Blending World Map Projections with Flex Projector
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Mollweide Projection and the ICA Logo Cartotalk, October 21, 2011 Institute of Geoinformation and Cartography, Research Group Cartography (Draft Paper)
Mollweide Projection and the ICA Logo CartoTalk, October 21, 2011 Institute of Geoinformation and Cartography, Research Group Cartography (Draft paper) Miljenko Lapaine University of Zagreb, Faculty of Geodesy, [email protected] Abstract The paper starts with the description of Mollweide's life and work. The formula or equation in mathematics known after him as Mollweide's formula is shown, as well as its proof "without words". Then, the Mollweide map projection is defined and formulas derived in different ways to show several possibilities that lead to the same result. A generalization of Mollweide projection is derived enabling to obtain a pseudocylindrical equal-area projection having the overall shape of an ellipse with any prescribed ratio of its semiaxes. The inverse equations of Mollweide projection has been derived, as well. The most important part in research of any map projection is distortion distribution. That means that the paper continues with the formulas and images enabling us to get some filling about the liner and angular distortion of the Mollweide projection. Finally, the ICA logo is used as an example of nice application of the Mollweide projection. A small warning is put on the map painted on the ICA flag. It seams that the map is not produced according to the Mollweide projection and is different from the ICA logo map. Keywords: Mollweide, Mollweide's formula, Mollweide map projection, ICA logo 1. Introduction Pseudocylindrical map projections have in common straight parallel lines of latitude and curved meridians. Until the 19th century the only pseudocylindrical projection with important properties was the sinusoidal or Sanson-Flamsteed. -
An Efficient Technique for Creating a Continuum of Equal-Area Map Projections
Cartography and Geographic Information Science ISSN: 1523-0406 (Print) 1545-0465 (Online) Journal homepage: http://www.tandfonline.com/loi/tcag20 An efficient technique for creating a continuum of equal-area map projections Daniel “daan” Strebe To cite this article: Daniel “daan” Strebe (2017): An efficient technique for creating a continuum of equal-area map projections, Cartography and Geographic Information Science, DOI: 10.1080/15230406.2017.1405285 To link to this article: https://doi.org/10.1080/15230406.2017.1405285 View supplementary material Published online: 05 Dec 2017. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tcag20 Download by: [4.14.242.133] Date: 05 December 2017, At: 13:13 CARTOGRAPHY AND GEOGRAPHIC INFORMATION SCIENCE, 2017 https://doi.org/10.1080/15230406.2017.1405285 ARTICLE An efficient technique for creating a continuum of equal-area map projections Daniel “daan” Strebe Mapthematics LLC, Seattle, WA, USA ABSTRACT ARTICLE HISTORY Equivalence (the equal-area property of a map projection) is important to some categories of Received 4 July 2017 maps. However, unlike for conformal projections, completely general techniques have not been Accepted 11 November developed for creating new, computationally reasonable equal-area projections. The literature 2017 describes many specific equal-area projections and a few equal-area projections that are more or KEYWORDS less configurable, but flexibility is still sparse. This work develops a tractable technique for Map projection; dynamic generating a continuum of equal-area projections between two chosen equal-area projections. -
Rcosmo: R Package for Analysis of Spherical, Healpix and Cosmological Data Arxiv:1907.05648V1 [Stat.CO] 12 Jul 2019
CONTRIBUTED RESEARCH ARTICLE 1 rcosmo: R Package for Analysis of Spherical, HEALPix and Cosmological Data Daniel Fryer, Ming Li, Andriy Olenko Abstract The analysis of spatial observations on a sphere is important in areas such as geosciences, physics and embryo research, just to name a few. The purpose of the package rcosmo is to conduct efficient information processing, visualisation, manipulation and spatial statistical analysis of Cosmic Microwave Background (CMB) radiation and other spherical data. The package was developed for spherical data stored in the Hierarchical Equal Area isoLatitude Pixelation (Healpix) representation. rcosmo has more than 100 different functions. Most of them initially were developed for CMB, but also can be used for other spherical data as rcosmo contains tools for transforming spherical data in cartesian and geographic coordinates into the HEALPix representation. We give a general description of the package and illustrate some important functionalities and benchmarks. Introduction Directional statistics deals with data observed at a set of spatial directions, which are usually positioned on the surface of the unit sphere or star-shaped random particles. Spherical methods are important research tools in geospatial, biological, palaeomagnetic and astrostatistical analysis, just to name a few. The books (Fisher et al., 1987; Mardia and Jupp, 2009) provide comprehensive overviews of classical practical spherical statistical methods. Various stochastic and statistical inference modelling issues are covered in (Yadrenko, 1983; Marinucci and Peccati, 2011). The CRAN Task View Spatial shows several packages for Earth-referenced data mapping and analysis. All currently available R packages for spherical data can be classified in three broad groups. The first group provides various functions for working with geographic and spherical coordinate systems and their visualizations. -
Unusual Map Projections Tobler 1999
Unusual Map Projections Waldo Tobler Professor Emeritus Geography department University of California Santa Barbara, CA 93106-4060 http://www.geog.ucsb.edu/~tobler 1 Based on an invited presentation at the 1999 meeting of the Association of American Geographers in Hawaii. Copyright Waldo Tobler 2000 2 Subjects To Be Covered Partial List The earth’s surface Area cartograms Mercator’s projection Combined projections The earth on a globe Azimuthal enlargements Satellite tracking Special projections Mapping distances And some new ones 3 The Mapping Process Common Surfaces Used in cartography 4 The surface of the earth is two dimensional, which is why only (but also both) latitude and longitude are needed to pin down a location. Many authors refer to it as three dimensional. This is incorrect. All map projections preserve the two dimensionality of the surface. The Byte magazine cover from May 1979 shows how the graticule rides up and down over hill and dale. Yes, it is embedded in three dimensions, but the surface is a curved, closed, and bumpy, two dimensional surface. Map projections convert this to a flat two dimensional surface. 5 The Surface of the Earth Is Two-Dimensional 6 The easy way to demonstrate that Mercator’s projection cannot be obtained as a perspective transformation is to draw lines from the latitudes on the projection to their occurrence on a sphere, here represented by an adjoining circle. The rays will not intersect in a point. 7 Mercator’s Projection Is Not Perspective 8 It is sometimes asserted that one disadvantage of a globe is that one cannot see all of the entire earth at one time. -
5–21 5.5 Miscellaneous Projections GMT Supports 6 Common
GMT TECHNICAL REFERENCE & COOKBOOK 5–21 5.5 Miscellaneous Projections GMT supports 6 common projections for global presentation of data or models. These are the Hammer, Mollweide, Winkel Tripel, Robinson, Eckert VI, and Sinusoidal projections. Due to the small scale used for global maps these projections all use the spherical approximation rather than more elaborate elliptical formulae. 5.5.1 Hammer Projection (–Jh or –JH) The equal-area Hammer projection, first presented by Ernst von Hammer in 1892, is also known as Hammer-Aitoff (the Aitoff projection looks similar, but is not equal-area). The border is an ellipse, equator and central meridian are straight lines, while other parallels and meridians are complex curves. The projection is defined by selecting • The central meridian • Scale along equator in inch/degree or 1:xxxxx (–Jh), or map width (–JH) A view of the Pacific ocean using the Dateline as central meridian is accomplished by running the command pscoast -R0/360/-90/90 -JH180/5 -Bg30/g15 -Dc -A10000 -G0 -P -X0.1 -Y0.1 > hammer.ps 5.5.2 Mollweide Projection (–Jw or –JW) This pseudo-cylindrical, equal-area projection was developed by Mollweide in 1805. Parallels are unequally spaced straight lines with the meridians being equally spaced elliptical arcs. The scale is only true along latitudes 40˚ 44' north and south. The projection is used mainly for global maps showing data distributions. It is occasionally referenced under the name homalographic projection. Like the Hammer projection, outlined above, we need to specify only -
Geographical Analysis on the Projection and Distortion of IN¯O's
International Journal of Geo-Information Article Geographical Analysis on the Projection and Distortion of INO’s¯ Tokyo Map in 1817 Yuki Iwai 1,* and Yuji Murayama 2 1 Graduate School of Life and Environmental Science, University of Tsukuba, Tsukuba 305-8572, Japan 2 Faculty of Life and Environmental Science, University of Tsukuba, Tsukuba 305-8572, Japan; [email protected] * Correspondence: [email protected]; Tel.: +81-29-853-5696 Received: 5 July 2019; Accepted: 10 October 2019; Published: 12 October 2019 Abstract: The history of modern maps in Japan begins with the Japan maps (called INO’s¯ maps) prepared by Tadataka Ino¯ after he thoroughly surveyed the whole of Japan around 200 years ago. The purpose of this study was to investigate the precision degree of INO’s¯ Tokyo map by overlaying it with present maps and analyzing the map style (map projection, map scale, etc.). Specifically, we quantitatively examined the spatial distortion of INO’s¯ maps through comparisons with the present map using GIS (geographic information system), a spatial analysis tool. Furthermore, by examining various factors that caused the positional gap and distortion of features, we explored the actual situation of surveying in that age from a geographical viewpoint. As a result of the analysis, a particular spatial regularity was confirmed in the positional gaps with the present map. We found that INO’s¯ Tokyo map had considerably high precision. The causes of positional gaps from the present map were related not only to natural conditions, such as areas and land but also to social and cultural phenomena. -
A Bevy of Area Preserving Transforms for Map Projection Designers.Pdf
Cartography and Geographic Information Science ISSN: 1523-0406 (Print) 1545-0465 (Online) Journal homepage: http://www.tandfonline.com/loi/tcag20 A bevy of area-preserving transforms for map projection designers Daniel “daan” Strebe To cite this article: Daniel “daan” Strebe (2018): A bevy of area-preserving transforms for map projection designers, Cartography and Geographic Information Science, DOI: 10.1080/15230406.2018.1452632 To link to this article: https://doi.org/10.1080/15230406.2018.1452632 Published online: 05 Apr 2018. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tcag20 CARTOGRAPHY AND GEOGRAPHIC INFORMATION SCIENCE, 2018 https://doi.org/10.1080/15230406.2018.1452632 A bevy of area-preserving transforms for map projection designers Daniel “daan” Strebe Mapthematics LLC, Seattle, WA, USA ABSTRACT ARTICLE HISTORY Sometimes map projection designers need to create equal-area projections to best fill the Received 1 January 2018 projections’ purposes. However, unlike for conformal projections, few transformations have Accepted 12 March 2018 been described that can be applied to equal-area projections to develop new equal-area projec- KEYWORDS tions. Here, I survey area-preserving transformations, giving examples of their applications and Map projection; equal-area proposing an efficient way of deploying an equal-area system for raster-based Web mapping. projection; area-preserving Together, these transformations provide a toolbox for the map projection designer working in transformation; the area-preserving domain. area-preserving homotopy; Strebe 1995 projection 1. Introduction two categories: plane-to-plane transformations and “sphere-to-sphere” transformations – but in quotes It is easy to construct a new conformal projection: Find because the manifold need not be a sphere at all. -
Bibliography of Map Projections
AVAILABILITY OF BOOKS AND MAPS OF THE U.S. GEOlOGICAL SURVEY Instructions on ordering publications of the U.S. Geological Survey, along with prices of the last offerings, are given in the cur rent-year issues of the monthly catalog "New Publications of the U.S. Geological Survey." Prices of available U.S. Geological Sur vey publications released prior to the current year are listed in the most recent annual "Price and Availability List" Publications that are listed in various U.S. Geological Survey catalogs (see back inside cover) but not listed in the most recent annual "Price and Availability List" are no longer available. Prices of reports released to the open files are given in the listing "U.S. Geological Survey Open-File Reports," updated month ly, which is for sale in microfiche from the U.S. Geological Survey, Books and Open-File Reports Section, Federal Center, Box 25425, Denver, CO 80225. Reports released through the NTIS may be obtained by writing to the National Technical Information Service, U.S. Department of Commerce, Springfield, VA 22161; please include NTIS report number with inquiry. Order U.S. Geological Survey publications by mail or over the counter from the offices given below. BY MAIL OVER THE COUNTER Books Books Professional Papers, Bulletins, Water-Supply Papers, Techniques of Water-Resources Investigations, Circulars, publications of general in Books of the U.S. Geological Survey are available over the terest (such as leaflets, pamphlets, booklets), single copies of Earthquakes counter at the following Geological Survey Public Inquiries Offices, all & Volcanoes, Preliminary Determination of Epicenters, and some mis of which are authorized agents of the Superintendent of Documents: cellaneous reports, including some of the foregoing series that have gone out of print at the Superintendent of Documents, are obtainable by mail from • WASHINGTON, D.C.--Main Interior Bldg., 2600 corridor, 18th and C Sts., NW. -
Portraying Earth
A map says to you, 'Read me carefully, follow me closely, doubt me not.' It says, 'I am the Earth in the palm of your hand. Without me, you are alone and lost.’ Beryl Markham (West With the Night, 1946 ) • Map Projections • Families of Projections • Computer Cartography Students often have trouble with geographic names and terms. If you need/want to know how to pronounce something, try this link. Audio Pronunciation Guide The site doesn’t list everything but it does have the words with which you’re most likely to have trouble. • Methods for representing part of the surface of the earth on a flat surface • Systematic representations of all or part of the three-dimensional Earth’s surface in a two- dimensional model • Transform spherical surfaces into flat maps. • Affect how maps are used. The problem: Imagine a large transparent globe with drawings. You carefully cover the globe with a sheet of paper. You turn on a light bulb at the center of the globe and trace all of the things drawn on the globe onto the paper. You carefully remove the paper and flatten it on the table. How likely is it that the flattened image will be an exact copy of the globe? The different map projections are the different methods geographers have used attempting to transform an image of the spherical surface of the Earth into flat maps with as little distortion as possible. No matter which map projection method you use, it is impossible to show the curved earth on a flat surface without some distortion. -
Maps and Cartography: Map Projections a Tutorial Created by the GIS Research & Map Collection
Maps and Cartography: Map Projections A Tutorial Created by the GIS Research & Map Collection Ball State University Libraries A destination for research, learning, and friends What is a map projection? Map makers attempt to transfer the earth—a round, spherical globe—to flat paper. Map projections are the different techniques used by cartographers for presenting a round globe on a flat surface. Angles, areas, directions, shapes, and distances can become distorted when transformed from a curved surface to a plane. Different projections have been designed where the distortion in one property is minimized, while other properties become more distorted. So map projections are chosen based on the purposes of the map. Keywords •azimuthal: projections with the property that all directions (azimuths) from a central point are accurate •conformal: projections where angles and small areas’ shapes are preserved accurately •equal area: projections where area is accurate •equidistant: projections where distance from a standard point or line is preserved; true to scale in all directions •oblique: slanting, not perpendicular or straight •rhumb lines: lines shown on a map as crossing all meridians at the same angle; paths of constant bearing •tangent: touching at a single point in relation to a curve or surface •transverse: at right angles to the earth’s axis Models of Map Projections There are two models for creating different map projections: projections by presentation of a metric property and projections created from different surfaces. • Projections by presentation of a metric property would include equidistant, conformal, gnomonic, equal area, and compromise projections. These projections account for area, shape, direction, bearing, distance, and scale. -
Map Projections and Coordinate Systems Datums Tell Us the Latitudes and Longi- Vertex, Node, Or Grid Cell in a Data Set, Con- Tudes of Features on an Ellipsoid
116 GIS Fundamentals Map Projections and Coordinate Systems Datums tell us the latitudes and longi- vertex, node, or grid cell in a data set, con- tudes of features on an ellipsoid. We need to verting the vector or raster data feature by transfer these from the curved ellipsoid to a feature from geographic to Mercator coordi- flat map. A map projection is a systematic nates. rendering of locations from the curved Earth Notice that there are parameters we surface onto a flat map surface. must specify for this projection, here R, the Nearly all projections are applied via Earth’s radius, and o, the longitudinal ori- exact or iterated mathematical formulas that gin. Different values for these parameters convert between geographic latitude and give different values for the coordinates, so longitude and projected X an Y (Easting and even though we may have the same kind of Northing) coordinates. Figure 3-30 shows projection (transverse Mercator), we have one of the simpler projection equations, different versions each time we specify dif- between Mercator and geographic coordi- ferent parameters. nates, assuming a spherical Earth. These Projection equations must also be speci- equations would be applied for every point, fied in the “backward” direction, from pro- jected coordinates to geographic coordinates, if they are to be useful. The pro- jection coordinates in this backward, or “inverse,” direction are often much more complicated that the forward direction, but are specified for every commonly used pro- jection. Most projection equations are much more complicated than the transverse Mer- cator, in part because most adopt an ellipsoi- dal Earth, and because the projections are onto curved surfaces rather than a plane, but thankfully, projection equations have long been standardized, documented, and made widely available through proven programing libraries and projection calculators. -
Representations of Celestial Coordinates in FITS
A&A 395, 1077–1122 (2002) Astronomy DOI: 10.1051/0004-6361:20021327 & c ESO 2002 Astrophysics Representations of celestial coordinates in FITS M. R. Calabretta1 and E. W. Greisen2 1 Australia Telescope National Facility, PO Box 76, Epping, NSW 1710, Australia 2 National Radio Astronomy Observatory, PO Box O, Socorro, NM 87801-0387, USA Received 24 July 2002 / Accepted 9 September 2002 Abstract. In Paper I, Greisen & Calabretta (2002) describe a generalized method for assigning physical coordinates to FITS image pixels. This paper implements this method for all spherical map projections likely to be of interest in astronomy. The new methods encompass existing informal FITS spherical coordinate conventions and translations from them are described. Detailed examples of header interpretation and construction are given. Key words. methods: data analysis – techniques: image processing – astronomical data bases: miscellaneous – astrometry 1. Introduction PIXEL p COORDINATES j This paper is the second in a series which establishes conven- linear transformation: CRPIXja r j tions by which world coordinates may be associated with FITS translation, rotation, PCi_ja mij (Hanisch et al. 2001) image, random groups, and table data. skewness, scale CDELTia si Paper I (Greisen & Calabretta 2002) lays the groundwork by developing general constructs and related FITS header key- PROJECTION PLANE x words and the rules for their usage in recording coordinate in- COORDINATES ( ,y) formation. In Paper III, Greisen et al. (2002) apply these meth- spherical CTYPEia (φ0,θ0) ods to spectral coordinates. Paper IV (Calabretta et al. 2002) projection PVi_ma Table 13 extends the formalism to deal with general distortions of the co- ordinate grid.