  
Unit Vectors ◽  
Polar Line

The position of the centre S of spherical curvature at a point P of a given curve C may be found in the following manner, regarding S as the limiting position of the centre of a sphere through four adjacent points P, P1, P2, P3 on the curve, as these points tend to coincidence at P. The centre of a sphere through P and P1 lies on the plane which is the perpendicular bisector of the chord PP1 and so on. Thus the centre of spherical curvature is the limiting position of the intersection of three normal planes at adjacent points. Let s be the arc-length of the curve C, r the position vector of the point P, and t, n, b unit vectors in the directions of the tangent, principal normal and binormal at P. Then if s is the position vector of the current point on the normal plane at P, the equation of this plane isSince r and t are functions of s, the limiting position of the line of intersection of the normal planes at P and an adjacent point (i.e. the polar line) is determined by (1) and the equation obtained by differentiating this with respect to s, viz.

A Range Free Acoustic Localization Algorithm in Wireless Sensor Networks

2014 ◽  
Vol 538 ◽  
pp. 502-507
Author(s):  
Jiang Shan Ai ◽  
Xiao Hong Chen
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Background Noise ◽  
Low Complexity ◽  
Wireless Sensor ◽  
Perpendicular Bisector ◽  
Localization Algorithm ◽  
Acoustic Localization ◽  
Localization Precision ◽  
Range Free

For accomplishing acoustic location in wireless sensor networks (WSNs), a range free acoustic localization algorithm based on perpendicular bisector partition is proposed, taking into account of reducing computation complexity and reduce the interference of the background noise. Adopting a range free perpendicular bisector partition, the proposed method can find the sub-region of the source, and the time complexity is much lower than that of existing methods. According to extensive analysis on noise, the concept of noise sensitive region is derived. Experimental results show that the proposed method has a high localization precision and low complexity.

scholarly journals Generalized Perpendicular Bisector and Circumcenter

2010 ◽  
pp. 1-10 ◽  
Author(s):  
Marc Rodríguez ◽  
Sere Abdoulaye ◽  
Gaëlle Largeteau-Skapin ◽  
Eric Andres
Keyword(s):  
Perpendicular Bisector

Perpendicular bisectors, duality and local symmetry of plane curves

1995 ◽  
Vol 125 (1) ◽  
pp. 181-194 ◽  
Author(s):  
Peter Giblin ◽  
Farid Tari
Keyword(s):  
Simple Closed Curve ◽  
Local Symmetry ◽  
Closed Curve ◽  
Critical Values ◽  
Parameter Family ◽  
Complete List ◽  
Plane Curves ◽  
Perpendicular Bisector ◽  
Symmetry Set

For a smooth, simple closed curve α in the plane, the perpendicular bisector map P associates to each pair of distinct points (p, q) on α the perpendicular bisector of the chord joining p and q. To a pair (p, p), the map P associates the normal to α at p. The set of critical values of this map is the union of the dual of the symmetry set of α and the dual of the evolute. (The symmetry set is the locus of the centres of circles bitangent to α.) We study the mapP and use it to give a complete list of the transitions which take place on the dual of the symmetry set and the dual of the evolute, as α varies in a generic one-parameter family of plane curves.

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