Basic GIS Coordinates © 2004 by CRC Press LLC C RC PR E S S Boca Raton London New York Washington, D.C. Basic GIS Coordinates Jan Van Sickle © 2004 by CRC Press LLC This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2004 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-415-30216-1 Library of Congress Card Number 2003069761 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Library of Congress Cataloging-in-Publication Data Van Sickle, Jan. Basic GIS coordinates / Jan Van Sickle. p. cm. Includes bibliographical references (p. ). ISBN 0-415-30216-1 1. Grids (Cartography) Data processing. 2. Geographic information systems. I. Title GA116.V36 2004 910 ¢ .285—dc22 2003069761 TF1625_C00.fm Page 4 Wednesday, April 28, 2004 10:08 AM © 2004 by CRC Press LLC Dedication To Sally Vitamvas TF1625_C00.fm Page 5 Wednesday, April 28, 2004 10:08 AM © 2004 by CRC Press LLC Preface Coordinates? Press a few keys on a computer and they are automatically imported, exported, rotated, translated, collated, annotated and served up in any format you choose with no trouble at all. There really is nothing to it. Why have a book about coordinates? That is actually a good question. Computers are astounding in their ability to make the mathematics behind coordinate manipulation transparent to the user. However, this book is not about that mathematics. It is about coordinates and coordinate systems. It is about how coordinates tie the real world to its electronic image in the computer. It is about understanding how these systems work, and how they sometimes do not work. It is about how points that should be in New Jersey end up in the middle of the Atlantic Ocean, even when the computer has done everything exactly as it was told — and that is, I suppose, the answer to the question from my point of view. Computers tend to be very good at repetition and very bad at interpre- tation. People, on the other hand, are poor at repetition. We tend to get bored. Yet we can be excellent indeed at interpretation, if we have the information to understand what we are interpreting. This book is about providing some of that information. TF1625_C00.fm Page 7 Wednesday, April 28, 2004 10:08 AM © 2004 by CRC Press LLC Author Biography Jan Van Sickle has been mapping since 1966. He created and led the GIS department at Qwest Communications International, a telecommunications company with a worldwide 25,000-mile fiber optic network. In the early 1990s, Jan helped prepare the GCDB, a nationwide GIS database of public lands, for the U.S. Bureau of Land Management. Jan’s experience with GPS began in the 1980s when he supervised control work using the Macrometer, the first commercially available GPS receiver. He was involved in the first GPS control survey of the Grand Canyon, the Eastern Transportation Corri- dor in the LA Basin, and the GPS control survey of more than 7000 miles of fiber optic facilities across the United States. A college-level teacher of sur- veying and mapping, Jan also authored the nationally recognized texts, GPS for Land Surveyors and 1,001 Solved Fundamental Surveying Problems . This most recent book, Basic GIS Coordinates , is the foundation text for his nation- wide seminars. TF1625_C00.fm Page 9 Wednesday, April 28, 2004 10:08 AM © 2004 by CRC Press LLC Contents Chapter one Foundation of a coordinate system Datums to the rescue 1 René Descartés 2 Cartesian coordinates 2 Attachment to the real world 4 Cartesian coordinates and the Earth 4 The shape of the Earth 6 Latitude and longitude 8 Between the lines 9 Longitude 10 Latitude 12 Categories of latitude and longitude 12 The deflection of the vertical 13 Directions 18 Azimuths 18 Bearings 18 Astronomic and geodetic directions 19 North 21 Magnetic north 21 Grid north 22 Polar coordinates 22 Summary 26 Exercises 28 Answers and explanations 31 Chapter two Building a coordinate system Legacy geodetic surveying 35 Ellipsoids 36 Ellipsoid definition 37 Ellipsoid orientation 40 The initial point 41 Five parameters 42 Datum realization 43 The terrestrial reference frame 44 A new geocentric datum 45 TF1625_bookTOC.fm Page 11 Wednesday, April 28, 2004 10:09 AM © 2004 by CRC Press LLC Geocentric three-dimensional Cartesian coordinates 48 The IERS 51 Coordinate transformation 53 Common points 55 Molodenski transformation 55 Seven-parameter transformation 56 Surface fitting 57 Exercises 59 Answers and explanations 61 Chapter three Heights Two techniques 67 Trigonometric leveling 67 Spirit leveling 69 Evolution of a vertical datum 71 Sea level 71 A different approach 73 The zero point 74 The GEOID 75 Measuring gravity 77 Orthometric correction 78 GEOID99 82 Dynamic heights 83 Exercises 85 Answers and explanations 87 Chapter four Two-coordinate systems State plane coordinates 91 Map projection 91 Polar map projections 94 Choices 98 SPCS27 to SPCS83 102 Geodetic lengths to grid lengths 105 Geographic coordinates to grid coordinates 113 Conversion from geodetic azimuths to grid azimuths 114 SPCS to ground coordinates 117 Common problems with state plane coordinates 119 UTM coordinates 120 Exercises 125 Answers and explanations 128 Chapter five The Rectangular System Public lands 133 The initial points 134 Quadrangles 137 Townships 140 TF1625_bookTOC.fm Page 12 Wednesday, April 28, 2004 10:09 AM © 2004 by CRC Press LLC Sections 143 The subdivision of sections 146 Township plats 147 Fractional lots 148 Naming aliquot parts and corners 151 System principles 153 Exercises 158 Answers and explanations 161 Bibliography 165 TF1625_bookTOC.fm Page 13 Wednesday, April 28, 2004 10:09 AM © 2004 by CRC Press LLC 1 chapter one Foundation of a coordinate system Coordinates are slippery. A stake driven into the ground holds a clear posi- tion, but it is awfully hard for its coordinates to be so certain, even if the figures are precise. For example, a latitude of 40º 25' 33.504" N with a lon- gitude of 108º 45' 55.378" W appears to be an accurate, unique coordinate. Actually, it could correctly apply to more than one place. An elevation, or height, of 2658.2 m seems unambiguous too, but it is not. In fact, this latitude, longitude, and height once pinpointed a control point known as Youghall, but not anymore. Oh, Youghall still exists. It is a bronze disk cemented into a drill hole in an outcropping of bedrock on Tanks Peak in the Colorado Rocky Mountains; it is not going anywhere. However, its coor- dinates have not been nearly as stable as the monument. In 1937 the United States Coast and Geodetic Survey set Youghall at latitude 40º 25' 33.504" N and longitude 108° 45' 55.378" W. You might think that was that, but in November 1997 Youghall suddenly got a new coordinate, 40º 25' 33.39258" N and 108º 45' 57.78374" W. That is more than 56 m, 185 ft, west, and 3 m, 11 ft, south, of where it started. But Youghall had not actually moved at all. Its elevation changed, too. It was 2658.2 m in 1937. It is 2659.6 m today; that means it rose 4 1 / 2 ft. Of course, it did no such thing; the station is right where it has always been. The Earth shifted underneath it — well, nearly. It was the datum that changed. The 1937 latitude and longitude for Youghall was based on the North American Datum 1927 (NAD27) . Sixty years later, in 1997 the basis of the coordinate of Youghall became the North American Datum 1983 (NAD83). Datums to the rescue Coordinates without a specified datum are vague. It means that questions like “Height above what?,” “Where is the origin?,” and “On what surface do they lie?” go unanswered. When that happens, coordinates are of no real use. An origin , or a starting place, is a necessity for them to be meaningful. Not only must they have an origin, they must be on a clearly defined surface. These foundations constitute the datum . TF1625_C01.fm Page 1 Wednesday, April 28, 2004 10:10 AM © 2004 by CRC Press LLC [...]... calculate: Distance = Distance = (X1 – X2 )2 + (Y1 – Y2 )2 (2,090,924.62 – 2,0 91, 064.07)2 + (1, 414 ,754.47 1, 418 ,088.47)2 © 2004 by CRC Press LLC TF1625_C 01. fm Page 6 Wednesday, April 28, 2004 10 :10 AM 6 Basic GIS Coordinates Distance = Distance = (13 9.45)2 + (3334.00)2 (19 , 445.3025) + (11 ,11 5,556.0000) Distance = 11 ,13 5,0 01. 30 Distance = 3336. 91 ft It is 3336. 91 ft The distance between these points... 311 ); therefore, where X1 = 295 Y1 = 220 and X2 = 405 Y2 = 311 Distance = [(X1 ) − (X 2 )]2 + [(Y1 ) − (Y2 )]2 Distance = [(295) − (405)]2 + [(220) − ( 311 )]2 Distance = ( 11 0)2 + (– 91) 2 Distance = 12 ,10 0 + 8,2 81 Distance = 20, 3 81 Distance = 14 2.76 © 2004 by CRC Press LLC TF1625_C 01. fm Page 4 Wednesday, April 28, 2004 10 :10 AM 4 Basic GIS Coordinates The system works It is convenient But unless it... P2 (405, 311 ) N,E ( 311 ,405) P1 295 X,Y (295,220) N,E (220,295) 311 220 +10 0 Origin -2 00 -1 00 East (X) +10 0 +200 +300 +400 +500 -1 00 -2 00 Figure 1. 1 The Cartesian coordinate system right — is positive Similarly, on the y-axis, any point north of the origin is positive, and south, negative If these principles are held, then the rules of Euclidean geometry are true and the off-the-shelf computer-aided design... h/cos f 10 8∞ 46' 00.08" – ( +1. 75")/cos 40°25' 33.39" 10 8∞ 46' 00.08" – ( +1. 75")/0.7 612 447 10 8∞ 46' 00.08" – (+2.30") 10 8∞ 45' 57.78" © 2004 by CRC Press LLC TF1625_C 01. fm Page 17 Wednesday, April 28, 2004 10 :10 AM Chapter one: Foundation of a coordinate system 17 E (-) N (+) S (-) W (+) = + 2.89" = + 1. 75" Astronomic Coordinates = 40° 25' 36.28" = 10 8° 46' 00.08" Deflection of the Geodetic Coordinates. .. be 18 0∞ from Greenwich in a part of the world mostly covered by ocean Even though the line does not actually follow the meridian exactly, it avoids dividing populated areas; © 2004 by CRC Press LLC TF1625_C 01. fm Page 11 Wednesday, April 28, 2004 10 :10 AM Chapter one: Foundation of a coordinate system 11 18 0° E,W -1 50° 15 0° -1 20° 12 0° 90° W 90° E (+90° ) (-9 0° ) West Longitude East Longitude 60° -6 0°... 45 15 '35" " N (+) '3 5 08" 15 ' 338 0° 45 1 5 Az = Azimuths 19 Az = Chapter one: 338 15 '08" 99°39'46" 90° W (-) E (+) 90° Az = 99 °39'46" 18 " Az = ' °53 245 245°53 '18 " S (-) 0° 21 44'52" 45 15 '35" 0° E '3 5° 15 N4 2"W N (+) 5" 5 °44' N 21 Bearings 1 4 90°W (-) 8"W 3 '1 5°5 65°53 '18 " S6 E (+) 90° 3 2 S80°20 '1 4"E 80°20 '14 " S (-) 0° Figure 1. 8 Azimuths and bearings 4 However, if the datum is on the ellipsoidal... Convergence (+00°00' 17 .2") Figure 1. 9 Approximate grid azimuth = geodetic azimuth – convergence 23 © 2004 by CRC Press LLC TF1625_C 01. fm Page 23 Wednesday, April 28, 2004 10 :10 AM Grid North Chapter one: Geodetic North TF1625_C 01. fm Page 24 Wednesday, April 28, 2004 10 :10 AM 24 Basic GIS Coordinates There are coordinates that are all distances — Cartesian coordinates, for example Other coordinates are... mapping, and cartography arrangement of polar coordinates, the radius vector always points out from the origin and is always positive (87,45°) M a y a l s o b e e x p r e s s e d a s : (87,405°) (87 ,-3 15 °) (-8 7,225°) E EE EEE Figure 1. 11 Four ways of noting one position © 2004 by CRC Press LLC TF1625_C 01. fm Page 26 Wednesday, April 28, 2004 10 :10 AM 26 Basic GIS Coordinates North (45° ,87) Clockwise rotation... Longitude Detail B Detail C Quadrangles are 1 of Longitude by 1 of Latitude 90° Detail A 89°W 1 N Latitude 68.7 Miles Detail B Figure 1. 2 Distances across 1 © 2004 by CRC Press LLC 90° 89°W 60°N Latitude 89°W 69 Miles 90° North Pole 34.5 Miles Detail C Equator 0°Latitude 69.4 Miles TF1625_C 01. fm Page 10 Wednesday, April 28, 2004 10 :10 AM 10 Basic GIS Coordinates increase in the size of a degree of latitude... coordinate In surveying, polar coordinates are very often a first step toward calculating coordinates in other systems © 2004 by CRC Press LLC 294°59' 07.2" Geodetic Azimuth Convergence (-0 0°00' 40.0") Fink 1, 185,508.04 Grid North Geodetic North 3,002,080.79 294°58' 50" Grid Azimuth Geodetic Azimuth 65°00' 30" Grid Azimuth 65° 01' 10 " D266 1, 182,289.56 2,995 ,17 2.59 7 W = 10 5°30' 00" ft 14 1 62 Foundation of a . +500 295 405 14 2.76 X,Y (295,220) X,Y (405, 311 ) N,E (220,295) N,E ( 311 ,405) 220 P 1 P 2 311 +200 -2 00 -2 00 +200 +10 0 -1 00 -1 00 +10 0 East X() North Y() [(X ) (X )] + [(Y ) (Y )] 12 2 12 2 −− [(295) (405)] + [(220) ( 311 )] 22 −− ( 11 0). than he thought. (13 9.45) + (3334.00) 22 (19 ,445.3025)+ (11 ,11 5,556.0000) 11 ,13 5,0 01. 30 TF1625_C 01. fm Page 6 Wednesday, April 28, 2004 10 :10 AM © 2004 by CRC Press LLC Chapter one: Foundation. (Y – Y ) 12 2 12 2 (2,090,924.62 – 2,0 91, 064.07) + (1, 414 ,754.47 1, 418 ,088.47) 22 TF1625_C 01. fm Page 5 Wednesday, April 28, 2004 10 :10 AM © 2004 by CRC Press LLC 6 Basic GIS Coordinates