Maps and Cartography: Map Projections a Tutorial Created by the Geospatial Resources & Map Collection
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Nicolas-Auguste Tissot: a Link Between Cartography and Quasiconformal Theory
NICOLAS-AUGUSTE TISSOT: A LINK BETWEEN CARTOGRAPHY AND QUASICONFORMAL THEORY ATHANASE PAPADOPOULOS Abstract. Nicolas-Auguste Tissot (1824{1897) published a series of papers on cartography in which he introduced a tool which became known later on, among geographers, under the name of the Tissot indicatrix. This tool was broadly used during the twentieth century in the theory and in the practical aspects of the drawing of geographical maps. The Tissot indicatrix is a graph- ical representation of a field of ellipses on a map that describes its distortion. Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane that are used in drawing geographical maps, and more generally he developed a theory for the distor- sion of mappings between general surfaces. His ideas are at the heart of the work on quasiconformal mappings that was developed several decades after him by Gr¨otzsch, Lavrentieff, Ahlfors and Teichm¨uller.Gr¨otzsch mentions the work of Tissot and he uses the terminology related to his name (in particular, Gr¨otzsch uses the Tissot indicatrix). Teichm¨ullermentions the name of Tissot in a historical section in one of his fundamental papers where he claims that quasiconformal mappings were used by geographers, but without giving any hint about the nature of Tissot's work. The name of Tissot is also missing from all the historical surveys on quasiconformal mappings. In the present paper, we report on this work of Tissot. We shall also mention some related works on cartography, on the differential geometry of surfaces, and on the theory of quasiconformal mappings. -
Mercator-15Dec2015.Pdf
THE MERCATOR PROJECTIONS THE NORMAL AND TRANSVERSE MERCATOR PROJECTIONS ON THE SPHERE AND THE ELLIPSOID WITH FULL DERIVATIONS OF ALL FORMULAE PETER OSBORNE EDINBURGH 2013 This article describes the mathematics of the normal and transverse Mercator projections on the sphere and the ellipsoid with full deriva- tions of all formulae. The Transverse Mercator projection is the basis of many maps cov- ering individual countries, such as Australia and Great Britain, as well as the set of UTM projections covering the whole world (other than the polar regions). Such maps are invariably covered by a set of grid lines. It is important to appreciate the following two facts about the Transverse Mercator projection and the grids covering it: 1. Only one grid line runs true north–south. Thus in Britain only the grid line coincident with the central meridian at 2◦W is true: all other meridians deviate from grid lines. The UTM series is a set of 60 distinct Transverse Mercator projections each covering a width of 6◦in latitude: the grid lines run true north–south only on the central meridians at 3◦E, 9◦E, 15◦E, ... 2. The scale on the maps derived from Transverse Mercator pro- jections is not uniform: it is a function of position. For ex- ample the Landranger maps of the Ordnance Survey of Great Britain have a nominal scale of 1:50000: this value is only ex- act on two slightly curved lines almost parallel to the central meridian at 2◦W and distant approximately 180km east and west of it. The scale on the central meridian is constant but it is slightly less than the nominal value. -
Map Projections
Map Projections Chapter 4 Map Projections What is map projection? Why are map projections drawn? What are the different types of projections? Which projection is most suitably used for which area? In this chapter, we will seek the answers of such essential questions. MAP PROJECTION Map projection is the method of transferring the graticule of latitude and longitude on a plane surface. It can also be defined as the transformation of spherical network of parallels and meridians on a plane surface. As you know that, the earth on which we live in is not flat. It is geoid in shape like a sphere. A globe is the best model of the earth. Due to this property of the globe, the shape and sizes of the continents and oceans are accurately shown on it. It also shows the directions and distances very accurately. The globe is divided into various segments by the lines of latitude and longitude. The horizontal lines represent the parallels of latitude and the vertical lines represent the meridians of the longitude. The network of parallels and meridians is called graticule. This network facilitates drawing of maps. Drawing of the graticule on a flat surface is called projection. But a globe has many limitations. It is expensive. It can neither be carried everywhere easily nor can a minor detail be shown on it. Besides, on the globe the meridians are semi-circles and the parallels 35 are circles. When they are transferred on a plane surface, they become intersecting straight lines or curved lines. 2021-22 Practical Work in Geography NEED FOR MAP PROJECTION The need for a map projection mainly arises to have a detailed study of a 36 region, which is not possible to do from a globe. -
How to Determine Latitude and Longitude from Topographic Maps
Oregon Department of Environmental Quality HOW TO DETERMINE LATITUDE AND LONGITUDE FROM TOPOGRAPHIC MAPS Latitude is the distance north or south of the equator. 2. For each location, construct a small rectangle around Longitude is the distance east or west of the prime the point with fine pencil lines connecting the nearest meridian (Greenwich, England). Latitude and longitude 2-1/2′ or 5′ graticules. Graticules are intersections of are measured in seconds, minutes, and degrees: latitude and longitude lines that are marked on the map edge, and appear as black crosses at four points in ″ ′ 60 (seconds) = 1 (minute) the interior of the map. 60′ (minutes) = 1° (degree) 3. Read and record the latitude and longitude for the To determine the latitude and longitude of your facility, southeast corner of the small quadrangle drawn in step you will need a topographic map from United States two. The latitude and longitude are printed at the edges Geological Survey (USGS). of the map. How to Obtain USGS Maps: 4. To determine the increment of latitude above the latitude line recorded in step 3: USGS maps used for determining latitude and longitude • Position the map so that you face its west edge; may be obtained from the USGS distribution center. These maps are available in both the 7.5 minute and l5 • Place the ruler in approximately a north-south minute series. For maps of the United States, including alignment, with the “0” on the latitude line recorded Alaska, Hawaii, American Samoa, Guam, Puerto Rico, in step 3 and the edge intersecting the point. -
Scale in GIS: What You Need to Know for GIS
Scale in GIS: What You Need To Know for GIS Y WHAT IS SCALE? CARTOGRAPHIC SCALE Scale is the scale represents the relationship of the distance on Scale in a cartographic sense (1 inch = 1 mile) is a the map/data to the actual distance on the ground. remnant of vector cartography, but still has importance • Map detail is determined by the source scale of the data: the for us in a digital world. finer the scale, the more detail. • Source scale is the scale of the data source (i.e. aerial photo Common cartographic scales: or satellite image) from which data is digitized (into boundaries, roads, landcover, etc. in a GIS). • Topographic Maps • In a GIS, zooming in on a small scale map does not o 1:24,000 7.5” quads increase its level of accuracy or detail. o 1:63,360 15” quads • Rule of thumb: Match the appropriate scale to the level of o 1:100,000 quads detail required in the project. o 1:250,000 quads 1:500 1:1200 • World Maps o 1:2,000,000 – DCW • Shoreline o 1:20,000 o 1:70,000 Above, right. Comparing fine and coarse source scales shows how the • Aerial Photography level of detail in the data is o 1:40,000 NHAP/NAPP determined by the spatial scale of the o 1:12,000 – 4,000 custom aerial photography source data. SPATIAL SCALE GUIDELINES X Spatial scale involves grain & extent: 1. Pay attention to source scale of your spatial data. Grain: the size of your pixel & the smallest resolvable unit. -
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. -
Space Oblique Mercator Projection Mathematical Development
Space Oblique Mercator Projection Mathematical Development GEOLOGICAL SURVEY BULLETIN 1518 '• I : ' Space Oblique Mercator Projection Mathematical Development By JOHN P. SNYDER GEOLOGICAL SURVEY BULLETIN 1 5 1 8 Refined and improved equations that can be applied to Landsat and other near-polar-orbiting satellite parameters to describe a continuous true-to-scale groundtrack and that will contribute to the automated production of image-base maps UNITED STATES DEPARTMENT OF THE INTERIOR JAMES G. WATT, Secretary GEOLOGICAL SURVEY Doyle G. Frederick, Acting DirectM Library of Congress catalog-card No. 81-607029 For sale by Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 FOREWORD On July 25, 1972, the National Aeronautics and Space Adminis tration launched an Earth-sensing satellite named ERTS-1. This sat ellite, which was renamed Landsat-1 early in 1975, circled the Earth in a near-polar orbit and scanned continuous 185-km-wide swaths of the surface. Landsat-1 was followed by Landsats-2 and -3 and thus established a pattern by which the Earth is to be viewed from space. Conventional map projections are based on a static Earth, but Land sat imagery involves the parameter of time and the need for a dynamic map projection for its cartographic portrayal. During 1973, U.S. Geological Survey personnel conceived a new dy namic map projection which would, in fact, display Landsat imagery with a minimum of scale distortion. Moreover, the map projection was continuous so that the 185-km swath would involve no zone boundaries regardless of its length. This was accomplished by defining the ground track of the satellite as the centerline of the map projection. -
MAP PROJECTION DESIGN Alan Vonderohe (January 2020) Background
MAP PROJECTION DESIGN Alan Vonderohe (January 2020) Background For millennia cartographers, geodesists, and surveyors have sought and developed means for representing Earth’s surface on two-dimensional planes. These means are referred to as “map projections”. With modern BIM, CAD, and GIS technologies headed toward three- and four-dimensional representations, the need for two-dimensional depictions might seem to be diminishing. However, map projections are now used as horizontal rectangular coordinate reference systems for innumerable global, continental, national, regional, and local applications of human endeavor. With upcoming (2022) reference frame and coordinate system changes, interest in design of map projections (especially, low-distortion projections (LDPs)) has seen a reemergence. Earth’s surface being irregular and far from mathematically continuous, the first challenge has been to find an appropriate smooth surface to represent it. For hundreds of years, the best smooth surface was assumed to be a sphere. As the science of geodesy began to emerge and measurement technology advanced, it became clear that Earth is flattened at the poles and was, therefore, better represented by an oblate spheroid or ellipsoid of revolution about its minor axis. Since development of NAD 83, and into the foreseeable future, the ellipsoid used in the United States is referred to as “GRS 80”, with semi-major axis a = 6378137 m (exactly) and semi-minor axis b = 6356752.314140347 m (derived). Specifying a reference ellipsoid does not solve the problem of mathematically representing things on a two-dimensional plane. This is because an ellipsoid is not a “developable” surface. That is, no part of an ellipsoid can be laid flat without tearing or warping it. -
Latitude-Longitude and Topographic Maps Reading Supplement
Latitude-Longitude and Topographic Maps Reading Supplement Latitude and Longitude A key geographical question throughout the human experience has been, "Where am I?" In classical Greece and China, attempts were made to create logical grid systems of the world to answer this question. The ancient Greek geographer Ptolemy created a grid system and listed the coordinates for places throughout the known world in his book Geography. But it wasn't until the middle ages that the latitude and longitude system was developed and implemented. This system is written in degrees, using the symbol °. Latitude When looking at a map, latitude lines run horizontally. Latitude lines are also known as parallels since they are parallel and are an equal distant from each other. Each degree of latitude is approximately 69 miles (111 km) apart; there is a variation due to the fact that the earth is not a perfect sphere but an oblate ellipsoid (slightly egg-shaped). To remember latitude, imagine them as the horizontal rungs of a ladder ("ladder-tude"). Degrees latitude are numbered from 0° to 90° north and south. Zero degrees is the equator, the imaginary line which divides our planet into the northern and southern hemispheres. 90° north is the North Pole and 90° south is the South Pole. Longitude The vertical longitude lines are also known as meridians. They converge at the poles and are widest at the equator (about 69 miles or 111 km apart). Zero degrees longitude is located at Greenwich, England (0°). The degrees continue 180° east and 180° west where they meet and form the International Date Line in the Pacific Ocean. -
Geoide Ssii-109
GEOIDE SSII-109 MAP DISTORTIONS MAP DISTORTION • No map projection maintains correct scale throughout • The intersection of any two lines on the Earth is represented on the map at the same or different angle • Map projections aim to either minimize scale, angular or area distortion TISSOT'S THEOREM At every point there are two orthogonal principal directions which are perpendicular on both the Earth (u) and the map (u') Figure (A) displays a point on the Earth and (B) a point on the projection TISSOT'S INDICATRIX • An infinitely small circle on the Earth will project as an infinitely small ellipse on any map projection (except conformal projections where it is a circle) • Major and minor axis of ellipse are directly related to the scale distortion and the maximum angular deformation SCALE DISTORTION • The ratio of an infinitesimal linear distance in any direction at any point on the projection and the corresponding infinitesimal linear distance on the Earth • h – scale distortion along the meridians • k – scale distortion along the parallels MAXIMUM ANGULAR DEFORMATION AND AREAL SCALE FACTOR • Maximum angular deformation, w is the biggest deviation from principle directions define on the Earth to the principle directions define on the map • Occurs in each of the four quadrants define by the principle directions of Tissot's indicatrix • Areal scale factor is the exaggeration of an infinitesimally small area at a particular point NOTES • Three tangent points are taken for a number of different azimuthal projection: (φ, λ)= (0°, 0°), (φ, -
Metric Guide for Federal Construction (4-93)
METRIC GUIDE FOR FEDERAL CONSTRUCTION First Edition The Construction Subcommittee of the Metrication Operating Committee of the Interagency on Metric Policy Published by the NATIONAL INSTITUTE OF BUILDING SCIENCES 1201 L Street N.W. Washington, D.C. 20005 Call 202-289-7800 for ordering information. Copyright © 1991, 1992, 1993, National Institute of Building Sciences. First printing, December 1991 Second printing, March 1992 Third printing, August 1992 Fourth printing, April 1993 METRIC GUIDE FOR FEDERAL CONSTRUCTION First Edition The Construction Subcommittee of the Metrication Operating Committee of the Interagency Council on Metric Policy ACKNOWLEDGEMENTS The Construction Subcommittee of the Interagency Council on Metric Policy's Metrication Operating Committee has prepared this guide to aid the federal agencies in implementing the metric system of measurement in the federal construction process. I would like to express my appreciation to Arnold Prima of the Office of the Secretary of Defense, who first voiced a need for the guide and initiated its development; to William Brenner of the National Institute of Building Scienc- es, who wrote it; to Claret Heider, its editor; and to reviewers William Aird of the State Department, Valerie Antoine and Louis Sokol of the U.S. Metric Association, Bruce Barrow of the Defense Information Systems Agency, Maria Grazi Bruschi of the American Society of Civil Engineers, Ronald Clevenger of the Tennessee Valley Authority, Amitabha Datta of the General Services Admin- istration, Troy Estes of the National Aeronautics and Space Administration, Luther Flouton of the Public Health Service, James Gross of the National Institute of Standards and Technology, Leslie Hegyi, Stan Jakuba of S.I.