Conformality, the Exponential Function, and World Map Projections

ABSTRACTBoth the media and the geosciences often use small-scale world maps for demonstrating global phenomena. The most important demands on the projection of these maps are: (1) the map distortions have to be reduced as much as possible; (2) the outline shape of the mapped Earth must remind the viewer of the Globe. If the map theme to be illustrated requires neither equivalency (nor, which rarely happens, conformality) nor prescriptions for the map graticule, an aphylactic non-conical projection with simultaneously minimized angular and area distortions is advisable. In this paper, a graticule transformation by a parameterizable function helps to convert minimum distortion pointed-polar pseudocylindrical projections for world maps into general non-conical projections with further minimized distortions. The maximum curvature of the outline shape will be moderated at the same time in order to obtain a definitely pointed-polar character.

In my talk I will attempt to draw some connections between complex analysis and computational geometry, particularly between conformal mappings, hyperbolic geometry, the medial axis and optimal meshing. The Riemann mapping theorem says that there is a conformal (angle preserving) map of the unit disk, DISK, to the interior Ω of any simple n-gon. How much work is needed to compute this map? [MR2671015] We can compute the conformal map f: DISK -> Ω to within error e in time O(n ⋅ log 1/e log log 1/e).The proof uses an iteration that converges quadratically to f, but it needs a good initial guess that is close to the correct answer. The medial axis allows us to quickly construct a map Ω -> DISK that is close to Riemann's map with precise estimates. This gives our starting point, and the time estimates in the theorem depend on time needed to compute the medial axis (linear by work of Chin, Snoeyink and Wang).Conversely, the proof of Theorem 1 uses ideas from analysis that may be of interest in CG. For example, the dome of a planar domain Ω is the surface in R+3 given by the upper envelope of all hemispheres whose base on R2 is a medial axis disk of Ω. Domes arise in the study of 3-manifolds, and a fundamental result of Dennis Sullivan about convex sets in hyperbolic 3-space lies at the heart of Theorem 1. The proof of Theorem 1 also introduces the thick-thin decomposition of a polygon, inspired by the thick-thin decomposition of a hyperbolic manifold. The thin parts correspond to long, narrow pieces of the polygon (but not in the obvious way; the precise definition uses a conformal invariant called extremal length). This decomposition plays in important role in proving: [MR2671014] Any simple n-gon has a O(n) quadrilateral mesh with all angles ≤ 120O and all new angles ≥ 60O.The sharp upper bound is due to Bern and Eppstein; the lower bound is the novel part here. The result is also true for PSLGs with at most n edges and vertices, although simple examples show the O(n) must be replaced by O(n2): [Bishop-Quad] Every PSLG has an O(n2) quadrilateral mesh with all angles ≤ 120O and all new angles ≥ 60O. Only O(n/e) angles satisfy |θ-90O| > e, for any e >0.

For the conformal map in two dimensions, the local rescaling factor of metric amounts to that of surface element. To avoid the restriction on the functional form of the metric in d dimensions (d > 2), we generalize the conformal map in such a way that the volume element (d form), not the metric (quadratic form), is invariant up to local rescaling. For d = 4, quaternions are available to formulate such a generalized conformal map, as complex analysis is useful for the conventional conformal map.

Cartographers have developed various techniques for deriving new projections from existing projections. The goal of these techniques is to substitute a disadvantageous trait of one of the source projections with the second source projection. This chapter discusses creating new projections by the juxtaposition and blending of two existing projections. It also presents a new approach for selectively combining projection characteristics. The emphasis in this chapter is on projections for world maps, as the described techniques are most useful for this scale.

A one-parameter linear group in a vector space L is a continuous map \( g:t\mapsto {g^t}=g(t) \) on the space of parameters ℝ, taking values in a space of the linear operators on L and having the following properties: (1) g 0 = E and (2) the sum of parameter values corresponds to the composition \( {g^{s+t }}={g^s}\circ {g^t} \). Thus, one-parameter linear groups in ℝ are the continuous solutions of the functional equations g(0) = 1, g(t + s) = g(t)g(s). As is well known, exponential functions satisfy these equations. After solving the first problems in this chapter, readers will find that the continuous solutions are all exponential and, thus, will extract the usual properties of exponential functions from these functional equations (which corresponds to a real historical development).The second part of the chapter is devoted to proper generalizations for one-parameter matrix groups. Readers will find that all of them are given by the matrix exponential series, as in the one-dimensional case. However, a multidimensional picture looks much more complicated. Working on these problems will familiarize readers with matrix analysis. In addition, readers may need familiarity with some tools used by number theory and the theory of differential equations, such as Euclid’s algorithm, Chinese remainder theorem, and Poincaré’s recurrence theorem; we introduce and discuss these tools.In the third part of the chapter readers will find some elementary classic applications of one-parameter matrix groups in differential equation theory (Liouville formula), in complex analysis (e.g., complex exponential functions, Euler’s formula), for finite-dimensional functional spaces (spaces of quasipolynomials), and others. Readers will see how to deal with these problems using elementary tools of analysis and linear algebra.Further, part of this chapter contains problems for readers possessing more advanced preliminary experience (although we introduce and discuss the necessary tools). While working on these problems, readers will come across many interesting things regarding analysis and linear algebra and become acquainted with important concepts (symplectic forms and others).Readers will encounter far-reaching advances and applications of the subjects considered in the present chapters in powerful mathematical theories: of differential equations, Lie groups, and group representations (a guide to the literature is provided).KeywordsProblem GroupNumerical SeriesGreat Common DivisorExponential SeriesLess Common MultipleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Numerial Methods for Volterra Integral Equations with Singular Kernels

Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely by using one well-known tool, Brownian motion, and a new construction, the Schramm-Loewner evolution (SLE). This book is an introduction to the conformally invariant processes that appear as scaling limits. The following topics are covered: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; the Schramm-Loewner evolution (SLE), which is a Loewner chain with a Brownian motion input; and applications to intersection exponents for Brownian motion. The prerequisites are first-year graduate courses in real analysis, complex analysis, and probability. The book is suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

A complex variable is a quantity that contains both a real part and an imaginary part. Complex analysis is an application of mathematics that analyzes the features of functions of complex variables. Physics, engineering, and computer science are just a few of the scientific disciplines that benefit from complex analysis. In order to tackle issues that are challenging or impossible to resolve using only real variables, complex analysis is crucial. This paper introduces an important theorem in complex analysis, which is Cauchy’s residue theorem. A relative general expression for a complex integral along a simple closed contour is provided by Cauchy's residue theorem. Cauchy's residue theorem can be used to select an acceptable closed contour for the calculation of some unusual definite integrals that may be highly complex and challenging to solve using traditional methods. The residue at the function's isolated singularities is then established. In the formula that is subtracted in Cauchy's residue theorem, the values of the residues are extracted. The integral along a simple closed contour can then be represented in two pieces, in which one along the real axis and the other along the circle.

A number of results concerning infinite series are presented in this chapter. The material spans convergence tests for series of constants, uniform convergence for series of functions, and power series. It is not the intention here to give a comprehensive account of this theory, but rather to review material that is relevant to later chapters. This chapter is intended to make the rest of the book more self contained and remind the reader of some basic results. It is assumed that the reader has a background in real analysis of a single variable and elementary complex analysis; consequently, no attempt is made to prove these results or provide examples. Proofs and examples can be found in textbooks on real analysis such as Apostol (Mathematical Analysis, Addison-Wesley, 2nd ed., 1974), Fulks (Advanced Calculus, 3rd ed., Wiley, 1978), and Little et al. (Real Analysis via Sequences and Series, Springer-Verlag, 2015) and complex analysis (Ahlfors, Complex Analysis, McGraw-Hill, 2nd ed., 1966; Brown and Churchill, Complex Variables and Applications, McGraw-Hill, 7th ed., 2004; Priestley, Introduction to Complex Analysis, Rev. ed., Oxford University Press, 1990). The exception is in the final section on double sequences and series. This material is often not part of courses on real or complex analysis. Here we provide proofs, examples and exercises.

Vertical downward flow of water through the tension-saturated capillary fringe above a water table is perturbed by the presence of cavities that may exclude or allow water entry over different parts of their surface, with some critical shape that just excludes water entry over their whole surface. The analysis by Philip and his colleagues of the seepage exclusion problem was for a soil with an exponential hydraulic conductivity function that does not show a tension-saturated capillary fringe and was for an infinite flow field. The analysis given here is for the critical condition of water exclusion from two-dimensional tunnel cavities located above a water table in the saturated capillary fringe with a horizontal upper boundary where the soil-water pressure is the air-entry value for the soil. Conformal mapping, used to solve Laplace's equation with these boundary conditions, gives the critical shape of the tunnel cavity and also the flux density distribution through the upper boundary of given height. The shape of the cavity near its top can be approximated by a parabola that over-estimates the width of the cavity at depth below the apex. This is in contrast to the exact parabolic shape obtained for soils with an exponential hydraulic conductivity. Upper and lower bounds for a uniform flux density that raises the top of the saturated region in the vicinity above the cavity are obtained for the calculated critical shape.

After the year 1960, mathematical analysis for electromagnetic scattering, diffraction and guided field with complicated boundaries have been developed, based on field analysis methods shown in the previous period around years of 1950. J. A. Kong, K. Hongo and we are belonging to such investigation generation. Millimeter wave and optical technologies have rapidly grown after the coherent millimeter wave devices and lasers have been investigated. Here, some typical studies are shown. Novel mathematical method of integral equation method with conformal mapping is discussed for asymptotic diffraction of electromagnetic field by complicated boundaries, such as an elliptic cylinder perturbed by the circular cylinder. Mode conversions of millimeter wave fields in the waveguide junction between over size circular waveguide and conical horn are investigated by the asymptotic diffraction theory. Field characteristics of the Fabry-Perot resonator with finite circular curved disks are analyzed by the vector integral equations. General scattering and transmission characteristics of beam waves in curved boundaries with wedges are analyzed by spectral function method.

The Numerical Solution of Integral Equations on the Half-Line

ABSTRACTThis paper sheds light on the question of whether globes or maps are best suited to providing an adequate understanding of global phenomena. Drawing upon evidence from the history of cartography, we show how the ancient, medieval and early sixteenth-century mapmakers had already recognized the importance of globes and world maps. An analysis of Ptolemy, Strabo and the negotiations around the Treaty of Tordesillas will demonstrate how our basic assumptions regarding the relationships of globes and world maps have been present for a long time. These assumptions are discussed in the light of current user studies on world maps, for example on distance and area estimations and the issue of peripheral continuity. As we can see from this review of empirical research, designing an appropriate world map is not only an issue of map projection; the edge of a world map also affects the knowledge users may generate. In terms of the importance of ‘globes vs. world maps’, these findings indicate that globes are prominent because of their undistorted, but also seamless visualization of global phenomena.

A Semi-Grapidcal construction for the conical orthomorphic projection with one standard parallel was described in the Empire Survey Review in 1954 by Pye and Camm. As with most discussions of the orthomorphic projections, it was based on the consideration that the scale at any point is the same in all directions, and I was interested to compare the method with a conjecture I had put forward concerning Thomas Hariot's attempt in about 1595 to compute the meridional parts for the Mercator projection. Hariot's attempt was made before the discovery of logarithms or the integral calculus, and any graphical or semi-graphical construction for an orthomorphic projection must also get round the difficulty of constructing or calculating the exponential functions involved. My conjecture, was that he had used an equiangular spiral to relate the Mercator projection to the steteographic and I was pleased to find that the spiral also lends itself to a straightforward construction for the conical orthomorphic with either one or two standard parallels, and, at the same time, provides a further insight into the geometrical properties of this interesting and important group of projections. Its application can be extended to computations either on the sphere or on the spheroid.