Conformal mapping of long quadrilaterals and thick doubly connected domains

Abstract

Abstract. In this paper we investigate theoretically an approximation technique for avoiding the crowding phenomenon in numerical conformal mapping. The method applies to conformal maps from rectangles to "long quadrilaterals," i.e., Jordan domains bounded by two parallel straight lines and two Jordan arcs, where the two arcs are far apart. We require that these maps take the four corners of the rectangle to the four corners of the quadrilateral.Our main theorem tackles a conformal mapping problem for doubly connected domains, and we derive from this our results for quadrilaterals. As a corollary, we extend the "domain decomposition" mapping technique of Papamichael and Stylianopoulos.Similar results hold for the inverse maps, from quadrilaterals to rectangles.

Conformal mappings of stretched polyominoes onto half-plane

“…From (16) we deduce that h is negative and decreases in (0, c). It decreases on (c, α 0 ) from +∞ to a ∈ (0, 1) and increases on (α 0 , 1) from a to 1.…”

“…One more important direction of investigations is development of approximate methods of calculating of conformal modules and capacities such as method of decomposition of domain, finite element method and others (see, e. g., [2], [3], [5], [14], [15], [16], and [17]).…”

Riemann–Schwarz Reflection Principle and Asymptotics of Modules of Rectangular Frames

“…The quadrilateral Q H could be considered as a generalized long quadrilateral. Asymptotics of the modules of long quadrilaterals was investigated in [16], [17], [15], [18], [19], and other papers where various methods for computing the modules were suggested. (…”