Latitude-Longitude and Topographic Maps Reading Supplement
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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. -
QUICK REFERENCE GUIDE Latitude, Longitude and Associated Metadata
QUICK REFERENCE GUIDE Latitude, Longitude and Associated Metadata The Property Profile Form (PPF) requests the property name, address, city, state and zip. From these address fields, ACRES interfaces with Google Maps and extracts the latitude and longitude (lat/long) for the property location. ACRES sets the remaining property geographic information to default values. The data (known collectively as “metadata”) are required by EPA Data Standards. Should an ACRES user need to be update the metadata, the Edit Fields link on the PPF provides the ability to change the information. Before the metadata were populated by ACRES, the data were entered manually. There may still be the need to do so, for example some properties do not have a specific street address (e.g. a rural property located on a state highway) or an ACRES user may have an exact lat/long that is to be used. This Quick Reference Guide covers how to find latitude and longitude, define the metadata, fill out the associated fields in a Property Work Package, and convert latitude and longitude to decimal degree format. This explains how the metadata were determined prior to September 2011 (when the Google Maps interface was added to ACRES). Definitions Below are definitions of the six data elements for latitude and longitude data that are collected in a Property Work Package. The definitions below are based on text from the EPA Data Standard. Latitude: Is the measure of the angular distance on a meridian north or south of the equator. Latitudinal lines run horizontal around the earth in parallel concentric lines from the equator to each of the poles. -
AIM: Latitude and Longitude
AIM: Latitude and Longitude Latitude lines run east/west but they measure north or south of the equator (0°) splitting the earth into the Northern Hemisphere and Southern Hemisphere. Latitude North Pole 90 80 Lines of 70 60 latitude are 50 numbered 40 30 from 0° at 20 Lines of [ 10 the equator latitude are 10 to 90° N.L. 20 numbered 30 at the North from 0° at 40 Pole. 50 the equator ] 60 to 90° S.L. 70 80 at the 90 South Pole. South Pole Latitude The North Pole is at 90° N 40° N is the 40° The equator is at 0° line of latitude north of the latitude. It is neither equator. north nor south. It is at the center 40° S is the 40° between line of latitude north and The South Pole is at 90° S south of the south. equator. Longitude Lines of longitude begin at the Prime Meridian. 60° W is the 60° E is the 60° line of 60° line of longitude west longitude of the Prime east of the W E Prime Meridian. Meridian. The Prime Meridian is located at 0°. It is neither east or west 180° N Longitude West Longitude West East Longitude North Pole W E PRIME MERIDIAN S Lines of longitude are numbered east from the Prime Meridian to the 180° line and west from the Prime Meridian to the 180° line. Prime Meridian The Prime Meridian (0°) and the 180° line split the earth into the Western Hemisphere and Eastern Hemisphere. Prime Meridian Western Eastern Hemisphere Hemisphere Places located east of the Prime Meridian have an east longitude (E) address. -
Latitude/Longitude Data Standard
LATITUDE/LONGITUDE DATA STANDARD Standard No.: EX000017.2 January 6, 2006 Approved on January 6, 2006 by the Exchange Network Leadership Council for use on the Environmental Information Exchange Network Approved on January 6, 2006 by the Chief Information Officer of the U. S. Environmental Protection Agency for use within U.S. EPA This consensus standard was developed in collaboration by State, Tribal, and U. S. EPA representatives under the guidance of the Exchange Network Leadership Council and its predecessor organization, the Environmental Data Standards Council. Latitude/Longitude Data Standard Std No.:EX000017.2 Foreword The Environmental Data Standards Council (EDSC) identifies, prioritizes, and pursues the creation of data standards for those areas where information exchange standards will provide the most value in achieving environmental results. The Council involves Tribes and Tribal Nations, state and federal agencies in the development of the standards and then provides the draft materials for general review. Business groups, non- governmental organizations, and other interested parties may then provide input and comment for Council consideration and standard finalization. Standards are available at http://www.epa.gov/datastandards. 1.0 INTRODUCTION The Latitude/Longitude Data Standard is a set of data elements that can be used for recording horizontal and vertical coordinates and associated metadata that define a point on the earth. The latitude/longitude data standard establishes the requirements for documenting latitude and longitude coordinates and related method, accuracy, and description data for all places used in data exchange transaction. Places include facilities, sites, monitoring stations, observation points, and other regulated or tracked features. 1.1 Scope The purpose of the standard is to provide a common set of data elements to specify a point by latitude/longitude. -
Why Do We Use Latitude and Longitude? What Is the Equator?
Where in the World? This lesson teaches the concepts of latitude and longitude with relation to the globe. Grades: 4, 5, 6 Disciplines: Geography, Math Before starting the activity, make sure each student has access to a globe or a world map that contains latitude and longitude lines. Why Do We Use Latitude and Longitude? The Earth is divided into degrees of longitude and latitude which helps us measure location and time using a single standard. When used together, longitude and latitude define a specific location through geographical coordinates. These coordinates are what the Global Position System or GPS uses to provide an accurate locational relay. Longitude and latitude lines measure the distance from the Earth's Equator or central axis - running east to west - and the Prime Meridian in Greenwich, England - running north to south. What Is the Equator? The Equator is an imaginary line that runs around the center of the Earth from east to west. It is perpindicular to the Prime Meridan, the 0 degree line running from north to south that passes through Greenwich, England. There are equal distances from the Equator to the north pole, and also from the Equator to the south pole. The line uniformly divides the northern and southern hemispheres of the planet. Because of how the sun is situated above the Equator - it is primarily overhead - locations close to the Equator generally have high temperatures year round. In addition, they experience close to 12 hours of sunlight a day. Then, during the Autumn and Spring Equinoxes the sun is exactly overhead which results in 12-hour days and 12-hour nights. -
General Rules for Contour Lines
6/30/2015 EASC111LabEFor Printing LAB E: Topographic Maps (for printing) MAP READING PRELAB: Any type of map is a twodimensional (flat) representation of Earth’s surface. Road maps, surveying maps, topographic maps, geologic maps can all cover the same territory but highlight different features of the area. Consider the following images from the same area in Illinois. Make a list of the types of features that are shown on each type of map. Topographic Maps The light brown lines on the topographic map are called contour lines. A contour line connects points of equal height above sea level, called elevation. For example, a 600’ (six hundred foot) contour line on a map means that every point on that contour line is 600’ above sea level. In order to understand contour lines better, imagine a box with a “mountain” in it with a clear plastic lid on top of the box. Assume the base of the mountain is at sea level. The box is slowly being filled with water. Since water automatically levels itself off, it will touch the “mountain” at the same height all the way around. If we were to peer down into the box from above, we could draw a line on the lid that marks where the water touches the “mountain”. This would be a contour line for that elevation. As the box continues to fill with water, we would draw contour lines at specific intervals, such as 1”. Each time the water level rises one inch, we will draw a line marking where the water touches the land. -
Maps and Charts
Name:______________________________________ Maps and Charts Lab He had bought a large map representing the sea, without the least vestige of land And the crew were much pleased when they found it to be, a map they could all understand - Lewis Carroll, The Hunting of the Snark Map Projections: All maps and charts produce some degree of distortion when transferring the Earth's spherical surface to a flat piece of paper or computer screen. The ways that we deal with this distortion give us various types of map projections. Depending on the type of projection used, there may be distortion of distance, direction, shape and/or area. One type of projection may distort distances but correctly maintain directions, whereas another type may distort shape but maintain correct area. The type of information we need from a map determines which type of projection we might use. Below are two common projections among the many that exist. Can you tell what sort of distortion occurs with each projection? 1 Map Locations The latitude-longitude system is the standard system that we use to locate places on the Earth’s surface. The system uses a grid of intersecting east-west (latitude) and north-south (longitude) lines. Any point on Earth can be identified by the intersection of a line of latitude and a line of longitude. Lines of latitude: • also called “parallels” • equator = 0° latitude • increase N and S of the equator • range 0° to 90°N or 90°S Lines of longitude: • also called “meridians” • Prime Meridian = 0° longitude • increase E and W of the P.M. -
Spherical Coordinate Systems
Spherical Coordinate Systems Exploring Space Through Math Pre-Calculus let's examine the Earth in 3-dimensional space. The Earth is a large spherical object. In order to find a location on the surface, The Global Pos~ioning System grid is used. The Earth is conventionally broken up into 4 parts called hemispheres. The North and South hemispheres are separated by the equator. The East and West hemispheres are separated by the Prime Meridian. The Geographic Coordinate System grid utilizes a series of horizontal and vertical lines. The horizontal lines are called latitude lines. The equator is the center line of latitude. Each line is measured in degrees to the North or South of the equator. Since there are 360 degrees in a circle, each hemisphere is 180 degrees. The vertical lines are called longitude lines. The Prime Meridian is the center line of longitude. Each hemisphere either East or West from the center line is 180 degrees. These lines form a grid or mapping system for the surface of the Earth, This is how latitude and longitude lines are represented on a flat map called a Mercator Projection. Lat~ude , l ong~ude , and elevalion allows us to uniquely identify a location on Earth but, how do we identify the pos~ion of another point or object above Earth's surface relative to that I? NASA uses a spherical Coordinate system called the Topodetic coordinate system. Consider the position of the space shuttle . The first variable used for position is called the azimuth. Azimuth is the horizontal angle Az of the location on the Earth, measured clockwise from a - line pointing due north. -
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. -
Latitude and Longitude
Latitude and Longitude Finding your location throughout the world! What is Latitude? • Latitude is defined as a measurement of distance in degrees north and south of the equator • The word latitude is derived from the Latin word, “latus”, meaning “wide.” What is Latitude • There are 90 degrees of latitude from the equator to each of the poles, north and south. • Latitude lines are parallel, that is they are the same distance apart • These lines are sometimes refered to as parallels. The Equator • The equator is the longest of all lines of latitude • It divides the earth in half and is measured as 0° (Zero degrees). North and South Latitudes • Positions on latitude lines above the equator are called “north” and are in the northern hemisphere. • Positions on latitude lines below the equator are called “south” and are in the southern hemisphere. Let’s take a quiz Pull out your white boards Lines of latitude are ______________Parallel to the equator There are __________90 degrees of latitude north and south of the equator. The equator is ___________0 degrees. Another name for latitude lines is ______________.Parallels The equator divides the earth into ___________2 equal parts. Great Job!!! Lets Continue! What is Longitude? • Longitude is defined as measurement of distance in degrees east or west of the prime meridian. • The word longitude is derived from the Latin word, “longus”, meaning “length.” What is Longitude? • The Prime Meridian, as do all other lines of longitude, pass through the north and south pole. • They make the earth look like a peeled orange. The Prime Meridian • The Prime meridian divides the earth in half too. -
Conventional Terrestrial Reference Frame
CHAPTER 3 CONVENTIONAL TERRESTRIAL REFERENCE FRAME Definition The Terrestrial Reference System adopted for either the analysis of individual data sets by techniques (VLBI, SLR, LLR, GPS...) or the combination of individual Solutions into a unified set of data (Station coordinates, Earth orientation parameters, etc..) follows these criteria (Boucher, 1991): a) It is geocentric, the center of mass being defined for the whole Earth, including oceans and atmosphere. b) Its scale is that of a local Earth frame, in the meaning of a relativistic theory of gravitation. c) Its orientation is given by the BIH orientation at 1984.0. d) Its time evolution in orientation will create no residual global rotation with regards to the crust. Realization When one wants to realize such a conventional terrestrial reference system through a reference frame i.e. a network of Station coordinates, it will be specified by Cartesian equatorial coordinates X, Y, and Z, by preference. If geographical coordi nates are needed, the GRS80 ellipsoid is recommended (see table 3.2) . Each analysis center compares its reference frame to a realization of the Conventional Terrestrial Reference System (CTRS), as described above. Within IERS, each Terrestrial Reference System (TRF) is either directly, or after transformation, expressed as a realization of the CTRS adopted by IERS as its ITRS. The position of a point located on the surface of the solid Earth should be expressed by X(t) = X0 + VQ(t-t0) +£AXi(ü), 2 where AXj are corrections to various time changing effects, and X0 and V0 are position and velocity at the epoch t0.