Circle Domain Research Articles

Study on linear conjugated combination of Zernike modes

We present linear conjugated combined aberration modes with a concentric pupil diameter of 4 mm. The combinations are according to the coupling relationship between Zernike modes over the concentric circle domain within the unit circle and the root mean square decreasing amplitude ratio of the corresponding aberration modes in the concentric pupil, in which the reconstruction pupil diameter is 6 mm. Each combined mode shares the characters of 2 radial orders apart, the same azimuth frequency, the same coefficient sign, and the prescribed amount, such as (C02 = 0.7\lambda, C04 = 0.3\lambda), (C-33 = 0.8\lambda, C-35 = 0.3\lambda), and so on. We also analyze the influence of the combined modes on optical quality. Simulations and experiments show improvement after combination; they also indicate that the influence of conjugated combination on optical quality has compensation and not superposition.

Wavefront Aberration Correction with Conjugated Combination Models of Zernike Modes

In view of the coupling relationship between the Zernike modes over the concentric circle domain within the unit circle and the wavefront counteraction of those coupling modes within the reconstruction pupil region, the linear combination models of Zernike modes are presented. The modes in each model share with negative correlation, same coefficient sign and prescribed amount. Finally, the experiments are implemented on AO system, the results show that it’s feasible and advantageous to use the combination models in the process of aberration correction.

Sketch Based Image Deformation and Editing with Guaranteed Feature Correspondence

The establishment of a good correspondence mapping is a key issue in planar animations such as image morphing and deformation. In this paper, we present a novel mapping framework for animation of complex shapes. We firstly let the user extract the outlines of the interested object and target interested area from the input images and specify some optional feature lines, and then we generate a sparse delaunay triangulation mesh taking the outlines and the feature lines of the source shape as constraints. Then we copy the topology from the source shape to the target shape to construct a valid triangulation in the target shape. After that, each triangle of this triangular mesh is further segmented into a dense mesh patch. Each mesh patch is parameterized onto a unit circle domain. With such parametrization, we can easily construct a correspondence mapping between the source patches and the corresponding target patches. Our framework can work well for various applications such as shape deformation and morphing. Pleasing results generated by our framework show that the framework works well.

Relative Riemann mapping criteria and hyperbolic convexity

Let R R be a simply-connected Riemann surface with a simply-connected subdomain U U . We give a criterion in terms of conformal reflections to determine whether R R can be embedded in the complex plane so that U U is mapped onto a disc. If it can, then U U is convex with respect to the hyperbolic metric of R R , by a theorem of Jørgensen. We discuss the close relationship of our criterion to two generalizations of Jørgensen’s theorem by Minda and Solynin. We generalize our criterion to the quasiconformal setting and also give a criterion for the multiply-connected case, where an embedding is sought that maps a given subdomain onto a circle domain.

A stabilized MLPG method for steady state incompressible fluid flow simulation

In this paper, the meshless local Petrov–Galerkin (MLPG) method is extended to solve the incompressible fluid flow problems. The streamline upwind Petrov–Galerkin (SUPG) method is applied to overcome oscillations in convection-dominated problems, and the pressure-stabilizing Petrov–Galerkin (PSPG) method is applied to satisfy the so-called Babuška–Brezzi condition. The same stabilization parameter τ( τ SUPG = τ PSPG ) is used in the present method. The circle domain of support, linear basis, and fourth-order spline weight function are applied to compute the shape function, and Bubnov–Galerkin method is applied to discretize the PDEs. The lid-driven cavity flow, backward facing step flow and natural convection in the square cavity are applied to validate the accuracy and feasibility of the present method. The results show that the stability of the present method is very good and convergent solutions can be obtained at high Reynolds number. The results of the present method are in good agreement with the classical results. It also seems that the present method (which is a truly meshless) is very promising in dealing with the convection- dominated problems.

A study of two-dimensional flow past regular polygons via conformal mapping

In this paper we study two-dimensional flow around regular polygons with an arbitrary but even number of edges N and one apex pointing to the free stream, with comparison to circular-cylinder flow. Both inviscid flow and low-Reynolds-number viscous flow are addressed. For inviscid flow, we obtained the exact solution for pure potential flow through Schwarz–Christoffel transformation, with the emphasis on the role of edge number, N, on the flow details. We also studied the behaviour, stationary lines and stability of vortex pair and found new stationary lines compared to circular cylinder. For viscous flow we derived the equation of stream function in the mapped (circle) domain, based on which approximate expressions for the critical Reynolds numbers and Strouhal number, as functions of the edge number, are obtained. The Reynolds number is based on the diameter of the circumscribed circle. For the steady flow, the first critical Reynolds number is a monotonically decreasing function of N, while N → ∞ corresponds to that for circular cylinder. The bifurcation point is ahead of the bifurcation point for circular cylinder. For unsteady flow, the critical Reynolds number for vortex shedding and the Strouhal number are both monotonically decreasing functions of N.

Radial and circular slit maps of unbounded multiply connected circle domains

Infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain, or to domains with both radial and circular slit boundary components are derived and implemented numerically and graphically. The formulae are generated by analytic continuation with the reflection principle. Convergence of the infinite products is proved for domains with sufficiently well-separated boundary components. Some recent progress in the numerical implementation of infinite product mapping formulae is presented.

Conformal welding and Koebe’s theorem

It is well known that not every orientation-preserving homeomorphism of the circle to itself is a conformal welding, but in this paper we prove several results which state that every homeomorphism is almost a welding in a precise way. The proofs are based on Koebe's circle domain theorem. We also give a new proof of the well known fact that quasisymmetric maps are conformal weldings.

On the models of the hyperbolic plane

Several geometric books introduce the models of the hyperbolic plane, the connections and the isomorphisms between them. This short article presents the connections of the most familiar models of the hyperbolic plane in the same figure. These connections are based on proper projections. Let the Poincare model (G) be defined on the hemisphere x + y + z=1, z > 0. In this model the lines are the half-circle sections whose planes are perpendicular to the xy-plane. (Detailed description in [BJ].) Let the Weierstrass model (W) be on the sheet z > 0 of the hyperboloid x + y− z = 1 ([FR]). The lines of this model are the hyperbola-branch sections whose planes pass through the point O(0, 0, 0). Let the Klein-Poincare circle model (K) be defined on the circle domain x + y < 1 on the plane z = 0, in this model the lines are the diameters and the circular arcs intersecting the base circle perpendicularly. The Cayley–Klein model (C) is defined on the circle domain x + y < 1 on the plane z = 1, where the lines are the chords of the base circle ([BJ], [SZP]).

Semi—circle theorem of unstable spectrum distribution of heterotropic perturbation and the upper bound estimation of its growth rate

Based on the theoretical analysis on the unstable spectrum distribution of heterotropic perturbation, the semi–circle theorem of such spectrum distribution is obtained — the spectrum distributed within an upper semi–circle domain with radius R0 and taking origin as its circle center in the complex plane, and the upper bound estimation of growth rate is given simultaneously. It is found that the smaller the horizontal scale of perturbation is, the higher the model top located, the greater the estimated value of upper bound of such growth rate is. Also the increase of vertical shear of wind and latitude have the positive contribution to the increase of such growth rate. Finally when stratified stability decreases, the relative maximal growth rate increases, while the maximal growth rate decreases instead.

Computer simulation on reconstruction of 3-D flame temperature distribution

To measure non-symmetric unsteady three dimensional temperature distribution in flame by simple, economic, fast and accurate means, and to apply a priori information to the measurement both sufficiently and efficiently, we conducted computer simulations. Simulation results proved that finite series-expansion reconstruction method is more suitable for measurement of temperature distribution in flame than transform method which is widely used in medical scanning and nondestructive testing. By comparing errors of simulations with different numbers of views, different domain shapes, different numbers of projections per view, different angles of views and different grid shapes, etc., we find that circle domain, triangular grid and sufficient number of projections per view, can improve the accuracy in the reconstruction of 3-D temperature distribution with limited views. With six views, errors caused by reconstruction computation are reduced, they are smaller than those caused by measurement. Therefore, a comparatively better means of measuring 3-D temperature distribution in flame with limited projection views by emission tomography is achieved. Experimental results also showed that the method we used was appropriate for measurement of 3-D temperature distribution with limited number of views [1].

Transboundary extremal length

We introduce two basic notions, ‘transboundary extremal length’ and ‘fat sets’, and apply these concepts to the theory of conformal uniformization of multiply connected planar domains. A new short proof is given to Koebe's conjecture in the countable case: every planar domain with countably many boundary components is conformally equivalent to a circle domain. This theorem is further generalized in two direction. We show that the same statement is true for a wide class of domains with uncountably many boundary components, in particular for domains bounded byK-quasicircles and points. Moreover, these domains admit more general uniformizations. For example, every circle domain is conformally equivalent to a domain whose complementary components are heart-shapes and points.

Rigidity of circle domains whose boundary has?-finite linear measure

Let Ω be a circle domain in the Riemann sphere whose boundary has σ-finite linear measure. We show that Ω is rigid in the sense that any conformal homeomorphism of Ω onto any other circle domain is equal to the restriction of a Mobius transformation. Previously, Kaufman and Bishop have independently found examples of non-rigid circle, domains whose boundary is a Cantor set of (Hausdorff) dimension one.

Uniform, Sobolev extension and quasiconformal circle domains

This paper contributes to the theory of uniform domains and Sobolev extension domains. We present new features of these domains and exhibit numerous relations among them. We examine two types of Sobolev extension domains, demonstrate their equivalence for bounded domains and generalize known sufficient geometric conditions for them. We observe that in the plane essentially all of these domains possess the trait that there is a quasiconformal self-homeomorphism of the extended plane which maps a given domain conformally onto a circle domain. We establish a geometric condition enjoyed by these plane domains which characterizes them among all quasicircle domains having no large and no small boundary components.

Quasiextremal distance domains and conformal mappings onto circle domains

This paper contributes to the theory of quasiextremal distance domains. We present some new properties for these domains and point out results concerning the extension of quasiconformal homeomorphisms.For example, we establish a continuity property for mod(E, F; D) and use this to demonstrate that mod(E, F; D) = cap(E, F; D) whenever D is a QED domain and E, F are disjoint compacta in D. Our final result is that if each boundary component of a plane domain is either a point or a Jordan curve and if the domain satisfies a boundary quasiextremal distance property, then there exists a quasiconformal self-homeomorphism of the entire plane which maps the given domain conformally onto a circle domain.