We study conformal mappings from the unit disk to circular-arc quadrilaterals with four right angles. The problem is reduced to a Sturm-Liouville boundary value problem on a real interval, with a nonlinear boundary condition, in which the coefficient functions contain the accessory parameters t,lambda of the mapping problem. The parameter lambda is designed in such a way that for fixed t, it plays the role of an eigenvalue of the Sturm-Liouville problem. Further, for each t a particular solution (an elliptic integral) is known a priori, as well as its corresponding spectral parameter lambda. This leads to insight into the dependence of the image quadrilateral on the parameters, and permits application of a recently developed spectral parameter power series (SPPS) method for numerical solution. Rate of convergence, accuracy, and computational complexity are presented for the resulting numerical procedure, which in simplicity and efficiency compares favorably with previously known methods for this type of problem.
Deep Dive into Conformal Mapping of Right Circular Quadrilaterals.
We study conformal mappings from the unit disk to circular-arc quadrilaterals with four right angles. The problem is reduced to a Sturm-Liouville boundary value problem on a real interval, with a nonlinear boundary condition, in which the coefficient functions contain the accessory parameters t,lambda of the mapping problem. The parameter lambda is designed in such a way that for fixed t, it plays the role of an eigenvalue of the Sturm-Liouville problem. Further, for each t a particular solution (an elliptic integral) is known a priori, as well as its corresponding spectral parameter lambda. This leads to insight into the dependence of the image quadrilateral on the parameters, and permits application of a recently developed spectral parameter power series (SPPS) method for numerical solution. Rate of convergence, accuracy, and computational complexity are presented for the resulting numerical procedure, which in simplicity and efficiency compares favorably with previously known method
A symmetric circular quadrilateral (s.c.q.) is a Jordan curve P in C formed of four circular arcs (or straight segments) with all four internal angles equal to π/2. Will assume that the vertices of P are situated in positions of the form ±A, ±A, where Re A > 0, Im A > 0.
Let D be a plane domain containing the origin and bounded by an s.c.q. The set of conformal types of such domains (considering the vertices as distinguished points) forms a two-dimensional space in a natural way. We consider conformal mappings f : D → D from the unit disk D to D. Since f extends continuously to the boundary P , there is a unique value t ∈ [0, 2π] such that f (e it ) = A; We will generally assume that f (0) = 0 and f ′ (0) > 0, which implies 0 < t < π/2.
The more general problem of mapping the disk to circular-arc polygon domains is treated in [Bj], [DT,Chapter 4], [He,Chapter 16], [Hi,Section 17.6], [Ho], [Ne,Chapter 5]. In particular, it is well known that the Schwarzian derivative of a conformal mapping of D onto a circular-arc polygon is a rational function of degree two. In this article we will develop further the work of P. Brown on the accessory-parameter problem for s.c.q.s. We follow much of the notation and copy several equations from [B] where it is verified that due to the symmetries of s.c.q.s, the Schwarzian derivative S f of f is of the specific form (1) below. This function is determined by two real parameters t, s and the relationship (2) must hold. As a partial converse, it is well known that due to the intimate relationship between the Schwarzian derivative and curvature, if the Schwarzian derivative of a holomorphic function f defined in D is of the form (1) and if (2) holds, then f is a local homeomorphism onto a (not necessarily schlicht) domain bounded by a (not necessarily simple) right circular-arc polygon.
A basic question is the following. Given P = ∂D (for example by specifying A and also the radius or the midpoint of one of the edges of P ), to find the parameters t, s of the Schwarzian derivative of f . In [B] Brown looked first at a simpler question, fixing t, normalizing f ′ (0) = 0 and calculating the remaining parameter s corresponding to a geometric characteristic of P , such as the radius of one of its edges. This reduces the problem to one real dimension. As with most methods which have been developed for conformal mapping of circular-arc polygons, in [B] the Schwarzian differential equation is solved numerically for trial values of s, the corresponding geometric characteristics of P = f (∂D) are calculated, and the process is repeated until a sufficiently close s is located.
Here we apply a technique developed in [KP] for dealing with Sturm-Liouville problems, the spectral parameter power series (SPPS) method, which permits a direct calculation in the sense that once certain auxiliary parameters are calculated for given t, one may evaluate the solution corresponding to any desired s without resorting to further integrations.
In a second article [B’] the full two-parameter problem is addressed. The problem is formulated with the normalization f (1) = 1, and the data is given in terms of the radius 1/κ 1 of the right edge of P and the midpoint p 2 of the upper edge. Brown’s solution involves a table of previously calculated values of these parameters in terms of (t, s). To apply it one looks for values reasonably close to the desired geometric parameters in this table, and then applies an iterative process to approximate the sought-after (t, s) to the desired accuracy. In this paper we apply our solution of the one-parameter problem, which is quite rapid, to this two-parameter problem in an iterative way. In part due to properties of an equivalent parameter λ which we use in place of s, our method does not require consultation of a table of prior values.
In the next two sections we set up a Sturm-Liouville boundary value problem whose solution relates the accessory parameter λ to the curvature κ of the right edge of P . It is seen that λ is a spectral parameter in a boundary value problem.
In Section 3 we describe the so-called canonical mapping f ∞ of D to a rectangle, and identify its parameter λ ∞ and the corresponding eigenfunction y ∞ of the Sturm-Liouville problem, which are used in the application of the SPPS method which is summarized in Section 4 and then applied in Section 5 to represent κ as a power series in λ for fixed t. An algorithm for the one-parameter problem κ → λ for fixed t is presented in Section 6, and for the two-parameter problem (κ 1 , p 2 ) → (t, λ) in Section 7.
We begin by setting up the classical second-order differential equation which governs the conformal mapping to s.c.q.s. The Schwarzian derivative of a holomorphic f , namely
is again holomorphic when f ′ does not vanish. For the conformal mapping of D to an s.c.q., formula [B, (3)], which we will refer to as (B3), says
(1) in which the parameter c ∈ C is subject to one restraint as follows. Wri
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