Spherical Map Projections
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Understanding Projection Systems
Understanding Projection Systems Understanding Projection Systems A Point: A point has no dimensions, a theoretical location that has neither length, width nor height. A point shows an exact location in space. It is important to understand that a point is not an object, but a position. We represent a point by placing a dot with a pencil. A Line: A line is a geometric object that has length and direction but no thickness. A line may be straight or curved. A line may be infinitely long. If a line has a definite length it is called a line segment or curve segment. A straight line is the shortest distance between two points which is known as the true length of the line. A line is named using letters to indicate its endpoints. B B A A AB - Straight Line Segment AB – Curved Line Segment A line may be seen as the locus of a point as it travels between two points. A B A line can graphically represent the intersection of two surfaces, the edge view of a surface, or the limiting element of a surface. B A Plane: A plane is a flat surface which is infinitely large with zero thickness. Just as a point generates a line, a line can generate a plane. A A portion of a plane is referred to as a lamina. A Plane may be defined in a number of different ways. - 1 - Understanding Projection Systems A plane may be defined by; (i) 3 non-linear points (ii) A line and a point (iii) Two intersecting lines (iv) Two Parallel Lines (The point can not lie on the line) Descriptive Geometry: refers to the representation of 3D objects in a 2D format using points, lines and planes. -
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. -
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. -
Polar Zenithal Map Projections
POLAR ZENITHAL MAP Part 1 PROJECTIONS Five basic by Keith Selkirk, School of Education, University of Nottingham projections Map projections used to be studied as part of the geography syllabus, but have disappeared from it in recent years. They provide some excellent practical work in trigonometry and calculus, and deserve to be more widely studied for this alone. In addition, at a time when world-wide travel is becoming increasingly common, it is un- fortunate that few people are aware of the effects of map projections upon the resulting maps, particularly in the polar regions. In the first of these articles we shall study five basic types of projection and in the second we shall look at some of the mathematics which can be developed from them. Polar Zenithal Projections The zenithal and cylindrical projections are limiting cases of the conical projection. This is illustrated in Figure 1. The A football cannot be wrapped in a piece of paper without either semi-vertical angle of a cone is the angle between its axis and a stretching or creasing the paper. This is what we mean when straight line on its surface through its vertex. Figure l(b) and we say that the surface of a sphere is not developable into a (c) shows cones of semi-vertical angles 600 and 300 respectively plane. As a result, any plane map of a sphere must contain dis- resting on a sphere. If the semi-vertical angle is increased to 900, tortion; the way in which this distortion takes place is deter- the cone becomes a disc as in Figure l(a) and this is the zenithal mined by the map projection. -
Notes on Projections Part II - Common Projections James R
Notes on Projections Part II - Common Projections James R. Clynch 2003 I. Common Projections There are several areas where maps are commonly used and a few projections dominate these fields. An extensive list is given at the end of the Introduction Chapter in Snyder. Here just a few will be given. Whole World Mercator Most common world projection. Cylindrical Robinson Less distortion than Mercator. Pseudocylindrical Goode Interrupted map. Common for thematic maps. Navigation Charts UTM Common for ocean charts. Part of military map system. UPS For polar regions. Part of military map system. Lambert Lambert Conformal Conic standard in Air Navigation Charts Topographic Maps Polyconic US Geological Survey Standard. UTM coordinates on margins. Surveying / Land Use / General Adlers Equal Area (Conic) Transverse Mercator For areas mainly North-South Lambert For areas mainly East-West A discussion of these and a few others follows. For an extensive list see Snyder. The two maps that form the military grid reference system (MGRS) the UTM and UPS are discussed in detail in a separate note. II. Azimuthal or Planar Projections I you project a globe on a plane that is tangent or symmetric about the polar axis the azimuths to the center point are all true. This leads to the name Azimuthal for projections on a plane. There are 4 common projections that use a plane as the projection surface. Three are perspective. The fourth is the simple polar plot that has the official name equidistant azimuthal projection. The three perspective azimuthal projections are shown below. They differ in the location of the perspective or projection point. -
3D Viewing Week 8, Lecture 15
CS 536 Computer Graphics 3D Viewing Week 8, Lecture 15 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University 1 Overview • 3D Viewing • 3D Projective Geometry • Mapping 3D worlds to 2D screens • Introduction and discussion of homework #4 Lecture Credits: Most pictures are from Foley/VanDam; Additional and extensive thanks also goes to those credited on individual slides 2 Pics/Math courtesy of Dave Mount @ UMD-CP 1994 Foley/VanDam/Finer/Huges/Phillips ICG Recall the 2D Problem • Objects exist in a 2D WCS • Objects clipped/transformed to viewport • Viewport transformed and drawn on 2D screen 3 Pics/Math courtesy of Dave Mount @ UMD-CP From 3D Virtual World to 2D Screen • Not unlike The Allegory of the Cave (Plato’s “Republic", Book VII) • Viewers see a 2D shadow of 3D world • How do we create this shadow? • How do we make it as realistic as possible? 4 Pics/Math courtesy of Dave Mount @ UMD-CP History of Linear Perspective • Renaissance artists – Alberti (1435) – Della Francesca (1470) – Da Vinci (1490) – Pélerin (1505) – Dürer (1525) Dürer: Measurement Instruction with Compass and Straight Edge http://www.handprint.com/HP/WCL/tech10.html 5 The 3D Problem: Using a Synthetic Camera • Think of 3D viewing as taking a photo: – Select Projection – Specify viewing parameters – Clip objects in 3D – Project the results onto the display and draw 6 1994 Foley/VanDam/Finer/Huges/Phillips ICG The 3D Problem: (Slightly) Alternate Approach • Think of 3D viewing as taking a photo: – Select Projection – Specify -
Comparison of Spherical Cube Map Projections Used in Planet-Sized Terrain Rendering
FACTA UNIVERSITATIS (NIS)ˇ Ser. Math. Inform. Vol. 31, No 2 (2016), 259–297 COMPARISON OF SPHERICAL CUBE MAP PROJECTIONS USED IN PLANET-SIZED TERRAIN RENDERING Aleksandar M. Dimitrijevi´c, Martin Lambers and Dejan D. Ranˇci´c Abstract. A wide variety of projections from a planet surface to a two-dimensional map are known, and the correct choice of a particular projection for a given application area depends on many factors. In the computer graphics domain, in particular in the field of planet rendering systems, the importance of that choice has been neglected so far and inadequate criteria have been used to select a projection. In this paper, we derive evaluation criteria, based on texture distortion, suitable for this application domain, and apply them to a comprehensive list of spherical cube map projections to demonstrate their properties. Keywords: Map projection, spherical cube, distortion, texturing, graphics 1. Introduction Map projections have been used for centuries to represent the curved surface of the Earth with a two-dimensional map. A wide variety of map projections have been proposed, each with different properties. Of particular interest are scale variations and angular distortions introduced by map projections – since the spheroidal surface is not developable, a projection onto a plane cannot be both conformal (angle- preserving) and equal-area (constant-scale) at the same time. These two properties are usually analyzed using Tissot’s indicatrix. An overview of map projections and an introduction to Tissot’s indicatrix are given by Snyder [24]. In computer graphics, a map projection is a central part of systems that render planets or similar celestial bodies: the surface properties (photos, digital elevation models, radar imagery, thermal measurements, etc.) are stored in a map hierarchy in different resolutions. -
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. -
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. -
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. -
U. S. Department of Agriculture Technical Release No
U. S. DEPARTMENT OF AGRICULTURE TECHNICAL RELEASE NO. 41 SO1 L CONSERVATION SERVICE GEOLOGY &INEERING DIVISION MARCH 1969 U. S. Department of Agriculture Technical Release No. 41 Soil Conservation Service Geology Engineering Division March 1969 GRAPHICAL SOLUTIONS OF GEOLOGIC PROBLEMS D. H. Hixson Geologist GRAPHICAL SOLUTIONS OF GEOLOGIC PROBLEMS Contents Page Introduction Scope Orthographic Projections Depth to a Dipping Bed Determine True Dip from One Apparent Dip and the Strike Determine True Dip from Two Apparent Dip Measurements at Same Point Three Point Problem Problems Involving Points, Lines, and Planes Problems Involving Points and Lines Shortest Distance between Two Non-Parallel, Non-Intersecting Lines Distance from a Point to a Plane Determine the Line of Intersection of Two Oblique Planes Displacement of a Vertical Fault Displacement of an Inclined Fault Stereographic Projection True Dip from Two Apparent Dips Apparent Dip from True Dip Line of Intersection of Two Oblique Planes Rotation of a Bed Rotation of a Fault Poles Rotation of a Bed Rotation of a Fault Vertical Drill Holes Inclined Drill Holes Combination Orthographic and Stereographic Technique References Figures Fig. 1 Orthographic Projection Fig. 2 Orthographic Projection Fig. 3 True Dip from Apparent Dip and Strike Fig. 4 True Dip from Two Apparent Dips Fig. 5 True Dip from Two Apparent Dips Fig. 6 True Dip from Two Apparent Dips Fig. 7 Three Point Problem Fig. 8 Three Point Problem Page Fig. Distance from a Point to a Line 17 Fig. Shortest Distance between Two Lines 19 Fig. Distance from a Point to a Plane 21 Fig. Nomenclature of Fault Displacement 23 Fig.