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(12) Patent Application Publication (10) Pub. No.: US 2015/0260527 A1 Fink (43) Pub
US 20150260527A1 (19) United States (12) Patent Application Publication (10) Pub. No.: US 2015/0260527 A1 Fink (43) Pub. Date: Sep. 17, 2015 (54) SYSTEMAND METHOD OF DETERMINING (52) U.S. Cl. A POSITION OF A REMOTE OBJECT VA CPC ...................................... G0IC 2 1/20 (2013.01) ONE ORMORE IMAGE SENSORS (57) ABSTRACT (71) Applicant: Ian Michael Fink, Austin, TX (US) In one or more embodiments, one or more systems, methods (72) Inventor: Ian Michael Fink, Austin, TX (US) and/or processes may determine a location of a remote object (e.g., a point and/or area of interest, landmark, structure that (21) Appl. No.: 14/716,817 “looks interesting, buoy, anchored boat, etc.). For example, the location of a remote object may be determined via a first (22) Filed: May 19, 2015 bearing, at a first location, and a second bearing, at a second O O location, to the remote object. For instance, the first and Related U.S. Application Data second locations can be determined via a position device, (63) Continuation of application No. 13/837,853, filed on Such as a global positioning system device. In one or more Mar. 15, 2013, now Pat. No. 9,074,892. embodiments, the location of the remote object may be based on the first location, the second location, the first bearing, and Publication Classification the second bearing. For example, the location of the remote object may be provided to a user via a map. For instance, (51) Int. Cl. turn-by-turn direction to the location of the remote object GOIC 2L/20 (2006.01) may be provided to the user. -
Spherical Geometry
Spherical Geometry Eric Lehman february-april 2012 2 Table of content 1 Spherical biangles and spherical triangles 3 § 1. The unit sphere . .3 § 2. Biangles . .6 § 3. Triangles . .7 2 Coordinates on a sphere 13 § 1. Cartesian versus spherical coordinates . 13 § 2. Basic astronomy . 16 § 3. Geodesics . 16 3 Spherical trigonometry 23 § 1. Pythagoras’ theorem . 23 § 2. The three laws of spherical trigonometry . 24 4 Stereographic projection 29 § 1. Definition . 29 § 2. The stereographic projection preserves the angles . 34 § 3. Images of circles . 36 5 Projective geometry 49 § 1. First description of the real projective plane . 49 § 2. The real projective plane . 52 § 3. Generalisations . 59 3 4 TABLE OF CONTENT TABLE OF CONTENT 1 Introduction The aim of this course is to show different aspects of spherical geometry for itself, in relation to applications and in relation to other geometries and other parts of mathematics. The chapters will be (mostly) independant from each other. To begin, we’l work on the sphere as Euclid did in the plane looking at triangles. Many things look alike, but there are some striking differences. The second viewpoint will be the introduction of coordinates and the application to basic astronomy. The theorem of Pytha- goras has a very nice and simple shape in spherical geometry. To contemplate spherical trigonometry will give us respect for our ancestors and navigators, but we shall skip the computations! and let the GPS do them. The stereographic projection is a marvellous tool to understand the pencils of coaxial circles and many aspects of the relation between the spherical geometry, the euclidean affine plane, the complex projective line, the real projec- tive plane, the Möbius strip and even the hyperbolic plane. -
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. -
Choosing the Best Atlas for Your Classroom William Slattery, Editor
Choosing the Best Atlas for Your Classroom William Slattery, Editor (Last updated August 14, 2008) As editor of Social Studies School Service's Geography Catalog, I frequently receive plaintive phone calls from teachers and department heads who are about to spend their hard-won money on an atlas (or on dozens or even hundreds of atlases), and they need more information than the one-paragraph descriptions our print catalogs give them. Often they ask which atlas is "best," but the real question that needs to be asked is "Which atlas is best for my particular needs?" Some important questions to consider here: Who is going to use the atlas? What will they use it for? What features are most important to you? How much do you plan to spend? For instance, if you will use the atlas for intensive work on understanding latitude and longitude, you might require an atlas that uses no other location system that might distract students or be used as a crutch. On the other hand, you may prefer an atlas that includes not only latitude and longitude but also alphanumeric grids, because then one system can serve as the answer key to the other. This article provides a series of descriptions that permit point-by-point comparisons between atlases. It discusses what types of maps are included, how color is used, how much detail is shown, and how up-to-date the atlases are. Note that, throughout these descriptions, the term "sub-national political units" refers to political entities one level below the national government (such as states, departments, provinces, and republics in a federation). -
Primer of Celestial Navigation
LIFEBOAT MANUAL By Lt. Comdr. A. E. Redifer, U.S.M.S. Lifeboat Training Officer, Avalon, Calif., Training Station *Indispensable to ordinary seamen preparing for the Lifeboat Examination. *A complete guide for anyone who may have to use a lifeboat. Construction, equipment, operation of lifeboats and other buoyant apparatus are here, along with the latest Government regulations which you need to know. 163 Pages More than 150 illustrations Indexed $2.00 METEOROLOGY WORKBOOK with Problems By Peter E. Kraght Senior Meteorologist, American Airlines, Inc. *Embodies author's experience in teaching hundreds of aircraft flight crews. *For self-examination and for classroom use. *For student navigator or meteorologist or weather hobbyist. *A good companion book to Mr. Kraght's Meteorology for Ship and Aircraft Operation, 0r any other standard reference work. 141 Illustrations Cloud form photographs Weather maps 148 Pages W$T x tl" Format $2.25 MERCHANT MARBMI OFFICERS' HANDBOOK By E. A. Turpin and W. A. MacEwen, Master Mariners "A valuable guide for both new candidates for . officer's license and the older, more experienced officers." Marine Engi- neering and Shipping Re$etu. An unequalled guide to ship's officers' duties, navigation, meteorology, shiphandling and cargo, ship construction and sta- bility, rules of the road, first aid and ship sanitation, and enlist- ments in the Naval Reserve all in one volume. 03 086 3rd Edition "This book will be a great help to not only beginners but to practical navigating officers." Harold Kildall, Instructor in Charge, Navigation School of the Washing- ton Technical Institute, "Lays bare the mysteries of navigation with a firm and sure touch . -
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. -
Great Circle If the Plane Passes Through the Centre of the Sphere, and a Small Circle If the Plane Does Not Pass Through the Centre of the Sphere
1 Assoc. Prof. Dr. Bahattin ERDOĞAN Assoc. Prof. Dr. Nursu TUNALIOĞLU Yildiz Technical University Faculty of Civil Engineering Department of Geomatic Engineering Lecture Notes 2 1. A sphere is a solid bounded by a surface every point of which is equally distant from a fixed point which is called the centre of the sphere. The straight line which joins any point of the surface with the centre is called a radius. A straight line drawn through the centre and terminated both ways by the surface is called a diameter. 3 2. The section of the surface of a sphere made by any plane is a circle. 4 5 » The section of the surface of a sphere by a plane is called a great circle if the plane passes through the centre of the sphere, and a small circle if the plane does not pass through the centre of the sphere. Thus the radius of a great circle is equal to the radius of the sphere. » On the globe, equator circle and circles of longitudes are the examples of great circle and, circles of latitudes are the example of small circle. 6 7 » Latitude: latitude lines are also known as parallels (since they are parallel to one another). The lines are actually full circles that extend around the earth and vary in length depending on where we are located. The biggest circle is at the equator and represents the earth's circumference. This line is also called a Great Circle. There can be infinitely many great circles, but only one that is a line of latitude (the equator). -
A Disorienting Look at Euler's Theorem on the Axis of a Rotation
A Disorienting Look at Euler’s Theorem on the Axis of a Rotation Bob Palais, Richard Palais, and Stephen Rodi 1. INTRODUCTION. A rotation in two dimensions (or other even dimensions) does not in general leave any direction fixed, and even in three dimensions it is not imme- diately obvious that the composition of rotations about distinct axes is equivalent to a rotation about a single axis. However, in 1775–1776, Leonhard Euler [8] published a remarkable result stating that in three dimensions every rotation of a sphere about its center has an axis, and providing a geometric construction for finding it. In modern terms, we formulate Euler’s result in terms of rotation matrices as fol- lows. Euler’s Theorem on the Axis of a Three-Dimensional Rotation. If R is a 3 × 3 orthogonal matrix (RTR = RRT = I) and R is proper (det R =+1), then there is a nonzero vector v satisfying Rv = v. This important fact has a myriad of applications in pure and applied mathematics, and as a result there are many known proofs. It is so well known that the general concept of a rotation is often confused with rotation about an axis. In the next section, we offer a slightly different formulation, assuming only or- thogonality, but not necessarily orientation preservation. We give an elementary and constructive proof that appears to be new that there is either a fixed vector or else a “reversed” vector, i.e., one satisfying Rv =−v. In the spirit of the recent tercentenary of Euler’s birth, following our proof it seems appropriate to survey other proofs of this famous theorem. -
Cylindrical Projections 27
MAP PROJECTION PROPERTIES: CONSIDERATIONS FOR SMALL-SCALE GIS APPLICATIONS by Eric M. Delmelle A project submitted to the Faculty of the Graduate School of State University of New York at Buffalo in partial fulfillments of the requirements for the degree of Master of Arts Geographical Information Systems and Computer Cartography Department of Geography May 2001 Master Advisory Committee: David M. Mark Douglas M. Flewelling Abstract Since Ptolemeus established that the Earth was round, the number of map projections has increased considerably. Cartographers have at present an impressive number of projections, but often lack a suitable classification and selection scheme for them, which significantly slows down the mapping process. Although a projection portrays a part of the Earth on a flat surface, projections generate distortion from the original shape. On world maps, continental areas may severely be distorted, increasingly away from the center of the projection. Over the years, map projections have been devised to preserve selected geometric properties (e.g. conformality, equivalence, and equidistance) and special properties (e.g. shape of the parallels and meridians, the representation of the Pole as a line or a point and the ratio of the axes). Unfortunately, Tissot proved that the perfect projection does not exist since it is not possible to combine all geometric properties together in a single projection. In the twentieth century however, cartographers have not given up their creativity, which has resulted in the appearance of new projections better matching specific needs. This paper will review how some of the most popular world projections may be suited for particular purposes and not for others, in order to enhance the message the map aims to communicate. -
Assignment #3 Map Projection
Assignment #3 Map Projection 1.) Mapping the world For assignment 3, I have used World Winkel Tripel Projection system to project map of the world. This projection is used for world maps to minimize shape and area distortion. It has been used by the National Geographic Society since 1998 for general and thematic world maps. It is a compromise projection used for world maps that averages the coordinates from the equirectangular (equidistant cylindrical) and Aitoff projections. This was developed by Oswald Winkel in 1921. Modified azimuthal—coordinates are the average of the Aitoff and equirectangular projections. Meridians are equally spaced and concave toward the central meridian. The central meridian is a straight line. Parallels are equally spaced curves, concave toward the poles. The poles are approximately 0.4 times the length of the equator. The length of the poles depends on the standard parallel chosen. Figure 1: Mapping world map using World Winkel Tripel NGS Projection 2.) Mapping the United States In this exercise, map of the United States showing states was selected. The properties of U.S cities was modified using “Query Builder” such that the capital cities of each state is considered with population greater than 10000. Symbology was edited and each capital cities were labeled. After that, the distance between two cities was measured using “Measure” tool in toolbar and “Magnifier” option in windows tool in toolbar. Later, for the same map projection was introduced. The map was projected using USA Albert Equal Area projection. After the projection was completed, the distance between same two cities was measured. -
Types of Projections
Types of Projections Conic Cylindrical Planar Pseudocylindrical Conic Projection In flattened form a conic projection produces a roughly semicircular map with the area below the apex of the cone at its center. When the central point is either of Earth's poles, parallels appear as concentric arcs and meridians as straight lines radiating from the center. Usually used for maps of countries or continents in the middle latitudes (30-60 degrees) Cylindrical Projection A cylindrical projection is a type of map in which a cylinder is wrapped around a sphere (the globe), and the details of the globe are projected onto the cylindrical surface. Then, the cylinder is unwrapped into a flat surface, yielding a rectangular-shaped map. Generally used for navigation, but this map is very distorted at the poles. Very Northern Hemisphere oriented. Planar (Azimuthal) Projection Planar projections are the subset of 3D graphical projections constructed by linearly mapping points in three-dimensional space to points on a two-dimensional projection plane. Generally used for polar maps. Focused on a central point. Outside edge is distorted Oval / Pseudo-cylindrical Projection Pseudo-cylindrical maps combine many cylindrical maps together. This reduces distortion. Each cylinder is focused on a particular latitude line. Generally used to show world phenomenon or movement – quite accurate because it is computer generated. Famous Map Projections Mercator Winkel-Tripel Sinusoidal Goode’s Interrupted Homolosine Robinson Mollweide Mercator The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. this map accurately shows the true distance and the shapes of landmasses, but as you move away from the equator the size and distance is distorted. -
The Project Gutenberg Ebook of Spherical Trigonometry, by I
The Project Gutenberg EBook of Spherical Trigonometry, by I. Todhunter This eBook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org. If you are not located in the United States, you'll have to check the laws of the country where you are located before using this ebook. Title: Spherical Trigonometry For the Use of Colleges and Schools Author: I. Todhunter Release Date: August 19, 2020 [EBook #19770] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK SPHERICAL TRIGONOMETRY *** Credit: K.F. Greiner, Berj Zamanian, Joshua Hutchinson andthe Online Distributed Proofreading Team at http://www.pgdp.net(This file was produced from images generously made availableby Cornell University Digital Collections SPHERICAL TRIGONOMETRY. ii SPHERICAL TRIGONOMETRY For the Use of Colleges and Schools: WITH NUMEROUS EXAMPLES. BY I. TODHUNTER, M.A., F.R.S., HONORARY FELLOW OF ST JOHN'S COLLEGE, CAMBRIDGE. FIFTH EDITION. London : MACMILLAN AND CO. 1886 [All Rights reserved.] ii Cambridge: PRINTED BY C. J. CLAY, M.A. AND SON, AT THE UNIVERSITY PRESS. PREFACE The present work is constructed on the same plan as my treatise on Plane Trigonometry, to which it is intended as a sequel; it contains all the propositions usually included under the head of Spherical Trigonometry, together with a large collection of examples for exercise.