Conic Projection Research Articles

Copying-lineages of portolan chart metrics and implications of their pre-medieval origin

ABSTRACT Seven portolan charts and atlases (including the anonymous Rex Tholomeus chart) made between 1311 and 1538 were cartometrically analysed. The findings indicate that their geometry was not improved chronologically and that the accuracy of Pietro Vesconte's charts (1311–13) was never surpassed. Portolan charts and atlases appear to be mosaics composed of subsections, each with variable metrics and accuracy levels that are, on average, twice as great in comparison to their Mediterranean and Black Sea areas (as a whole), and it is highly unlikely that they are genuine late medieval products, which is also supported by historical evidence. Their anticlockwise tilts fit well with the tilt of the Gibraltar-Antioch line on modern maps that closely approximate Ptolemy I and Ptolemy II projections centred near Gibraltar and with their meridian tilts for areas around Genoa. If late medieval Italian cartographers had acquired a collection of maps or charts from before the Middle Ages, it is theoretically feasible that they could have achieved the typical geometry of portolan charts by graphically combining their manuscript copies of large-scale regional maps in Mercator or Mercator-like projection, using a small-scale map in conic or pseudoconic projection (on which the convergence of meridians is displayed) as a template.

Conformal mapping at elevation with implications for low-distortion projections

This article develops the theory of the conformal mapping of the surface of constant height above the ellipsoid (ChS), which has implications for the design of low-distortion projections. This paper gives a development of the Mercator, polar stereographic, Lambert conformal conic (LCC) and transverse Mercator (TM) projections based on the ChS, a better Earth model than the reference ellipsoid (RE) for positions at height. This article provides new formulas for the LCC and TM inverse scale problems – when given a value for the local scale function, where does it occur? These formulas apply to both traditional mapping on the RE and novel mapping on the ChS.

Connection of Conic and Cylindrical Map Projections

In previous papers that have dealt with cylindrical map projections as limiting cases of conical projections, standard or equidistant parallels were used in the derivations. This paper shows that this is not necessary and that it is sufficient to use parallels that preserve length. In addition, unlike other approaches, in this article the limiting cases of conic projections are derived in the most natural way, by deriving the equations of cylindrical projections from the equations of conic projections in a rectangular system in the projection plane using a mathematical concept of limits. It is shown that such an approach is possible, but not always, so it should be used carefully, or even better, avoided in teaching and studying map projections.

Cylindrical Projections as a Limiting Case of Conic Projections

Lambert (1772) derived the equation of the Mercator projection as a limiting case of a conformal conic projection. In this paper, we give a derivation for equidistant, equal-area, conformal and perspective cylindrical projections as limiting cases of equidistant, equal-area, conformal and perspective conic projections. In this article the conic and cylindrical projections are not projections on a cone or a cylinder whose surfaces are cut and developed into a plane, but rather mappings of the sphere directly into the plane. Exceptions are projections that are defined as mappings on the surface of a cone or plane, as is the case with perspective projections. In the end, we prove that it is not always possible to obtain a corresponding cylindrical projection as a limiting case from a conic projection, as one might conclude at first glance. Therefore, the final conclusion is that it is not advisable to interpret cylindrical projections as limiting cases of conic projections.

From Conic to Cylindrical Map Projections

In books and textbooks on map projections, cylindrical, conic and azimuthal projections are usually considered separately. It is sometimes mentioned that cylindrical and azimuthal projections can be interpreted as limiting cases of conic, but this is rarely proven. The goal of this article is to show in a rigorous and systematic way how to generally approach solving the problem of transition from a conic to a corresponding cylindrical projection. This article points to the fact that J. H. Lambert showed as early as 1772 that a conformal cylindrical projection is created from a conformal conic projection. Following his idea, this paper shows that not only conformal, but also equal-area and equidistant cylindrical projections can be derived from corresponding conic map projections. Although it seems that the paper deals with quite well known and intuitive property of conic projections, it will also show that the transition from the conic to the corresponding cylindrical projection is not always possible.

Scalable Distributed Optimization Combining Conic Projection and Linear Programming for Energy Community Scheduling

Scalable Distributed Optimization Combining Conic Projection and Linear Programming for Energy Community Scheduling

Map Projections Classification

Many books, textbooks and papers have been published in which the classification of map projections is based on auxiliary (developable) surfaces and projections are divided into conic, cylindrical and azimuthal projections. We argue that such a classification of map projections is unacceptable and give many reasons for that. Many authors wrote in more detail about the classification of map projections, and our intention is to give a new refined and rectified insight into the classification of map projections. Our approach can be included in map projection publications of general and thematic cartography. Doing this, misconceptions and unnecessary insistence on conceptuality instead of reality will be avoided.

Conic Projections with Three or More Standard Parallels

Abstract. The basic property of all map projections is the distribution of inevitable distortions. Conic projections with one or two standard parallels are mentioned in the literature. These are parallels with the property that the distortion of length, area and angles equals zero at each of their points. It turns out that there are conic projections with no standard parallels, as well as those with more than two standard parallels. Such projections exist not only in theory, but examples of such projections can also be constructed.

Proposed single-zone map projection system for Turkey

AbstractThe coordinate base of the maps or sheets produced is the Universal Transversal Mercator (UTM) conformal projection, and it is not possible to work in a single coordinate system in Turkey. Therefore, a transition from UTM to other conformal projections is required. For the countries extending in an east–west UTM zone width like Turkey, composite projection (CP), a double standard paralleling Lambert Conformal Conic (LCC) and double map projections (DP) are used widely. However, this process causes increase in working load and processing errors by users. This study aims to determine a common projection system that can be used in the whole country. In this context, a composite projection from UTM and LCC projection has been defined for the first time. According to the results obtained, map projection CP with the least distortion values in both east–west and north–south directions has been chosen. With the CP selection, a single coordinate system has been determined for medium- and large-scale maps. Projection correction formulas, scale factor and false origin have been determined for map coordinates in CP. These distortions are obtained with a difference of less than 1 cm for 1 km long sides and less than 0.003″ for the azimuth value of this side, when the correction formulas are used.

Lower Pitch Voice Produce by Effort Sound in Previously Uvulectomized Children

The palatine uvula is conic projection composing of soft connective tissue which has a number of racemose glands. It hangs from the posterior edge of the middle of the soft palate of the roof of the mouth. It has circular muscle fibres at its base and helps in closing the nasopharynx during deglutition. Copious secretion from the uvula helps moisten the larynx and vocal cord to maintain quality speech. In the Hausas’ culture in Northern Nigeria, the uvula is highly demonized and blamed for a lot of health problems and risk of death and so all infants must ‘proactively’ undergo uvulectomy on the 7th day after birth, with few exceptions. This study aims at evaluating the effect of uvulectomy on pitch quality of effort sounds of young children between 2 to 11 years of age. Eighty subjects were divided into two groups of 40 uvulectomized and 40 non-uvulectomized children which were further subdivided into various age ranges of equal mix of male and female. Audio frequency analyzer software is used to conduct the test and the result shows a reduction in frequency of the sounds produced by uvulectomized subjects, p<0.05. Uvulectomy does affect the quality of sound produced in children giving a significant lower pitch quality. Keywords: Uvula, Uvulectomy, Effort Sound, Frequency, Pitch DOI: 10.7176/JHMN/90-01 Publication date: June 30 th 2021

Investigating the effect of cartographic properties on updating cadastral maps

Cadastral maps are the main documents of ownership and plots of land, as it contribute to preserving the property rights of individuals and institutions. It indicates the size and shape of each parcel and reveals geographic relationships that affect property value. The Iraqi cadastral maps are in old coordinate system AL-nahrwan 1934 and lambert conformal conic projection. Therefore these maps are old and unfit for use. The main objective of this paper is to investigate the effect of cartographic properties on updating cadastral maps. This depends on studying the effect of conversion the projection and the datum of the cadastral maps of the study area from (datum: nahrwan34, projection: lambert conformal conic) to the (datum WGS84, projection: UTM). The results indicated that the distortions are very small in small areas and distance, but it increases with increasing areas and distances. While, there is no distortion in directions. The Affine (2D) and Molodensky (3D) transformations were used in datum transformation. The total root mean square error (TRMSE) of affine transformation was ±0.895m, while it was±0.651m for Molodensky results. Therefore, the Molodensky method was used to transform the datum for all cadastral maps.

Analysis of hyperboloid cooling tower projection on 2D shape

This paper undertakes the problem of mapping a hyperboloid cooling tower on a single plane. Measurements performed by ground-based laser scanning technology quickly deliver substantial amounts of geometric data of the tower’s outer wall. The essence of the article is projection of the rotational hyperboloid on a plane. The shape of the hyperboloid cooling tower is not directly expandable to a single plane. Mapping a hyperboloid shape on a plane is, therefore, associated with distortions. This paper presents a comparison between cylindrical and conic projection of a hyperboloid cooling tower. The most popular method of mapping hyperboloid is cylindrical projection. The cylinder’s side surface is easily developed on the drawing sheet. For the hyperboloid cooling tower, the biggest distortions occur in the latitudinal direction and reach the highest values at the top and bottom edges. The equation (13) describe distortion for the cylindrical projection. The equation (18) describe distortion for the conical projection. This paper presents results obtained from the performed measurement. The analysis found that cone mapping produces less distortion than cylindrical projection for the hyperboloid cooling tower. We think, that in conical projection, the shape of a hyperboloid cooling tower and theoretical conic shape have better corresponding together than in cylindrical projection.

Cartography Through the Years – Personal Views About a Young Science

Abstract. Although cartographic products have been produced for thousands of years, cartography as a science has only been established in the early 20th century. Great works of cartography include, for instance, the conic map projections by Ptolemy, the Tabula Rogeriana by Idrisi, the Waldseemüller map, and the Mercator map. Numerous cartographers, predominantly mathematicians, have shaped the theory of map projections throughout the centuries.With the advent of geographic information systems (GIS) in the 1960s and the rapid developments of digital technologies, cartography found itself in the middle of an identity crisis. For some time, it was not clear whether cartography would become obsolete and be replaced by GIS mapping technologies or whether GIS is a novel manifestation of cartography. During this period various misconceptions about the role of maps and mapping as well as uncertainty about the future developments of mapping in general added to this confusion.This contribution elaborates the major characteristics of cartography versus other disciplines, in particular geography and GIS, and takes a look at possible future directions and developments with regard to the theory of cartography as well as novel and future display technologies. Personal observations of the author during his professional life since the early 1980s illustrate these developments.

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

A Lambert Conformal Conic (LCC) projection with two true-scale parallels of latitudes ϕl and ϕu can be recast in a LCC projection with one standard parallel of latitude ϕ0 and scale k0 , 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 ϕ0 and k0 in terms ϕl and ϕu 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.

Ten ensembles with hourly high spatial resolution datasets for severe precipitation events in Egypt.

In this article, we created a new dataset comprising 10 ensembles for severe rainfall simulations at a high spatial resolution (10 km grid spacing, 120 × 120 grid points) in Egypt using the Weather Research and Forecast model Version 3.8 (WRF 3.8). The vertical grid had over 41 levels, extending from the surface to 10 hPa. The defined domain, a Lambert conformal conic projection, started from 24°E to 36°E. The ensembles were generated using 10 different microphysics schemes within the WRF 3.8. The severe rainfall event occurred between October 26 and 29, 2016. Final analysis data from National Center for Environmental Predictions were used for the initial and boundary conditions every 6 h at a spatial resolution of 1° × 1°. The geographical static input data, such as land use, albedo, and terrain height, were interpolated and prepared using a geogrid program in the WRF preprocessing system. This dataset is the first of its kind. It is addressing a need for this type of high resolution data over Egypt using physically- based numerical weather prediction models.

Optimalna konformna projekcija za vseevropsko kartiranje

There is an increasing need for pan-European spatial datasets, mainly to support the common European Union policies. This has inevitably raised demands for adopting pan-European cartographic projections to visualize the spatial data. The Map Projection Workshop organized by EuroGeographics in 2001 provided a recommendation to the European Commission to adopt the Lambert conformal conic projection for conformal pan-European mapping at the scales smaller or equal to 1:500,000. This paper discusses if the projection is an optimal solution in terms of linear deformations over the mapping region. An optimized CAMPREL (conformal adaptive mapping projection of rotation ellipsoid) projection for the area of interest is proposed as an alternative solution. The projection quality criteria were calculated and compared with those of the Lambert conformal conic projection. The maximal possible absolute linear distortion for conformal mapping of the pan-European area is also given. It has been shown that the CAMPREL projection designed for pan-European mapping better meets the projection selection criteria.

ОСОБЛИВОСТІ КАРТ НАТО

The article describes basic data about the features of creating and operating of NATO maps. The article provides informationabout scale standards for NATO topographic maps. The structure of topographic maps, the order of their creation, mainpurposes, tasks, requirements according to NATO standards are considered. Topographic maps at scales of 1:25 000, 1:50 000and 1: 100 000 are created by NATO countries in accordance with national requirements while maintaining their traditionaltransition to the creation of topographic maps, but adhering to a single NATO standard for mandatory mapping of WGS -84 andUTM grids, the printing of explanation symbols and abbreviations in English and application of geographical names in Latin. Atpresent, there is a coherent NATO geopolitics for the creation of topographic special maps (including digital maps), the basicprinciple of which is that each NATO member is responsible for providing the necessary cartographic materials to its troops andNATO forces on its territory and to the globe. for planning and conducting military operations. A 1: 250,000 scale map is used tostudy and evaluate in detail individual, relatively small but important areas, when crossing water obstacles, during hostilities inlarge settlements, as well as when designing and constructing large engineering structures. Projections of topographic maps ofscale 1: 250 000 are considered, specifics of delineation and designations adopted for the map, features of the content of th etopographic map according to NATO standards. The maps are created in the Universal Transversal Mercator Projection (UTM),the Universal Polar Stereographic Projection (UPS) and the Lambert Conformal Conic Projection. The article presents a system ofgraphic symbols and symbols of NATO.

Bifurcations from a center at infinity in 3D piecewise linear systems with two zones

We consider continuous piecewise linear systems in R3 with two zones under the assumption of having a linear center in the invariant manifold of the point at infinity. A specific conic projection is introduced, so that it is possible to analyze in a convenient way the dynamics near such a center at infinity in two qualitative different situations.In the semi-homogeneous case, such a center is associated to the existence of a continuum of invariant semi-cones sharing the vertex at the origin; perturbing the configuration it is possible to detect the bifurcation of a limit cycle at infinity leading to the bifurcation of isolated invariant semi-cones.For the non-homogeneous case, the cycles at infinity does not imply invariant semi-cones. However, the non-generic case when the center at infinity is associated to the existence of invariant cylinders for one of the involved vector fields, becomes rather interesting. It is possible then, by perturbing the other vector field, to get the bifurcation of a big limit cycle from infinity without destroying the center.

Proyección Cónica Equivalente de Albers: propiedades y aplicación para México

La cartografía nacional de México emplea de manera oficial tres proyecciones: la Universal Transversa de Mercator, utilizada en las cartas topográficas para escalas medianas y grandes; la Cónica Conforme de Lambert, adaptada a las condiciones del país para su representación en escalas pequeñas; y la Normal de Mercator, empleada con fines de navegación marítima. El uso por largo tiempo de estas proyecciones ha sido asimilado por la comunidad científica y se emplean de forma regular; no obstante, la investigación cartográfica relacionada con esta temática se ha detenido y no se ha buscado la aplicación de otras proyecciones cartográficas en el afán de alcanzar una mejor representación de los diversos temas de investigación, lo que impacta también en la precisión de los datos obtenidos. El presente trabajo tiene por objetivo mostrar el empleo de la proyección Cónica Equivalente de Albers con las adaptaciones de parámetros necesarios; en este sentido, se presentan las formulaciones que posibilitan la proyección de las coordenadas, el análisis de las deformaciones y la concepción una cartografía nacional a escala 1:500,000; subsecuentemente, el trabajo también pretende estimular la reflexión en torno a las posibilidades que tiene esta proyección que no preserva el ortomorfismo, considerado hasta ahora como única propiedad en la cartografía nacional mexicana.

Секущие параллели в азимутальных проекциях

Map projections are commonly approached as mapping onto developable surfaces; cylindrical projections onto the lateral surface of a cylinder, conic projections onto the lateral surface of a cone, and azimuthal projections onto a plane. If an intermediate developable surface intersects the Earth’s sphere or ellipsoid, the projection is referred to as a secant projection. The intersection of a developable surface and the Earth’s sphere or ellipsoid, e.g. secant parallel is considered a standard parallel. In this paper the definitions of secant and standard parallel in azimuthal projections are given. The first conclusion is that the secant and standard parallels are two distinct notions. The second one is that a standard parallel, if such a parallel exists in an azimuthal projection, is a secant parallel, while the converse statement is not true in general. Furthermore, it is shown that there are azimuthal projections with only one secant parallel that is not standard, with only one standard parallel which is also secant one, with two different secant parallels, and with one standard and one secant parallel.