Fibonacci lattices for the evaluation and optimization of map projections

[EN] Latitude-longitude grids are frequently used in geosciences for global numerical modelling although they are remarkably inhomogeneous due to meridian convergence. In contrast, Fibonacci lattices are highly isotropic and homogeneous so that the area represented by each lattice point is virtually the same. In the present paper we show the higher performance of Fibonacci versus latitude-longitude lattices for evaluating distortion coefficients of map projections. In particular, we obtain first a typical distortion for the Lambert Conformal Conic projection with their currently defined parameters and geographic boundaries for Europe that has been adopted as standard by the INSPIRE directive. Further, we optimize the defining parameters of this projection, lower and upper standard parallel latitudes, so that the typical distortion for Europe is reduced a 10% when they are set to 36 degrees and 61.5 degrees, respectively. We also apply the optimization procedure to the determination of the best standard parallels for using this projection in Spain, whose values remained unspecified by the National decree that commanded its official adoption, and obtain optimum values of 37 degrees and 42 degrees and a resulting typical distortion of 828 ppm.Baselga Moreno, S. (2018). Fibonacci lattices for the evaluation and optimization of map projections. Computers & Geosciences. 117:1-8. https://doi.org/10.1016/j.cageo.2018.04.012S1811

TestGrids: Evaluating and Optimizing Map Projections

[EN] In the study of map projections, it is relatively simple to obtain meaningful estimators of distortion for a small area. The definition and especially the evaluation of global distortion measures (i.e., estimators representing the distortion worldwide or in a continent-like area) are undoubtedly more troublesome. Therefore, it is relatively common to find that recommendations for the parameters to use in a particular map projection, be it devised for a continent or a country, are based on simple rules (like the one-sixth rule of thumb for conic projections), with no possibility of further improvement in terms of resulting distortions and sometimes even with no knowledge at all of the sizes of these distortions. Although the choice of map defining parameters is normally made for reasons other than distortion minimization, such as ease of use (e.g., integer or half-integer numbers may be preferable), preservation of conventional or traditional definitions, and uniformity of parameters between neighboring regions, it is always worthwhile to know the optimal set of parameters in terms of minimal distortion. Then, the cartographer may mindfully deviate from this optimal set, documenting the differences in defining parameters and in the resulting distortions. The present research provides a means to do this by extending a related work presented in a previous contribution, where the evaluation and optimization of distortions were studied for a single map projection and only two areas of interest. To this end, a new tool has been developed and presented in this paper. This tool allows users to evaluate several measures of distortion for the most common conformal and equal-area projections within user-defined geographic boundaries of interest. Also embedded in the tool and transparent to users are global optimization techniques operating on Fibonacci grids, which permit the optimization of parameters for the particular map projection and area of interest under two possible criteria: minimization of typical distortion or minimization of extreme distortions. This tool and the associated techniques are applied to several official projections to analyze their original performance and to propose new parameters that significantly improve the resulting distortions while leaving room for users to easily evaluate and optimize the tool for the lowest distortions of these projections within their regions of interest.Baselga Moreno, S. (2019). TestGrids: Evaluating and Optimizing Map Projections. Journal of Surveying Engineering. 145(3):1-8. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000279S18145

Optimising 2-parameter Lambert Conformal Conic projections for ground-to-grid distortions

[EN] A Lambert Conformal Conic (LCC) projection with two true-scale parallels of latitudes phi(l)and phi(u)can be recast in a LCC projection with one standard parallel of latitude phi(0)and scalek(0), having the practical advantage that the same type of definition can be used for the two conformal projections universally used: LCC and Transverse Mercator (TM). While equations giving phi(0)andk(0)in terms phi(l)and phi(u)can be found in the literature, inverse relationships are not readily found. They are derived in the present paper. These may be necessary in views of the planned future definition of the United States State Plane Coordinate System (SPCS) 2022 for the users of particular mapping software requiring to specify the two latitude values instead of the central latitude and central scale. While map projection parameters are customary selected to minimise ellipsoid-to-grid distortions for a region, in some cases it could be more convenient to study and minimise ground-to-grid distortions. Also bearing in mind the design of SPCS 2022, we discuss the advantages and disadvantages of working with each type of distortion definition.Baselga Moreno, S. (2021). Optimising 2-parameter Lambert Conformal Conic projections for ground-to-grid distortions. Survey Review. 53(380):415-421. https://doi.org/10.1080/00396265.2020.17973394154215338

A Square Equal-area Map Projection

A novel square equal-area map projection is proposed. The projection combines closed-form forward and inverse solutions with relatively low angular distortion and minimal cusps, a combination of properties not manifested by any previously published square equal-area projection. Thus, the new projection has lower angular distortion than any previously published square equal-area projection with a closed-form solution. Utilizing a quincuncial arrangement, the new projection places the north pole at the center of the square and divides the south pole between its four corners; the projection can be seamlessly tiled. The existence of closed-form solutions makes the projection suitable for real-time visualization applications, both in cartography and in other areas, such as for the display of panoramic images.Comment: 15 pages, 5 figures, 1 tabl

Anyonic entanglement renormalization

We introduce a family of variational ansatz states for chains of anyons which optimally exploits the structure of the anyonic Hilbert space. This ansatz is the natural analog of the multi-scale entanglement renormalization ansatz for spin chains. In particular, it has the same interpretation as a coarse-graining procedure and is expected to accurately describe critical systems with algebraically decaying correlations. We numerically investigate the validity of this ansatz using the anyonic golden chain and its relatives as a testbed. This demonstrates the power of entanglement renormalization in a setting with non-abelian exchange statistics, extending previous work on qudits, bosons and fermions in two dimensions.Comment: 19 pages, 10 figures, v2: extended, updated to match published versio

Design and implementation of a modular interface to integrate CLP and tabled execution

Logic programming (LP) is a family of high-level programming languages which provides high expressive power. With LP, the programmer writes the properties of the result and / or executable specifications instead of detailed computation steps. Logic programming systems which feature tabled execution and constraint logic programming have been shown to increase the declarativeness and efficiency of Prolog, while at the same time making it possible to write very expressive programs. Tabled execution avoids infinite failure in some cases, while improving efficiency in programs which repeat computations. CLP reduces the search tree and brings the power of solving (in)equations over arbitrary domains. Similarly to the LP case, CLP systems can also benefit from the power of tabling. Previous implementations which take ful advantage of the ideas behind tabling (e.g., forcing suspension, answer subsumption, etc. wherever it is necessary to avoid recomputation and terminate whenever possible) did not offer a simple, well-documented, easy-to-understand interface. This would be necessary to make the integratation of arbitrary CLP solvers into existing tabling systems possible. This clearly hinders a more widespread usage of the combination of both facilities. In this thesis we examine the requirements that a constraint solver must fulfill in order to be interfaced with a tabling system. We propose and implement a framework, which we have called Mod TCLP, with a minimal set of operations (e.g., entailment checking and projection) which the constraint solver has to provide to the tabling engine. We validate the design of Mod TCLP by a series of use cases: we re-engineer a previously existing tabled constrain domain (difference constraints) which was connected in an ad-hoc manner with the tabling engine in Ciao Prolog; we integrateHolzbauer’s CLP(Q) implementationwith Ciao Prolog’s tabling engine; and we implement a constraint solver over (finite) lattices. We evaluate its performance with several benchmarks that implement a simple abstract interpreter whose fixpoint is reached by means of tabled execution, and whose domain operations are handled by the constraint over (finite) lattices, where TCLP avoids recomputing subsumed abstractions.---ABSTRACT---La programación lógica con restricciones (CLP) y la tabulación son extensiones de la programación lógica que incrementan la declaratividad y eficiencia de Prolog, al mismo tiempo que hacen posible escribir programasmás expresivos. Las implementaciones anteriores que integran completamente ambas extensiones, incluyendo la suspensión de la ejecución de objetivos siempre que sea necesario, la implementación de inclusión (subsumption) de respuestas, etc., en todos los puntos en los que sea necesario para evitar recomputaciones y garantizar la terminación cuando sea posible, no han proporcionan una interfaz simple, bien documentada y fácil de entender. Esta interfaz es necesaria para permitir integrar resolutores de CLP arbitrarios en el sistema de tabulación. Esto claramente dificulta un uso más generalizado de la integración de ambas extensiones. En esta tesis examinamos los requisitos que un resolutor de restricciones debe cumplir para ser integrado con un sistema de tabulación. Proponemos un esquema (y su implementación), que hemos llamadoMod TCLP, que requiere un reducido conjunto de operaciones (en particular, y entre otras, entailment y proyección de almacenes de restricciones) que el resolutor de restricciones debe ofrecer al sistema de tabulación. Hemos validado el diseño de Mod TCLP con una serie de casos de uso: la refactorización de un sistema de restricciones (difference constraints) previamente conectado de un modo ad-hoc con la tabulación de Ciao Prolog; la integración del sistema de restricciones CLP(Q) de Holzbauer; y la implementación de un resolutor de restricciones sobre retículos finitos. Hemos evaluado su rendimiento con varios programas de prueba, incluyendo la implementación de un intérprete abstracto que alcanza su punto fijo mediante el sistema de tabulación y en el que las operaciones en el dominio son realizadas por el resolutor de restricciones sobre retículos (finitos) donde TCLP evita la recomputación de valores abstractos de las variables ya contenidos en llamadas anteriores

Automated map projection selection for GIS

The selection of an appropriate map projection has a fundamental impact on the visualization and analysis of geographic information. Distortion is inevitable and the decision requires simultaneous consideration of several different factors; a process which can be confusing for many cartographers and GIS users. The last few decades have seen numerous attempts to create automated map projection selection solutions based on traditional classification and selection guidelines, but there are no existing tools directly accessible to users of GIS software when making projection selection decisions. This paper outlines key elements of projection selection and distortion theory, critically reviews the previous solutions, and introduces a new tool developed for ESRI’s ArcGIS, employing an original selection method tailored to the specific purpose and geographical footprint characteristics of a GIS project. The tool incorporates novel quantitative projection distortion measures which are currently unavailable within existing GIS packages. Parameters are optimized for certain projections to further reduce distortions. A set of candidate projected coordinate systems are generated that can be applied to the GIS project; enabling a qualitative visual assessment to facilitate the final user selection. The proposed tool provides a straightforward application which improves understanding of the projection selection process and assists users in making more effective use of GIS