DOCSLIB.ORG

A Bevy of Area Preserving Transforms for Map Projection Designers.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. an existing conformal projection and apply any complex Some of the techniques combine both kinds of trans- analytic function to it. Voilà, new conformal projection! formations or use the plane as an intermediary. In practice, finding the right function for the purpose In the case of plane-to-plane transformations, the pre- usually takes some work, but nevertheless, the toolbox is sumption is that the manifold has been projected to the rich and powerful. Not so for equal-area projections. A plane already in a way that preserves areas. Hence, the few techniques have been exploited over the centuries, consideration is purely planar, and regardless of what even fewer explicitly described, and in general the domain manifold was designated as the original surface before remains lightly explored. No area-preserving analog to projecting, the only question is whether the plane-to- complex analysis is known to exist. plane transformation itself preserves area measure. This work doesn’t solve such big problems. Instead, it collects together in one place a body of techniques map projection designers can use to efficiently generate new equal area map projections. Some are obvious; 1.1. Terminology some are simple but clever; some are a little more Terminology is not well standardized for equal-area pro- involved. Their inventions range from centuries ago jections. More in line with the field of differential geometry to being introduced in this paper. I give examples than with mathematical cartography, the title of this paper with particular attention to a novel replacement uses the term area-preserving transformation so as to be scheme for the Web Mercator. readily understandable to the widest audience. However, I Since the advent of digital computers, several itera- do not use that term in the body of this text. To avoid the tive methods for minimizing distortion across a cho- awkwardness of noun forms of equal-area,Iwillsome- sen region have appeared in the literature. Iterative times use equivalence.Thetermauthalic also appears in systems are specifically outside the purview of this the literature, but is redundant to my needs here and will work. They are adequately described by their authors not be used. in any case. I invite the interested reader to peruse, for example, Dyer and Snyder (1989) or Canters (2002, pp. 115–244). 1.2. Symbols For simplicity, I discuss the manifold to be projected as a sphere, but the techniques I describe generally do ● λ refers to a geodetic longitude; not depend on this specificity. The techniques fall into ● φ refers to a geodetic latitude; CONTACT Daniel “daan” Strebe [email protected] © 2018 The Author. 2 D. STREBE ● θ refers to a counterclockwise angle from positive of the map, and can adjust the proportions and distortion x axis on the cartesian plane; characteristics of the map to better suit the purpose. A ● ρ refers to a distance from origin on the cartesian common example is based on the cylindric equal-area plane; presented by Lambert (Tobler, 1972), with primitive ● 0 added to a variable name denotes a variable formulae transformed from the variable of the same name x ¼ λ that lacks the prime symbol. L ¼ : (2:4) cea y ¼ sinðÞφ In its original form, the projection has no distortion along the equator, and “correct” scale there. By scaling the 2. Transformations φ φ height by sec 0 and the width by cos 0,achosenlatitude φ 2.1. Scaling 0 canbemadetohavenodistortionandtorepresent nominal scale. No less than seven variants of Lambert’s As noted by Strebe (2016), the equal-area property for original that use this technique have been formally a mapping from sphere to plane can be defined either described independently by nine others, including: strictly or loosely: strictly in that @ @ @ @ ● Gall (1885)(φ ¼ 45 , “Gall orthographic,” now y x À y x ¼ 2 φ; : 0 @φ @λ @λ @φ R cos (2 1) usually “Gall–Peters,” presentedpffiffiffiffiffiffiffiffi in 1855) ● φ ¼ =π “ ’ Smyth (1870)( 0 arccos 2 , Smyth s equal- (with R being the radius of the generating globe) or surface”) loosely in that ● φ ¼ Behrmann (1910)( 0 30 ) @y @x @y @x ● Craster (1929) (identical to Smyth equal-surface) À ¼ s cos φ; (2:2) ● φ ¼ @φ @λ @λ @φ Balthasart (1935)( 0 50 ) ● φ ¼ 0 “ ” Edwards (1953)( 0 37 24 , Trystan Edwards ), with s being any non-zero, finite, real value. The ratio- ● Peters (1983) (identical to Gall orthographic, first nale for the strict case is that the projection maps public mention in 1967) pffiffiffiffiffiffiffiffi regions on the globe to the regions on the plane such ● φ ¼ =π Tobler and Chen (1986)(0 arccos 1 , that the area measure of any mapped region remains “Tobler’s world in a square”) the same as the area measure of the unmapped region. ● φ ¼ 0 “ – ” Abramms (2006)( 0 37 30 , Hobo Dyer, com- The rationale for the looser case is that relative areas missioned in 2002) are preserved throughout the map, regardless of how their area measure might be scaled with respect to the Each of these projections can be described as an affine generating globe. Simple, isotropic scaling is an area- transformation on Lambert’s cylindric equal-area: preserving transformation by the looser condition. φ On its own, scaling is not terribly interesting. cos 0 0 Á : : φ Lcea (2 5) Combined with other techniques, however, it becomes 0 sec 0 powerful, as we shall see in subsequent sections. Other examples of this technique can be found in pseudocylindric projections when the simplest form of the generating formulae does not yield the desired 2.2. Affine transformation latitude to be free of distortion along the central mer- Also noted by Strebe (2016), affine transformation idian. Boggs (1929), for example, stretches x to slightly preserves areas. That is, given x and y as the planar more than twice the primitive formulae’s results, and coordinates of an equal-area map projection, compresses y by the reciprocal. Affine transformation relates to the scaling constant ab x x 0 ¼ 0 (2:3) s reported in Equation (2.2). Letting cd y y 0 0 ab x ; y is also equal-area, provided the matrix is not singular A ¼ (2:6) and a; b; c; d are constant. When b ¼ 0; c ¼ 0; a ¼ d,the cd affine transformation degenerates to the simple, isotropic such that A is not singular and a; b; c; d are constant scaling described in Section 2.1.Withb ¼ 0; c ¼ 0; aÞd, when applying A to the projection, then the affine transformation describes a scaling such that the x ¼ A; : and y directions scale differently. This can be useful, parti- s det (2 7) cularly when d ¼ 1=a. This last case preserves overall area with det being the determinant. CARTOGRAPHY AND GEOGRAPHIC INFORMATION SCIENCE 3 I found no instance of affine transformation with Let us say we have a detailed base map prepared for shear components exploited in cartographic maps before use in a service deployed on the World Wide Web. We Strebe (2017). As will be seen in Section 2.9,general wish to accept arbitrary data sets to represent and affine transformations can serve useful purposes in con- overlay onto the base map in order to augment the junction with other transformations. However, perhaps map with information customized for a user’s needs. the value of affine transformation on its own has been The bulk of the rendering work and detail needed for underappreciated, as discussed next. the complete map has already gone into the base map, Consider an arbitrary, non-conformal projection P. and we do not wish to render the base map anew for As described by Tissot (1881), the distortion undergone every custom map because we want to conserve com- by a geographical point p when projected by P can be putational resources and to reduce delivery time. described as the projection of an infinitesimal circle The scenario turns out to be common: Practically around p into an infinitesimal ellipse on the plane.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Map Projections Paper 4 (Th.) UNIT : I ; TOPIC : 3 …Introduction
    FOR SEMESTER 3 GE Students , Geography Map Projections Paper 4 (Th.) UNIT : I ; TOPIC : 3 …Introduction Prepared and Compiled By Dr. Rajashree Dasgupta Assistant Professor Dept. of Geography Government Girls’ General Degree College 3/23/2020 1 Map Projections … The method by which we transform the earth’s spheroid (real world) to a flat surface (abstraction), either on paper or digitally Define the spatial relationship between locations on earth and their relative locations on a flat map Think about projecting a see- through globe onto a wall Dept. of Geography, GGGDC, 3/23/2020 Kolkata 2 Spatial Reference = Datum + Projection + Coordinate system Two basic locational systems: geometric or Cartesian (x, y, z) and geographic or gravitational (f, l, z) Mean sea level surface or geoid is approximated by an ellipsoid to define an earth datum which gives (f, l) and distance above geoid gives (z) 3/23/2020 Dept. of Geography, GGGDC, Kolkata 3 3/23/2020 Dept. of Geography, GGGDC, Kolkata 4 Classifications of Map Projections Criteria Parameter Classes/ Subclasses Extrinsic Datum Direct / Double/ Spherical Triple Surface Spheroidal Plane or Ist Order 2nd Order 3rd Order surface of I. Planar a. Tangent i. Normal projection II. Conical b. Secant ii. Transverse III. Cylindric c. Polysuperficial iii. Oblique al Method of Perspective Semi-perspective Non- Convention Projection perspective al Intrinsic Properties Azimuthal Equidistant Othomorphic Homologra phic Appearance Both parallels and meridians straight of parallels Parallels straight, meridians curve and Parallels curves, meridians straight meridians Both parallels and meridians curves Parallels concentric circles , meridians radiating st. lines Parallels concentric circles, meridians curves Geometric Rectangular Circular Elliptical Parabolic Shape 3/23/2020 Dept.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • A Pipeline for Producing a Scientifically Accurate Visualisation of the Milky
    A pipeline for producing a scientifically accurate visualisation of the Milky Way M´arciaBarros1;2, Francisco M. Couto2, H´elderSavietto3, Carlos Barata1, Miguel Domingos1, Alberto Krone-Martins1, and Andr´eMoitinho1 1 CENTRA, Faculdade de Ci^encias,Universidade de Lisboa, Portugal 2 LaSIGE, Faculdade de Ci^encias,Universidade de Lisboa, Portugal 3 Fork Research, R. Cruzado Osberno, 1, 9 Esq, Lisboa, Portugal [email protected] Abstract. This contribution presents the design and implementation of a pipeline for producing scientifically accurate visualisations of Big Data. As a case study, we address the data archive produced by the European Space Agency's (ESA) Gaia mission. The satellite was launched in Dec 19 2013 and is repeatedly monitoring almost two billion (2000 million) astronomical sources during five years. By the end of the mission, Gaia will have produced a data archive of almost a Petabyte. Producing visually intelligible representations of such large quantities of data meets several challenges. Situations such as cluttering, overplot- ting and overabundance of features must be dealt with. This requires careful choices of colourmaps, transparencies, glyph sizes and shapes, overlays, data aggregation and feature selection or extraction. In addi- tion, best practices in information visualisation are sought. These include simplicity, avoidance of superfluous elements and the goal of producing esthetical pleasing representations. Another type of challenge comes from the large physical size of the archive, which makes it unworkable (or even impossible) to transfer and process the data in the user's hardware. This requires systems that are deployed at the archive infrastructure and non-brute force approaches. Because the Gaia first data release will only happen in September 2016, we present the results of visualising two related data sets: the Gaia Uni- verse Model Snapshot (Gums), which is a simulation of Gaia-like data; the Initial Gaia Source List (IGSL) which provides initial positions of stars for the satellite's data processing algorithms.
    [Show full text]
  • 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.
    [Show full text]
  • View Preprint
    1 Remaining gaps in open source software 2 for Big Spatial Data 1 3 Lu´ısM. de Sousa 1ISRIC - World Soil Information, 4 Droevendaalsesteeg 3, Building 101 6708 PB Wageningen, The Netherlands 5 Corresponding author: 1 6 Lu´ıs M. de Sousa 7 Email address: [email protected] 8 ABSTRACT 9 The volume and coverage of spatial data has increased dramatically in recent years, with Earth observa- 10 tion programmes producing dozens of GB of data on a daily basis. The term Big Spatial Data is now 11 applied to data sets that impose real challenges to researchers and practitioners alike. As rule, these 12 data are provided in highly irregular geodesic grids, defined along equal intervals of latitude and longitude, 13 a vastly inefficient and burdensome topology. Compounding the problem, users of such data end up 14 taking geodesic coordinates in these grids as a Cartesian system, implicitly applying Marinus of Tyre’s 15 projection. 16 A first approach towards the compactness of global geo-spatial data is to work in a Cartesian system 17 produced by an equal-area projection. There are a good number to choose from, but those supported 18 by common GIS software invariably relate to the sinusoidal or pseudo-cylindrical families, that impose 19 important distortions of shape and distance. The land masses of Antarctica, Alaska, Canada, Greenland 20 and Russia are particularly distorted with such projections. A more effective approach is to store and 21 work with data in modern cartographic projections, in particular those defined with the Platonic and 22 Archimedean solids.
    [Show full text]
  • 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.
    [Show full text]
  • Estimation of an Unknown Cartographic Projection and Its Parameters from the Map
    Estimation of an Unknown Cartographic Projection and its Parameters from the Map Faculty of Sciences, Charles University in Prague , Tel.: ++420-221-951-400 Abstract This article presents a new off-line method for the detection, analysis and estimation of an unknown carto- graphic projection and its parameters from a map. Several invariants are used to construct the objective function φ that describes the relationship between the 0D, 1D, and 2D entities on the analyzed and reference maps. It is min- imized using the Nelder-Mead downhill simplex algorithm. A simplified and computationally cheaper version of the objective function φ involving only 0D elements is also presented. The following parameters are estimated: a map projection type, a map projection aspect given by the meta pole K coordinates [ϕk,λk], a true parallel latitude ϕ0, central meridian longitude λ0, a map scale, and a map rotation. Before the analysis, incorrectly drawn elements on the map can be detected and removed using the IRLS. Also introduced is a new method for computing the L2 distance between the turning functions Θ1, Θ2 of the corresponding faces using dynamic programming. Our ap- proach may be used to improve early map georeferencing; it can also be utilized in studies of national cartographic heritage or land use applications. The results are presented both for real cartographic data, representing early maps from the David Rumsay Map Collection, and for the synthetic tests. Keywords: digital cartography, map projection, analysis, simplex method, optimization, Voronoi diagram, outliers detection, early maps, georeferencing, cartographic heritage, meta data, Marc 21, MapAnalyst. 1 Introduction the proposed solution, this step can be performed semi- automatically and with a higher degree of relevance us- ing our method.
    [Show full text]
  • 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.
    [Show full text]
  • Mapping the Drowned World by Tracey Clement
    Mapping The Drowned World by Tracey Clement A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy Sydney College of The Arts The University of Sydney 2017 Statement This volume is presented as a record of the work undertaken for the degree of Doctor of Philosophy at Sydney College of the Arts, University of Sydney. Table of Contents Acknowledgments iii List of images iv Abstract x Introduction: In the beginning, The End 1 Chapter One: The ABCs of The Drowned World 6 Chapter Two: Mapping The Drowned World 64 Chapter Three: It's all about Eve 97 Chapter Four: Soon it would be too hot 119 Chapter Five: The ruined city 154 Conclusion: At the end, new beginnings 205 Bibliography 210 Appendix A: Timeline for The Drowned World 226 Appendix B: Mapping The Drowned World artist interviews 227 Appendix C: Experiment and process documentation 263 Work presented for examination 310 ii Acknowledgments I wish to acknowledge the support of my partner Peter Burgess. He provided installation expertise, help in wrangling images down to size, hard-copies of multiple drafts, many hot dinners, and much, much more. Thanks also to my supervisor Professor Bradford Buckley for his steady guidance. In addition, the technical staff at SCA deserve a mention. Thanks also to all my friends, particularly those who let themselves be coerced into assisting with installation: you know who you are. And last, but certainly not least, thank you so much to the librarians at all of the University of Sydney libraries, and especially at SCA.
    [Show full text]