Voronoi diagram for multiply-connected polygonal domains I: Algorithm

1987 ◽  
Vol 31 (3) ◽  
pp. 361-372 ◽  
Author(s):  
Vijay Srinivasan ◽  
Lee R. Nackman
Keyword(s):  
Voronoi Diagram ◽  
Polygonal Domains ◽  
Multiply Connected

scholarly journals Generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains

2014 ◽  
Vol 470 (2166) ◽  
pp. 20130848 ◽  
Author(s):  
Giovani L. Vasconcelos
Keyword(s):  
Conformal Mapping ◽  
Circular Domain ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Slit Domain ◽  
Combined Use ◽  
Polygonal Region ◽  
Upper Half Plane ◽  
Indefinite Integral ◽  
Detailed Derivation

A generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains, namely, a rectilinear slit domain and a bounded circular domain. The conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions, which is the same irrespective of the preimage slit domain, and a prefactor function that depends on the choice of the rectilinear slit domain. A detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half-plane with radial slits. Representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain.

scholarly journals The Motion of a Point Vortex in Multiply-Connected Polygonal Domains

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1175
Author(s):  
El Mostafa Kalmoun ◽  
Mohamed M. S. Nasser ◽  
Khalifa A. Hazaa
Keyword(s):  
Cross Sections ◽  
Single Point ◽  
Vortex Motion ◽  
Point Vortex ◽  
Multiply Connected Domains ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Contour Lines ◽  
Circular Domains ◽  
Prime Function

We study the motion of a single point vortex in simply- and multiply-connected polygonal domains. In the case of multiply-connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal domains onto circular domains. Then, we employ the Schottky-Klein prime function to compute the Hamiltonian governing the point vortex motion in circular domains. We compare between the topological structures of the contour lines of the Hamiltonian in symmetric and asymmetric domains. Special attention is paid to the interaction of point vortex trajectories with the polygonal obstacles. In this context, we discuss the effect of symmetry breaking, and obstacle location and shape on the behavior of vortex motion.

scholarly journals QUADRILATERAL MESHING BY CIRCLE PACKING

2000 ◽  
Vol 10 (04) ◽  
pp. 347-360 ◽  
Author(s):  
MARSHALL BERN ◽  
DAVID EPPSTEIN
Keyword(s):  
Voronoi Diagram ◽  
Circle Packing ◽  
Polygonal Domains ◽  
Quadrilateral Meshes ◽  
Quadrilateral Meshing

We use circle-packing methods to generate quadrilateral meshes for polygonal domains, with guaranteed bounds both on the quality and the number of elements. We show that these methods can generate meshes of several types: (1) the elements form the cells of a Voronoï diagram, (2) all elements have two opposite 90° angles, (3) all elements are kites, or (4) all angles are at most 120°. In each case the total number of elements is O(n), where n is the number of input vertices.

The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains

2005 ◽  
Vol 461 (2061) ◽  
pp. 2653-2678 ◽  
Author(s):  
Darren Crowdy
Keyword(s):  
Mapping Function ◽  
Circular Domain ◽  
Schottky Group ◽  
Polygonal Domain ◽  
Schottky Groups ◽  
Image Domain ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Connected Circular Domain ◽  
Natural Way

A formula for the generalized Schwarz–Christoffel mapping from a bounded multiply connected circular domain to a bounded multiply connected polygonal domain is derived. The theory of classical Schottky groups is employed. The formula for the derivative of the mapping function contains a product of powers of Schottky–Klein prime functions associated with a Schottky group relevant to the circular pre-image domain. The formula generalizes, in a natural way, the known mapping formulae for simply and doubly connected polygonal domains.

scholarly journals Multiple steadily translating bubbles in a Hele-Shaw channel

2014 ◽  
Vol 470 (2163) ◽  
pp. 20130698 ◽  
Author(s):  
Christopher C. Green ◽  
Giovani L. Vasconcelos
Keyword(s):  
Steady Motion ◽  
Flow Region ◽  
Circular Domain ◽  
Multiply Connected Domains ◽  
Moving Frames ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Region Exterior ◽  
Prime Function ◽  
Connected Circular Domain

Analytical solutions are constructed for an assembly of any finite number of bubbles in steady motion in a Hele-Shaw channel. The solutions are given in the form of a conformal mapping from a bounded multiply connected circular domain to the flow region exterior to the bubbles. The mapping is written as the sum of two analytic functions—corresponding to the complex potentials in the laboratory and co-moving frames—that map the circular domain onto respective degenerate polygonal domains. These functions are obtained using the generalized Schwarz–Christoffel formula for multiply connected domains in terms of the Schottky–Klein prime function. Our solutions are very general in that no symmetry assumption concerning the geometrical disposition of the bubbles is made. Several examples for various bubble configurations are discussed.

Sign in / Sign up

Export Citation Format

Share Document