The Universal Transverse Mercator (UTM) Grid System and Topographic Maps —
An Introductory Guide for Scientists and Engineers Joe S. Depner First edition: 2008 Jun 02 This edition: 2010 Aug 12
c 2008 – 2010 Joe S. Depner. All rights reserved. Permission is hereby Copyright. Copyright granted to share this material for noncommercial educational purposes, but only in its entirety and without additions or alterations. Reproduction for any other use, including commercial use, is prohibited without written permission from the author. Disclaimer. This document has not been peer reviewed. The information presented here may contain errors and/or inaccuracies, and may be unsuitable for some purposes. The author makes no warranty regarding the correctness, accuracy, completeness, or suitability for any purpose, of the information. The author assumes no liability for damages, whether direct or indirect, caused by the use or misuse of the information. Contact Information. The author has provided this document as a public service. You can help improve its quality by reporting errors and suggesting changes. If you have comments or questions about this document, please contact the author via e-mail at the address below: Joe S. Depner
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Contents Preface
ix
Abbreviations and Symbols
xi
1 Introduction 1.1 Knowledge Prerequisites . . . . . . . . . . . 1.2 Motivation . . . . . . . . . . . . . . . . . . 1.3 History of the UTM Grid System . . . . . . 1.3.1 About Map Projections . . . . . . . 1.3.2 The Mercator Projection . . . . . . 1.3.3 The Transverse Mercator Projection 1.3.4 A Universal System . . . . . . . . .
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1 1 2 2 3 3 4 5
2 The UTM Grid 2.1 Area of Definition . . . . . . . . . . . . . . . . . . . . 2.2 Longitude and Latitude Zones . . . . . . . . . . . . . . 2.2.1 Longitude Zones . . . . . . . . . . . . . . . . . 2.2.2 Latitude Zones . . . . . . . . . . . . . . . . . . 2.2.3 Zone Specification . . . . . . . . . . . . . . . . 2.2.4 Irregular Longitude Zones . . . . . . . . . . . . 2.2.5 Points on Zone Boundaries . . . . . . . . . . . 2.3 Easting and Northing Coordinates . . . . . . . . . . . 2.3.1 Easting Coordinates . . . . . . . . . . . . . . . 2.3.2 Northing Coordinates . . . . . . . . . . . . . . 2.3.3 Easting and Northing Coordinate Specifications 2.3.4 Coordinate Gridlines . . . . . . . . . . . . . . . 2.4 UTM Coordinate Specifications . . . . . . . . . . . . . 2.5 Map Projections, Datums, and the Graticule . . . . . 2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . .
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7 7 7 7 7 11 14 17 18 19 19 21 22 22 25 26
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29 29 29 29 31 35 35 36
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3 The UTM Grid and USGS Topographic Maps 3.1 Map Elements Supporting Use of the UTM Grid . . . . 3.1.1 Horizontal Datum Identifier . . . . . . . . . . . . 3.1.2 UTM Longitude Zone Identifier . . . . . . . . . . 3.1.3 UTM Grid Tick Marks and Coordinate Labels . 3.1.4 UTM Grid Declination Information . . . . . . . . 3.1.5 UTM Gridlines . . . . . . . . . . . . . . . . . . . 3.2 Using Topographic Maps Without Preprinted Gridlines iii
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iv
CONTENTS
3.3
3.4 3.5 3.6
3.2.1 Visual Estimation of Gridline Positions . . . . . . . . 3.2.2 Overlaying a UTM Grid Transparency . . . . . . . . . 3.2.3 Drawing UTM Gridlines on Maps . . . . . . . . . . . Determining the UTM Coordinates of a Point on a Map . . . 3.3.1 Basic Procedure . . . . . . . . . . . . . . . . . . . . . 3.3.2 Measuring Projected Easting and Northing Distances Plotting a Point with Known UTM Coordinates on a Map . . Software for Using the UTM Grid with Topographic Maps . . Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . .
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4 Horizontal Distance and Bearing Determination 4.1 Points in the Same UTM Longitude Zone and Hemisphere . . . 4.1.1 Measurement Using a Paper Map . . . . . . . . . . . . . 4.1.2 Calculation Using Plane Geometry . . . . . . . . . . . . 4.2 Points Not in the Same UTM Longitude Zone and Hemisphere 4.2.1 Spherical Earth . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Ellipsoidal Earth . . . . . . . . . . . . . . . . . . . . . . 4.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . .
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36 38 39 40 40 43 47 51 51
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53 53 53 54 58 58 59 59
References
61
Appendices
64
A Approximate Ranges for UTM Easting Coordinates A.1 Crude Approximation, for Nonspecific Latitude . . . . . . . . . . . . . . . . . . . A.2 Refined Approximation, for Nonspecific Latitude . . . . . . . . . . . . . . . . . . A.3 Crude Approximation, for Specific Latitude . . . . . . . . . . . . . . . . . . . . .
67 67 68 70
B Approximate Ranges for UTM Northing Coordinates B.1 Northern Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Southern Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73 73 74
C Measuring the Distance from a Point to a Gridline C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Measuring the Distance from a Point to a Fully Displayed Gridline C.2.1 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . C.2.2 Indirect Method . . . . . . . . . . . . . . . . . . . . . . . . C.2.3 Combined Method for Measuring Easting and Northing . . C.3 Measuring the Distance from a Point to a Marked-only Gridline . .
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75 75 75 76 76 76 76
D Obtaining an Appropriate USGS Topographic Map D.1 Basic Procedure . . . . . . . . . . . . . . . . . . . . . D.2 Resources for Identifying Relevant Topographic Maps D.3 Selecting an Appropriate Map Scale . . . . . . . . . . D.4 Acquiring USGS Topographic Maps . . . . . . . . . . D.5 Verifying the Area of Coverage . . . . . . . . . . . . .
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List of Tables 1
Examples Illustrating Two Conventions for Grouping Digits . . . . . . . . . . . .
2.1 2.2 2.3 2.4 2.5
UTM Longitude Zones Spanning the U.S. . . . . . . . . . UTM Latitude Zones Spanning the U.S. . . . . . . . . . . Minimum and Maximum UTM Easting Coordinates . . . Minimum and Maximum UTM Northing Coordinates . . Summary of UTM Map Projections and Local Coordinate
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11 14 19 21 27
4.1 4.2
Data for Distance and Bearing Calculation Example – 1st of 2 . . . . . . . . . . . Data for Distance and Bearing Calculation Example – 2nd of 2 . . . . . . . . . .
55 57
A.1 Output from NGS Utility, for NAD 83 . . . . . . . . . . . . . . . . . . . . . . . . A.2 Output from NGS Utility, for NAD 27 . . . . . . . . . . . . . . . . . . . . . . . .
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vi
CONTENTS
List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7
Area of Definition for UTM Grid System . . . . . . . . . . . . . . . Regular UTM Longitude Zones – Equatorial Aspect . . . . . . . . Regular UTM Longitude Zones – Oblique Aspect . . . . . . . . . . UTM Latitude Zones – Equatorial Aspect . . . . . . . . . . . . . . UTM Latitude Zones – Oblique Aspect . . . . . . . . . . . . . . . Global Distribution of UTM Longitude Zones and Latitude Zones . Regular UTM Longitude Zone in Northern Hemisphere . . . . . .
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8 9 10 12 13 16 20
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
UTM Grid Information – Older Map . . . . . . . . . . . . . . . . . . . . . UTM Grid Information – Newer Map . . . . . . . . . . . . . . . . . . . . A Basic Corner Ruler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using a Corner Ruler with a Topographic Map . . . . . . . . . . . . . . . Using a Corner Ruler to Measure UTM Coordinates on a Map – 1st of 4 . Using a Corner Ruler to Measure UTM Coordinates on a Map – 2nd of 4 Using a Corner Ruler to Measure UTM Coordinates on a Map – 3rd of 4 . Using a Corner Ruler to Measure UTM Coordinates on a Map – 4th of 4 .
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30 30 37 43 45 46 48 49
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viii
Preface This guide provides a comprehensive introduction to the Universal Transverse Mercator (UTM) grid system (also called the UTM coordinate system or the GPS grid system) and its use with topographic maps. The intended audience is primarily scientists and engineers. It assumes the reader has basic knowledge of the following: • spatial coordinate systems, including Cartesian (rectangular) and geodetic (geographic) coordinate systems, • general cartographic principles, and • topographic maps. Section 1.1 (Knowledge Prerequisites) gives a more detailed list of the prerequisite subjects. Much information about the UTM grid system is available in many forms, including books, reports, articles, and websites. These range from the most basic, which assume little knowledge of mapping and navigation, to the advanced, which assume specialized knowledge in one or more subfields of geomatics (e.g., analytical cartography, geodesy, geographic information systems, global positioning system (GPS)). This guide takes a middle path. It provides more depth than the most basic materials, without requiring as much specialized knowledge as the advanced materials. It attempts to make explicit much of the information that’s implicit in some of the more terse references on the subject, such as Defense Mapping Agency (DMA) [1989]. This guide is intended primarily, but not exclusively, for civilian readers in the United States (U.S.). For instance, it discusses only those topographic maps produced by the U.S. Geological Survey (USGS). However, much of the material presented here, such as the basic description of the UTM grid system, is applicable worldwide. Additionally, the discussion of topographic maps likely applies, to some extent, to topographic maps produced by other agencies (e.g., U.S. Department of Defense) and to maps produced in other countries. Numerous resources are available for working with the UTM grid system and topographic maps. These range from the technologically primitive (e.g., paper map, straightedge, corner ruler) to the technologically advanced (computer and software, digital dataset, worldwide web, GPS). The current trend is toward the increased use of advanced resources. However, certain fundamental concepts underlie the competent use of even the most primitive resources, and such concepts can be explained quite naturally in terms of paper maps, rulers, and plane geometry. In contrast, explanations in terms of more advanced resources are likely to suffer from the distractions imposed by the complexities of the particular technologies, and to be less universally applicable. For these reasons, and not because of any anti-technology bias, I’ve formulated my explanations primarily in terms of map and straightedge rather than computer and worldwide web. Numerous examples are included to illustrate and reinforce the ideas presented here. ix
x
PREFACE
Table 1: Examples Illustrating Two Conventions for Grouping Digits Other U.S. Documents
This Document
8, 861
8861
56, 774.0
56 774.0
4, 936.2005
4936.2005
9, 385.70323
9385.703 23
8, 800, 512
8 800 512
In an attempt to appeal to an international audience, this document generally follows the convention recommended by Taylor [1995] for the grouping of digits: Because the comma is widely used as the decimal marker outside the United States, it should not be used to separate digits into groups of three. Instead, digits should be separated into groups of three, counting from the decimal marker towards the left and right, by the use of a thin, fixed space. However, this practice is not usually followed for numbers having only four digits on either side of the decimal marker except when uniformity in a table is desired. This convention eliminates potential confusion about interpretation of commas, without sacrificing readability of long numeric strings. A period serves as the decimal marker (point). Table 1 gives examples. This convention conforms to the recommendations of the International Union of Pure and Applied Chemistry (IUPAC) [IUPAC, 2006]. A different convention is followed when listing the formal UTM coordinate specification of one or more points. In that case, no commas or spaces are used. See Chapter 2 for details.
Abbreviations and Symbols Symbol
Description
a
Length of major semi-axis (semimajor axis) of ellipsoid
arccos
Inverse cosine function
arcsin
Inverse sine function
arctan
Inverse tangent function
b
Length of minor semi-axis (semiminor axis) of ellipsoid
CI
Contour interval
cm
Centimeter(s)
cm
Central meridian
Co.
Company
cos
Cosine function
D
Distance
DD
Degree(s) of arc
DMA
Defense Mapping Agency (U.S.)
DOI
Digital object identifier
DRG
Digital raster graphic
E
Easting coordinate identifier
E.
East
E-gridline
Easting gridline
FGDC
Federal Geographic Data Committee (U.S.)
ft
Foot (feet)
xi
xii
ABBREVIATIONS AND SYMBOLS
List of Abbreviations and Symbols (continued) Symbol
Description
GN
Grid north
GOI
Gridline of interest
GPO
Government Printing Office
GPS
Global Positioning System
hemis.
Hemisphere
in.
Inch(es)
IUGG
International Union of Geodesy and Geophysics
IUPAC
International Union of Pure and Applied Chemistry
km
Kilometer(s)
lat.
Latitude
lon.
Longitude
m
Meter(s)
MGRS
Military Grid Reference System (U.S.)
MM
Minute(s) of arc
mm
Millimeter(s)
MN
Magnetic north
N
Northing coordinate identifier
N.
North
NAD 27
North American Datum of 1927
NAD 83
North American Datum of 1983
NATO
North Atlantic Treaty Organization
N-gridline
Northing gridline
NGS
National Geodetic Survey (U.S.)
NIMA
National Imagery and Mapping Agency (U.S.)
NIST
National Institute of Standards and Technology (U.S.)
ABBREVIATIONS AND SYMBOLS
List of Abbreviations and Symbols (continued) Symbol
Description
NOAA
National Oceanic and Atmospheric Administration (U.S.)
NOS
National Ocean Service (U.S.)
OMNR
Ontario Ministry of Natural Resources (Canada)
PDF
Portable Document Format
PLSS
Public Land Survey System (U.S.)
POI
point of interest
r
Radius of Earth
rad
Radian(s) of arc
S
Length of arc on the surface of the ellipsoid
S.
South
SI
International System of Units
SS
Second(s) of arc
sin
Sine function
SPCS
State Plane Coordinate System (U.S.)
tan
Tangent function
u
Map scale
UPS
Universal Polar Stereographic
URI
Worldwide-web uniform resource indicator
US, U.S.
United States
USC&GS
United States Coast and Geodetic Survey
USGS
United States Geological Survey
USNG
United States National Grid
UTM
Universal Transverse Mercator
W.
West
WA
Washington State (U.S.)
xiii
xiv
ABBREVIATIONS AND SYMBOLS
List of Abbreviations and Symbols (continued) Symbol
Description
WGS 84
World Geodetic System of 1984
x
UTM easting coordinate
xcm
UTM easting coordinate of central meridian
xE−gridline
UTM easting coordinate of UTM easting gridline
xPOI
UTM easting coordinate of point of interest
∆x
Projected easting distance (as horizontal ground distance)
∆xmap
Projected easting distance (as map distance)
y
UTM northing coordinate
yN−gridline
UTM northing coordinate of UTM northing gridline
yPOI
UTM northing coordinate of point of interest
∆y
Projected northing distance (as horizontal ground distance)
∆ymap
Projected northing distance (as map distance)
β
Bearing (angle)
φ
Latitude coordinate (angle)
λ
Longitude coordinate (angle)
π
The mathematical constant, π = 3.14159265 . . .
:
(colon separating two integers) Numerical ratio
◦
Degree(s) of arc
0
Minute(s) of arc
Chapter 1
Introduction 1.1
Knowledge Prerequisites
To get the maximum benefit from this document, readers should have a basic familiarity with the topics and concepts listed in the following paragraphs. Each paragraph corresponds to a main topic, which is given by the paragraph heading. Keywords corresponding to associated subtopics, concepts, and terms follow in alphabetical order. Spatial coordinate systems coordinate axes, coordinate grid, coordinate grid tick mark, coordinate numerical values and units, coordinate origin, coordinate specification, orthogonal coordinates, orthogonal curvilinear coordinates, rectilinear coordinates, spatial coordinates Cartesian coordinate systems abscissa, Cartesian coordinates, distance coordinates, easting (or x) coordinate, horizontal coordinate, northing (or y) coordinate, ordered pair, ordered triplet, ordinate, vertical (or z) coordinate Geodetic/geographic coordinate systems angle, angle coordinates, degrees of arc, equator, geodetic coordinates, geographic coordinates, globe, graticule, great circle, great-circle distance, Greenwich Meridian, hemispheres, International Date Line, latitude, latitude lines (parallels), longitude, longitude lines (meridians), minutes of arc, polar regions, pole, Prime Meridian, radians, seconds of arc, small circle General cartographic principles direction, distance, explanatory material, explanatory text, horizontal datum, index map, labels, legend, locator map, map border, map collar, map sheet, map-sheet margin, neatline, orientation indicator, projection information, publication information (publisher name, year, copyright), scale, scale indicator (graphic, numerical, verbal), source note, title and subtitles, vertical datum Topographic maps contour interval (CI), CI indicator, contour lines, contour map, declination diagram, elevation, hypsography, magnetic declination, topographic map, topographic map symbols, topography, USGS 7.5-minute and 15-minute series quadrangles 1
2
CHAPTER 1. INTRODUCTION
1.2
Motivation
Coordinate systems provide effective means of communicating and analyzing information about position. Multiple coordinate systems have been developed for various uses, each with its own particular advantages. To extract the maximum value from data collection and analysis efforts, it’s important to choose the most appropriate coordinate system for the particular situation. Generally this requires knowing the following about coordinate systems: • their definitions and basic characteristics; • their strengths and weaknesses; • which ones are best suited to particular applications; and • how to convert coordinate data for one system to coordinate data for another system. The UTM grid system has advantages over other coordinate systems. For instance, unlike State Plane Coordinate Systems (SPCSs), which are defined over relatively small regions, the UTM grid system is defined worldwide exclusive of the polar regions. Cole [1977] and Grubb and Eakle [1988] summarize some of the advantages of the UTM grid system relative to other existing coordinate systems. The UTM grid system is conceptually simple to use, effectively requiring one to apply a local Cartesian (xy) coordinate system. This makes it easier to learn and more convenient to use than, say, the Public Land Survey System (PLSS; also sometimes referred to as the SectionTownship-Range System, or the Cadastral System). In the U.S., both military and civilian government agencies use what amount to extended forms of the UTM grid system for georeferencing. Hence, learning the UTM grid system is a logical first step toward learning these extended systems. The U.S. military’s worldwide georeferencing system is called the Military Grid Reference System (MGRS) [DMA, 1990]. U.S. Air Force [2001] gives a clear description of the MGRS. The MGRS applies two separate coordinate systems to their respective areas of definition. The UTM grid system is defined within the area of the globe between 80◦ S. lat. and 84◦ N. lat. A companion system, the Universal Polar Stereographic (UPS) grid system, is defined for the polar regions [DMA, 1989]. The MGRS also overlays additional location elements on the UTM grid. The system used by various local, state, and federal civilian agencies in the U.S. is called the U.S. National Grid (USNG) [Federal Geographic Data Committee (FGDC), 2001]. Like the MGRS, the USNG overlays additional location elements on the UTM grid. Within the U.S., the USNG is interoperable with the MGRS [FGDC, 2001]. In addition to its use for military purposes, the UTM grid system is widely used for surveying, mapping, and land and sea navigation [Langley, 1998]. The UTM grid system is used with the Global Positioning System (GPS), and the U.S. Geological Survey (USGS) projects most of its digital products on the UTM grid [Moore, 1997]. This makes the UTM grid system useful for both non-scientific applications (e.g., outdoor recreation, search-and-rescue operations) and scientific applications (e.g., environmental investigation, natural-resource management).
1.3
History of the UTM Grid System
This section summarizes the historical development of the UTM grid system.
1.3. HISTORY OF THE UTM GRID SYSTEM
1.3.1
3
About Map Projections
A map projection is a means by which one graphically represents points on the surface of the earth, a three-dimensional surface, as points on a map, a two-dimensional surface. For any given type of map projection, the particular way in which one projects the points is defined by geometrical construction, mathematical equations, or some combination of the two. Dana [2007] and Dean [2007] give good introductions to, and overviews of, map projections. For a comprehensive, technical reference on map projections, see Snyder [1987] or Snyder and Voxland [1989]. In practical applications the globe is approximated by an ellipsoid of revolution for which the equator is a great circle. A further simplification sometimes employed is to approximate the globe as a sphere, a particular type of ellipsoid of revolution. In the special case where the ellipsoid is spherical, the corresponding projections are known as spherical forms; otherwise they’re known as ellipsoidal forms. One family of map projections – the cylindrical projections – is central to the development of the UTM grid system. Conceptually, a cylindrical projection may be viewed as a projection of points onto an elliptical (in some cases circular) cylinder (the projection cylinder ) which is wrapped around the globe. Two particular subfamilies of cylindrical map projection are especially important in the development of the UTM grid system – the Mercator projection and the transverse Mercator projection. Both of these are defined by mathematical equations.
1.3.2
The Mercator Projection
The Flemish cartographer Gerhardus Mercator was the first to apply the projection that bears his name (i.e., the Mercator projection) when he produced his famous world chart in 1569 [OMNR, 1981]. The Mercator projection can take one of two forms based on the configuration of the projection cylinder. In the tangent form, the cylinder intersects the globe at the equator (i.e., the cylinder is tangent to the ellipsoid at the equator). In the secant form, the cylinder intersects the globe at two parallels of latitude equidistant from the equator (the standard parallels) (i.e., the cylinder is secant to the ellipsoid). The Mercator projection has the following characteristics: - The meridians of longitude are represented by straight lines oriented parallel to one another. For a given longitude increment, the distance between successive meridians is constant. - The parallels of latitude are represented by straight lines oriented parallel to one another. For a given latitude increment, the distance between successive parallels increases with their distance from the equator. - The meridians of longitude are orthogonal to the parallels of latitude. - The scale is the same in all directions. - The scale varies with location. In the tangent form, the projection is true to scale only at the equator. In the secant form, the projection is true to scale only at the two standard parallels. - The scale becomes infinite at the poles.
4
CHAPTER 1. INTRODUCTION - Any small area is represented in its true shape (i.e., the projection is conformal ). - Rhumb lines (i.e., lines of constant azimuth, lines of true constant bearing) appear as straight lines.
This last characteristic makes Mercator charts useful for global navigation. In 1910 the former U.S. Coast and Geodetic Survey (now the National Ocean Service) adopted the Mercator projection as the standard projection for the nautical charts it prepares [Shalowitz, 1964, p. 302].
1.3.3
The Transverse Mercator Projection
Conceptually, the transverse Mercator projection may be viewed as a projection of points onto an elliptical projection cylinder that is wrapped around the globe, with the axis of the cylinder lying in the equatorial plane. Like the Mercator projection, the transverse Mercator projection can take one of two forms based on the configuration of the cylinder. In the tangent form, the cylinder intersects the globe at the central meridian of the mapped area; that is, the cylinder is tangent to the ellipsoid at the central meridian. In the secant form, the cylinder intersects the globe at two arcs (the standard lines) parallel to and equidistant from the central meridian (i.e., the cylinder is secant to the ellipsoid). The transverse Mercator projection has the following characteristics: - The equator, the central meridian, and each meridian 90 degrees (90◦ ) from the central meridian are represented by straight lines. - Other meridians and parallels are represented by complex curves. - The meridians of longitude are orthogonal to the parallels of latitude. - The scale is the same in all directions. - The scale varies with location. In the tangent form the projection is true to scale only at the central meridian. In the secant form the projection is true to scale only at the standard lines [OMNR, 1981]. - The scale becomes infinite 90◦ from the central meridian. - Both the spherical and ellipsoidal forms of the projection are conformal [Snyder, 1987]. - Rhumb lines don’t appear as straight lines. The tangent form maps the central meridian and nearby regions on either side of it with low distortion [Snyder 1987]. Similarly, the secant form maps the two standard lines and the regions near them with low distortion. It follows that if the two standard lines are sufficiently close together, the region between them will have low distortion. Consequently the transverse Mercator projection typically is applied to long narrow bands. The Alsatian mathematician and cartographer Johann Heinrich Lambert invented the transverse Mercator projection in its spherical form [Snyder, 1987]. In 1772 Lambert presented the projection in his classic work, Beitr¨ age [Lambert, 1772]. While Lambert only indirectly discussed the ellipsoidal form of the transverse Mercator projection, Johann Karl Friedrich Gauss analyzed it further in 1822 [Snyder, 1987]. In 1912 and 1919 L. Kr¨ uger published, for the first time, results for the ellipsoidal form of the transverse
1.3. HISTORY OF THE UTM GRID SYSTEM
5
Mercator projection; for this reason it is sometimes called the Gauss-Kr¨ uger projection [O’Brien, 1986]. Others, including L.P. Lee of New Zealand, also contributed to the development of the ellipsoidal form [Snyder and Voxland, 1989]. In 1936 the International Union of Geodesy and Geophysics (IUGG) proposed the universal adoption of the transverse Mercator projection in 6◦ bands [OMNR, 1981].
1.3.4
A Universal System
After years of consideration, in 1947 the U.S. Army adopted the Universal Transverse Mercator (UTM) grid system as their standard for designating rectangular coordinates on large-scale military maps throughout the world [OMNR, 1981; Snyder, 1987]. Dean [2007] describes the context and rationale for the U.S. Army’s decision. Dracup [2007] provides details pertinent to the U.S. Army’s adoption and implementation of the UTM grid system. The UTM grid system applies the ellipsoidal, secant form of the transverse Mercator projection individually to bands 6◦ wide (in longitude), with additional modifications. These include the following [OMNR, 1981]: • a scale reduction of 1 part in 2500 (i.e., a scale factor of 0.9996) at the central meridian, • a definition of the area of coverage between 80◦ S. lat. and 84◦ N. lat., and • the use of metric units (meters). Subsequently, the North Atlantic Treaty Organization (NATO) and many other countries have adopted the UTM grid system as their official grid system for military purposes [OMNR, 1981].
6
CHAPTER 1. INTRODUCTION
Chapter 2
The UTM Grid The UTM grid system is, in effect, a hybrid coordinate system. It combines elements of the geographic coordinate system (i.e., longitude and latitude zones defined in terms of the graticule) with numerous, local Cartesian coordinate systems (i.e., easting and northing coordinates within each UTM longitude zone and hemisphere).
2.1
Area of Definition
The UTM grid system is defined over that portion of the earth’s surface between latitudes 80◦ S. and 84◦ N. (Figure 2.1). The UTM grid system isn’t defined for the polar regions (i.e., latitudes south of 80◦ S. and latitudes north of 84◦ N.).
2.2 2.2.1
Longitude and Latitude Zones Longitude Zones
The UTM grid divides the earth into 60 contiguous, non-overlapping longitude zones, each one 6◦ wide (as measured along a parallel). Each longitude zone is bounded on the east and on the west by meridians of longitude (see Figures 2.2 and 2.3). This document will refer to these as the zone’s bounding meridians. UTM longitude zones are also called grid zones, longitude zones, UTM zones, or zones. Each UTM longitude zone is identified by a one- or two-digit integer. The zones are numbered consecutively, beginning with “1” or “01” at the zone corresponding to 180◦ W. lon. - 174◦ W. lon., and increasing as one moves eastward to “60” at the zone corresponding to 174◦ E. lon. 180◦ E. lon. Hence, UTM longitude zones 01 through 30 lie in the western hemisphere, while UTM longitude zones 31 through 60 lie in the eastern hemisphere. Consequently, the Prime Meridian (0◦ lon.) separates UTM longitude zones 30 and 31, while the International Date Line (meridian of 180◦ lon.) separates UTM longitude zones 60 and 01. Each longitude zone is bounded on the north by the parallel of 84◦ N. lat. and on the south by the parallel of 80◦ S. lat. Table 2.1 summarizes the distribution of UTM longitude zones across the U.S.
2.2.2
Latitude Zones
The UTM grid divides the region of the earth that lies between the latitudes of 80◦ S. and 84◦ N. into 20 contiguous, non-overlapping latitude zones – 10 in each of the northern and southern hemispheres. Each latitude zone is bounded on the north and the south by parallels of latitude. 7
8
CHAPTER 2. THE UTM GRID
Figure 2.1: Area of Definition for UTM Grid System
2.2. LONGITUDE AND LATITUDE ZONES
Figure 2.2: Regular UTM Longitude Zones – Equatorial Aspect
9
10
CHAPTER 2. THE UTM GRID
Figure 2.3: Regular UTM Longitude Zones – Oblique Aspect
2.2. LONGITUDE AND LATITUDE ZONES
11
Table 2.1: UTM Longitude Zones Spanning the U.S.
Region
Longitude Range (approximate)
Longitude Zones
Number of Zones
Lower 48
124◦ 46’ W. – 66◦ 57’ W.
10 - 19
10
Alaska
172◦ 26’ E. – 130◦ 00’ W.
59, 60, and 1 - 9
11
Hawaii
178◦ 22’ W. – 154◦ 48’ W.
1-5
5
Entire U.S.
172◦ 26’ E. – 66◦ 57’ W.
59, 60, and 1 - 19
21
Notes: (1) “Lower 48” designates the 48 conterminous states and the District of Columbia. (2) “Entire U.S.” designates all 50 states and the District of Columbia. (3) Sources for longitude information: Wikipedia [2007b, c, d]
All of the latitude zones are 8◦ wide (as measured along a meridian), except the most northerly latitude zone, which is 12◦ wide (see Figures 2.4 and 2.5). The UTM latitude zones encircle the globe, from the meridian of 180◦ W. lon. eastward to the meridian of 180◦ E. lon. Each UTM latitude zone is identified by a single uppercase letter of the Latin alphabet. The latitude zones are lettered consecutively, beginning with “C” at the southernmost zone (i.e., 80◦ S. lat. - 72◦ S. lat.), and progressing alphabetically as one moves northward to zone “X” (i.e., 72◦ N. lat. - 84◦ N. lat.). To minimize the potential for confusion with the numerals “1” and “0”, respectively, the letters “I” and “O” aren’t used. Hence, latitude zones C through M, excluding I, lie in the southern hemisphere, while latitude zones N through X, excluding O, lie in the northern hemisphere. The equator separates UTM latitude zones M and N. Table 2.2 summarizes the distribution of UTM latitude zones across the U.S.
2.2.3
Zone Specification
When both the UTM longitude zone and the UTM latitude zone of a point are specified, normally their respective designations are combined into a single alphanumeric string consisting of the following elements, written from left to right in the order listed: • the word “zone” or “Zone”, • a single space, • the one- or two-digit numeric designation for the longitude zone, and • the single-letter alphabetic designation for the latitude zone.
12
CHAPTER 2. THE UTM GRID
Figure 2.4: UTM Latitude Zones – Equatorial Aspect
2.2. LONGITUDE AND LATITUDE ZONES
Figure 2.5: UTM Latitude Zones – Oblique Aspect
13
14
CHAPTER 2. THE UTM GRID
Table 2.2: UTM Latitude Zones Spanning the U.S.
Region
Latitude Range (approximate)
Latitude Zones
Number of Zones
Lower 48
24◦ 31’ N. – 49◦ 23’ N.
R, S, T, and U
4
Alaska
51◦ 12’ N. – 71◦ 23’ N.
U, V, and W
3
Hawaii
18◦ 55’ N. – 28◦ 27’ N.
Q and R
2
Entire U.S.
18◦ 55’ N. – 71◦ 23’ N.
Q through W
7
Notes: (1) “Lower 48” designates the 48 conterminous states and the District of Columbia. (2) “Entire U.S.” designates all 50 states and the District of Columbia. (3) Sources for latitude information: Wikipedia [2007b, c, d]
Example: Formats for Reporting Combined UTM Longitude/Latitude Zones Problem: Parse each of the following combined UTM longitude/latitude zone designations into its respective UTM longitude zone and UTM latitude zone: “Zone 01H” (or, equivalently, “zone 1H”) “Zone 17N” (or, equivalently, “zone 17N”) “Zone 51P” (or, equivalently, “zone 51P”) Solution: “Zone 01H” (or, equivalently, “zone 1H”) designates UTM longitude zone 1, UTM latitude zone H. “Zone 17N” (or, equivalently, “zone 17N”) designates UTM longitude zone 17, UTM latitude zone N. “Zone 51P” (or, equivalently, “zone 51P”) designates UTM longitude zone 51, UTM latitude zone P.
2.2.4
Irregular Longitude Zones
The scheme described above for defining the boundaries of the UTM longitude and latitude zones is valid everywhere between the latitudes of 80◦ S. and 84◦ N., with the exception of the two areas described below [DMA, 1990]. The first area is on or near the southwest coast of Norway, between latitudes 56◦ N. and ◦ 64 N. (i.e., in UTM latitude zone V). UTM zones 31V and 32V are 3◦ and 9◦ wide, respectively, rather than the usual 6◦ wide. UTM zones 31V and 32V extend from 0◦ E. lon. to 3◦ E. lon., and from 3◦ E. lon. to 12◦ E. lon., respectively. Normally UTM longitude zones 31 and 32 extend from 0◦ E. lon. to 6◦ E. lon., and from 6◦ E. lon. to 12◦ E. lon., respectively. The second area is around Svalbard, between latitudes 72◦ N. and 84◦ N. (i.e., in UTM
2.2. LONGITUDE AND LATITUDE ZONES
15
latitude zone X). Svalbard is an archipelago in the Arctic Ocean north of mainland Europe, approximately midway between Norway and the North Pole [Central Intelligence Agency, 2007]. UTM zones 31X and 37X are 9◦ wide, zones 33X and 35X are 12◦ wide, and zones 32X, 34X, and 36X are undefined. Consequently, the four UTM zones 31X, 33X, 35X, and 37X cover the same area that would have been covered by the seven zones 31X to 37X, had these zones been defined on a regular grid. The schematic diagram in Figure 2.6 shows the relative positions of UTM longitude zones and latitude zones.
NORTH LATITUDE (degrees)
EQUATOR
80
72
64
56
48
40
1
1C
1D
1E
1F
1G
1H
1J
1K
1L
1M
1N
1P
1Q
1R
1S
1T
1U
1V
1W
1X
Irregular UTM Zone
Gray Fill
6
Regular UTM Zone
150
6C
6D
6E
6F
6G
6H
6J
6K
6L
6M
6N
6P
6Q
6R
6S
6T
6U
6V
6W
6X
6
UTM Latitude Zone
5
5C
5D
5E
5F
5G
5H
5J
5K
5L
5M
5N
5P
5Q
5R
5S
5T
5U
5V
5W
5X
150
UTM Longitude Zone
4
4C
4D
4E
4F
4G
4H
4J
4K
4L
4M
4N
4P
4Q
4R
4S
4T
4U
4V
4W
4X
5
White Fill
3
3C
3D
3E
3F
3G
3H
3J
3K
3L
3M
3N
3P
3Q
3R
3S
3T
3U
3V
3W
3X
4
Blue Letter
2
2C
2D
2E
2F
2G
2H
2J
2K
2L
2M
2N
2P
2Q
2R
2S
2T
2U
2V
2W
2X
3
Red Number
KEY
180
C
D
E
F
G
H
J
K
L
M
N
P
Q
R
S
T
U
V
W
X
2
7
7C
7D
7E
7F
7G
7H
7J
7K
7L
7M
7N
7P
7Q
7R
7S
7T
7U
7V
7W
7X
7
8
8C
8D
8E
8F
8G
8H
8J
8K
8L
8M
8N
8P
8Q
8R
8S
8T
8U
8V
8W
8X
8
9
9C
9D
9E
9F
9G
9H
9J
9K
9L
9M
9N
9P
9Q
9R
9S
9T
9U
9V
9W
9X
9
120
11
11C
11D
11E
11F
11G
11H
11J
11K
11L
11M
11N
11P
11Q
11R
11S
11T
11U
11V
11W
13
14
14C
14D
14E
14F
14G
14H
14J
14K
14L
14M
14N
14P
14Q
14R
14S
14T
14U
14V
14W
90
15
15C
15D
15E
15F
15G
15H
15J
15K
15L
15M
15N
15P
15Q
15R
15S
15T
15U
15V
90
15W
15X
15
16
16C
16D
16E
16F
16G
16H
16J
16K
16L
16M
16N
16P
16Q
16R
16S
16T
16U
16V
16W
16X
16
17
17C
17D
17E
17F
17G
17H
17J
17K
17L
17M
17N
17P
17Q
17R
17S
17T
17U
17V
17W
17X
17
Latitude Zone F
Longitude Zone 54
18
18
18C
18D
18E
18F
18G
18H
18J
18K
18L
18M
18N
18P
18Q
18R
18S
18T
18U
18V
18W
18X
WEST LONGITUDE (degrees)
13C
13D
13E
13F
13G
13H
13J
13K
13L
13M
13N
13P
13Q
13R
13S
13T
13U
13V
13W
14X
14
WEST LONGITUDE (degrees)
13X
13
UTM Grid Code "54F" Denotes:
12
12C
12D
12E
12F
12G
12H
12J
12K
12L
12M
12N
12P
12Q
12R
12S
12T
12U
12V
12W
12X
12
19
19C
19D
19E
19F
19G
19H
19J
19K
19L
19M
19N
19P
19Q
19R
19S
19T
19U
19V
19W
19X
19
60
60
21
21C
21D
21E
21F
21G
21H
21J
21K
21L
21M
21N
21P
21Q
21R
21S
21T
21U
21V
21W
21X
21
22
22C
22D
22E
22F
22G
22H
22J
22K
22L
22M
22N
22P
22Q
22R
22S
22T
22U
22V
22W
22X
22
23
23C
23D
23E
23F
23G
23H
23J
23K
23L
23M
23N
23P
23Q
23R
23S
23T
23U
23V
23W
23X
23
24
24C
24D
24E
24F
24G
24H
24J
24K
24L
24M
24N
24P
24Q
24R
24S
24T
24U
24V
24W
24X
24
30
25
25C
25D
25E
25F
25G
25H
25J
25K
25L
25M
25N
25P
25Q
25R
25S
25T
25U
25V
30
25W
25X
25
26
26C
26D
26E
26F
26G
26H
26J
26K
26L
26M
26N
26P
26Q
26R
26S
26T
26U
26V
26W
26X
26
27
27C
27D
27E
27F
27G
27H
27J
27K
27L
27M
27N
27P
27Q
27R
27S
27T
27U
27V
27W
27X
27
28
28C
28D
28E
28F
28G
28H
28J
28K
28L
28M
28N
28P
28Q
28R
28S
28T
28U
28V
28W
28X
28
UTM Grid Zones 32X, 34X, and 36X are not defined.
Equirectangular projection of the graticule.
Notes:
20
20C
20D
20E
20F
20G
20H
20J
20K
20L
20M
20N
20P
20Q
20R
20S
20T
20U
20V
20W
20X
20
30
30C
30D
30E
30F
30G
30H
30J
30K
30L
30M
30N
30P
30Q
30R
30S
30T
30U
30V
30W
30X
30
0
31W
0
31
31C
31D
31E
31F
31G
31H
31J
31K
31L
31M
31N
31P
31Q
31R
31S
31T
31U
32W
32
32
32C
32D
32E
32F
32G
32H
32J
32K
32L
32M
32N
32P
32Q
32R
32S
32T
32U
32V
31X
31
GREENWICH MERIDIAN
29
29C
29D
29E
29F
29G
29H
29J
29K
29L
29M
29N
29P
29Q
29R
29S
29T
29U
29V
29W
29X
29
GREENWICH MERIDIAN
33
33C
33D
33E
33F
33G
33H
33J
33K
33L
33M
33N
33P
33Q
33R
33S
33T
33U
33V
33W
33X
33
34
34C
34D
34E
34F
34G
34H
34J
34K
34L
34M
34N
34P
34Q
34R
34S
34T
34U
34V
34W
34
30
35
35C
35D
35E
35F
35G
35H
35J
35K
35L
35M
35N
35P
35Q
35R
35S
35T
35U
35V
30
35W
35X
35
36
36C
36D
36E
36F
36G
36H
36J
36K
36L
36M
36N
36P
36Q
36R
36S
36T
36U
36V
36W
36
37
37C
37D
37E
37F
37G
37H
37J
37K
37L
37M
37N
37P
37Q
37R
37S
37T
37U
37V
37W
37X
37
38
38C
38D
38E
38F
38G
38H
38J
38K
38L
38M
38N
38P
38Q
38R
38S
38T
38U
38V
38W
38X
38
39
39C
39D
39E
39F
39G
39H
39J
39K
39L
39M
39N
39P
39Q
39R
39S
39T
39U
39V
39W
39X
39
60
40
40C
40D
40E
40F
40G
40H
40J
40K
40L
40M
40N
40P
40Q
40R
40S
40T
40U
40V
41
41C
41D
41E
41F
41G
41H
41J
41K
41L
41M
41N
41P
41Q
41R
41S
41T
41U
41V
41W
41X
41
42
42C
42D
42E
42F
42G
42H
42J
42K
42L
42M
42N
42P
42Q
42R
42S
42T
42U
42V
42W
42X
42
43
43C
43D
43E
43F
44
44C
44D
44E
44F
44G
44H
44J
44K
44L
44M
44N
44P
44Q
44R
44S
44T
44U
44V
90
45
45C
45D
45E
45F
45G
45H
45J
45K
45L
45M
45N
45P
45Q
45R
45S
45T
45U
45V
90
45W
45X
45
46
46C
46D
46E
46F
46G
46H
46J
46K
46L
46M
46N
46P
46Q
46R
46S
46T
46U
46V
46W
46X
46
47
47C
47D
47E
47F
47G
47H
47J
47K
47L
47M
47N
47P
47Q
47R
47S
47T
47U
47V
47W
47X
47
48X
48
48
48C
48D
48E
48F
48G
48H
48J
48K
48L
48M
48N
48P
48Q
48R
48S
48T
48U
48V
48W
EAST LONGITUDE (degrees)
43G
43H
43J
43K
43L
43M
43N
43P
43Q
43R
43S
43T
43U
43V
44W
44X
44
EAST LONGITUDE (degrees)
43W
43X
43
© 2008 Joe Depner
60
40W
40X
40
49
49C
49D
49E
49F
49G
49H
49J
49K
49L
49M
49N
49P
49Q
49R
49S
49T
49U
49V
49W
49X
49
50
51
51C
51D
51E
51F
51G
51H
51J
51K
51L
51M
51N
51P
51Q
51R
51S
51T
51U
51V
51W
120
50C
50D
50E
50F
50G
50H
50J
50K
50L
50M
50N
50P
50Q
50R
50S
50T
50U
50V
50W
51
51X
120
50X
50
52
52C
52D
52E
52F
52G
52H
52J
52K
52L
52M
52N
52P
52Q
52R
52S
52T
52U
52V
52W
52X
52
Figure 2.6: Global Distribution of UTM Longitude Zones and Latitude Zones
54F
Example:
10
10C
10D
10E
10F
10G
10H
10J
10K
10L
10M
10N
10P
10Q
10R
10S
10T
10U
10V
10W
11
11X
120
10X
10
53
53C
53D
53E
53F
53G
53H
53J
53K
53L
53M
53N
53P
53Q
53R
53S
53T
53U
53V
53W
53X
53
54
54C
54D
54E
54F
54G
54H
54J
54K
54L
54M
54N
54P
54Q
54R
54S
54T
54U
54V
54W
54X
54
55
56
56C
56D
56E
56F
56G
56H
56J
56K
56L
56M
56N
56P
56Q
56R
56S
56T
56U
56V
56W
150
55C
55D
55E
55F
55G
55H
55J
55K
55L
55M
55N
55P
55Q
55R
55S
55T
55U
55V
55W
56
56X
150
55X
55
57
57C
57D
57E
57F
57G
57H
57J
57K
57L
57M
57N
57P
57Q
57R
57S
57T
57U
57V
57W
57X
57
58
58C
58D
58E
58F
58G
58H
58J
58K
58L
58M
58N
58P
58Q
58R
58S
58T
58U
58V
58W
58X
58
59
59C
59D
59E
59F
59G
59H
59J
59K
59L
59M
59N
59P
59Q
59R
59S
59T
59U
59V
59W
59X
59
N
P
Q
R
S
T
U
V
60
C
D
E
F
G
H
J
K
L
180
60C
60D
60E
60F
60G
60H
60J
60K
60L
60M M
60N
60P
60Q
60R
60S
60T
60U
60V
60W W
X
180
60X
60
80
72
64
56
48
40
32
24
16
08
00
08
16
24
32
40
48
56
64
72
84
EQUATOR
32
24
16
08
00
08
16
24
32
40
48
56
64
72
84
1
NORTH LATITUDE (degrees)
SOUTH LATITUDE (degrees)
31V
180
16 CHAPTER 2. THE UTM GRID
SOUTH LATITUDE (degrees)
2.2. LONGITUDE AND LATITUDE ZONES
2.2.5
17
Points on Zone Boundaries
The UTM zone specifications for points on the boundaries between adjacent zones are nonunique. These points fall into one of the following categories: • points on the boundaries (meridians) between two adjacent UTM longitude zones; • points on the boundaries (parallels) between two adjacent UTM latitude zones, including - points not on the equator (this case is only relevant in those situations where the latitude-zone form of the UTM coordinate specification is used); - points on the equator (i.e., the boundary between the northern and southern hemispheres); and • points where two of the above boundaries intersect. Every point on the boundary between two or more adjacent zones can be considered a member of all of the corresponding adjacent zones. In these cases, the UTM zone specification corresponding to any of the adjacent zones can be used without introducing positional ambiguity. The following examples illustrate the concept. Example: Point on Boundary between Two Adjacent Longitude Zones Problem: Determine the UTM latitude and longitude zones corresponding to the point whose geographic coordinates are 19◦ S. lat. and 120◦ W. lon. Solution: Refer to Figure 2.6. The point is in both of UTM zones 10K and 11K because it lies at their intersection.
Example: Point on Boundary between Two Adjacent Latitude Zones Problem: Determine the UTM latitude and longitude zones corresponding to the point whose geographic coordinates are 72◦ N. lat. and 86◦ E. lon. Solution: Refer to Figure 2.6. The point is in both of UTM zones 45W and 45X, because it lies at their intersection.
Example: Point on Boundary between Three Adjacent Zones Problem: Determine the UTM latitude and longitude zones corresponding to the point whose geographic coordinates are 72◦ N. lat. and 33◦ E. lon. Solution: Refer to Figure 2.6. The point is in all three of UTM zones 36W, 35X and 37X, because it lies at their intersection.
18
CHAPTER 2. THE UTM GRID
Example: Point on Boundary between Four Adjacent Zones Problem: Determine the UTM latitude and longitude zones corresponding to the point whose geographic coordinates are 24◦ S. lat. and 162◦ W. lon. Solution: Refer to Figure 2.6. The point is in all four of UTM zones 3J, 3K, 4J, and 4K, because it lies at their intersection.
Example: Point(s) with Apparently Different UTM Zone Specifications Problem: Field notes by one field technician report the location of a particular monitoring station as UTM zone 11S. Field notes by a second technician report the location of the same monitoring station as UTM zone 12T. The station has not been moved. Is it possible that both technicians are correct? Solution: Refer to Figure 2.6. UTM zones 11S and 12T intersect at a corner point, which therefore lies in both zones. Therefore, it’s possible that both technicians are correct.
Example: Point(s) with Apparently Different UTM Zone Specifications Problem: Field notes by one technician report the location of a particular monitoring station as UTM zone 12S. Field notes by a second technician report the location of the same monitoring station as UTM zone 12U. The monitoring station has not been moved. Is it possible that both technicians are correct? Solution: Refer to Figure 2.6. UTM zones 12S and 12U do not intersect, so no points lie in both zones. Thus, in this case it’s not possible that both technicians are correct; either one or both are incorrect.
2.3
Easting and Northing Coordinates
Every UTM longitude zone has a particular Cartesian coordinate system associated with it (i.e., a local xy coordinate system). The UTM easting and northing coordinates are the x and y coordinates, respectively, of this system. UTM easting and northing coordinates are numerical, and are reported as base-ten integers (Arabic numerals, no decimals or fractions). The numerical values are written without commas, spaces, or decimal points; and in non-exponential notation (e.g., neither in engineering notation nor in scientific notation). The coordinate definitions (see below) imply that the numerical values are nonnegative. Therefore, the coordinates are written as unsigned numbers (i.e., without “+” or “−” signs). UTM easting and northing coordinates are reported in units of meters. Some non-technical publications on civilian navigation and the GPS use a nonstandard convention in which the easting and northing coordinates are reported in kilometers, but the use of kilometers doesn’t conform to the standard defined by DMA [1989] and therefore isn’t recommended.
2.3. EASTING AND NORTHING COORDINATES
19
Table 2.3: Minimum and Maximum UTM Easting Coordinates
Latitude
Horizontal Datum
UTM Easting Coordinates (meters) Minimum
Maximum
Difference
NAD 27
166 018
833 982
667 964
NAD 83
166 022
833 978
667 956
NAD 27
441 866
558 134
116 268
NAD 83
441 868
558 132
116 264
NAD 27
465 004
534 996
69 992
NAD 83
465 006
534 994
69 988
0◦ N./S.
80◦ N./S.
84◦ N./S.
Source: Decimal values of minimum and maximum easting coordinates were obtained from National Geodetic Survey [2007], and then were rounded up and down, respectively, to the nearest integer coordinate corresponding to points within a UTM longitude zone.
2.3.1
Easting Coordinates
The easting (x) coordinate increases continuously as one moves eastward. Each longitude zone has a central meridian midway between its two bounding meridians (See Figure 2.7). The central meridian of each longitude zone is assigned the easting coordinate 500000m (i.e., x = 500 000 m). Consequently, the easting coordinate has the following characteristics: • It’s local to the corresponding particular longitude zone; • It’s non-negative; and • It’s also referred to as false easting. Table 2.3 lists the minimum and maximum UTM easting coordinates for regular (6◦ wide) UTM longitude zones, at various latitudes and for the two most commonly used datums in the U.S. The full range is realized only at the equator, where the UTM longitude zones are widest. At higher latitudes the range generally is narrower, because the UTM longitude zones narrow with increasing latitude due to convergence of the meridians. The ranges in Table 2.3 correspond to regular UTM longitude zones; for irregular zones the ranges differ from these. From these results it follows that the UTM easting coordinate of every point within every regular UTM longitude zone is a six-digit integer. It turns out that this is also true for irregular UTM longitude zones (see Appendix A).
2.3.2
Northing Coordinates
The northing (y) coordinate increases continuously as one moves northward. In the northern hemisphere the equator is assigned the northing coordinate 0mN (i.e., y = 0 m). In the southern
20
CHAPTER 2. THE UTM GRID
Figure 2.7: Regular UTM Longitude Zone in Northern Hemisphere
2.3. EASTING AND NORTHING COORDINATES
21
Table 2.4: Minimum and Maximum UTM Northing Coordinates
Hemisphere
Horizontal Datum
UTM Northing Coordinates (meters) Minimum
Maximum
Difference
NAD 27
0
9 328 895
9 328 895
NAD 83
0
9 329 005
9 329 005
NAD 27
1 117 046
10 000 000
8 882 954
NAD 83
1 116 916
10 000 000
8 883 084
Northern
Southern
Source: Decimal values of minimum and maximum northing coordinates were obtained from National Geodetic Survey [2007], and then were rounded up and down, respectively, to the nearest integer coordinate corresponding to points within a UTM longitude zone.
hemisphere the equator is assigned the northing coordinate 10000000mN (i.e., y = 10 000 000 m). Consequently the northing coordinate has the following characteristics: • It’s local to the corresponding particular hemisphere. • It’s non-negative. • It’s also referred to as false northing. Table 2.4 lists the minimum and maximum UTM northing coordinates for any given UTM longitude zone, for the two most commonly used datums in the U.S. The full range isn’t realized near the bounding meridians of each zone, due to meridian convergence. From these results it follows that the UTM northing coordinate of every point within every UTM longitude zone is a one- to seven-digit integer (see Appendix B also).
2.3.3
Easting and Northing Coordinate Specifications
The conventional format for reporting UTM easting and northing coordinates is rather specific. The numerical value of the coordinate is written first (leftmost), immediately followed by the lowercase “m” abbreviation for meters, with or without a single space separating the two. An uppercase “E” or “N” immediately follows the “m”, to indicate whether the coordinate is an easting or a northing, respectively. The following examples illustrate the convention.
22
CHAPTER 2. THE UTM GRID
Examples: Conventional Format for Reporting UTM Easting Coordinates “500000mE” or “500000 mE” (e.g., the central meridian) “566785mE” or “566785 mE” “177003mE” or “177003 mE” “792324mE” or “792324 mE”
Examples: Conventional Format for Reporting UTM Northing Coordinates “0mN” or “0 mN” (e.g., the equator) “353mN” or “353 mN” “8315466mN” or “8315466 mN” “10000000mN” or “10000000 mN” (e.g., the equator)
2.3.4
Coordinate Gridlines
Within each UTM longitude zone, two sets of gridlines are defined – a set of UTM easting gridlines and a set of UTM northing gridlines. The easting gridlines are orthogonal to the northing gridlines. Each set is described below. Within each UTM longitude zone, the UTM easting gridlines form a set of contour lines. Each UTM easting gridline connects those points on the earth’s surface that have the same UTM easting coordinate. The easting gridline corresponding to 500 000 mE (i.e., the central meridian) extends from 80◦ S. lat. to 84◦ N. lat. As one moves poleward from the equator, each longitude zone becomes narrower, so the easting gridlines corresponding to the more extreme easting coordinates don’t extend as far poleward as the central meridian does (see Figure 2.7). Rather, these easting gridlines only extend northward to the points where they intersect the zone’s bounding meridians. Within any particular UTM longitude zone, the easting gridlines never intersect. Within any particular UTM longitude zone, the UTM northing gridlines also form a set of contour lines. Each UTM northing gridline connects those points on the earth’s surface that have the same UTM northing coordinate. The northing gridlines within each longitude zone extend from one bounding meridian to the other, across 6◦ of longitude. As one moves poleward from the equator, the longitude zones become narrower, so the northing gridlines become shorter (see Figure 2.7). Within any particular UTM longitude zone, the northing gridlines never intersect.
2.4
UTM Coordinate Specifications
Unambiguous determination of position using the UTM grid system generally requires specification of the following five elements: • horizontal datum, • UTM longitude zone, • hemisphere, or UTM latitude zone,
2.4. UTM COORDINATE SPECIFICATIONS
23
• UTM easting coordinate, and • UTM northing coordinate. Example: Conventional Format for UTM Coordinate Specification Problem: Consider the following UTM coordinate specification for a point: NAD 83, UTM Zone 11, N. hemis., 450300mE, 5291192mN What is the horizontal datum? In which UTM longitude zone is the point located? In which hemisphere is the point located? Solution: The horizontal datum is the North American Datum of 1983 (NAD 83). The location is within UTM longitude zone 11. The location is in the northern hemisphere.
In some situations it may be acceptable to omit the datum, zone or hemisphere from the specification, but only if the omitted elements are clearly implied by the context. Why is it necessary to specify all five elements of the UTM coordinate specification? Let’s consider each element in turn. The horizontal datum effectively defines the position and orientation of the graticule, relative to which each UTM grid system is defined. Therefore, the horizontal datum is an essential element in the definition of the UTM grid system. UTM coordinates defined with respect to one horizontal datum differ from those defined with respect to another horizontal datum. For instance, the UTM grid system defined with respect to the North American Datum of 1927 (NAD 27) and that defined with respect to NAD 83 are different grid systems. The two grid systems bear a superficial resemblance to one another because they’re structured similarly (i.e., both use 6◦ longitude zones and 8◦ latitude zones, etc.). However, they’re different grid systems because the positions and orientations of the graticules for the two systems differ. This distinction isn’t a mere technicality. For instance, according to the Department of the Army [2001], UTM coordinates for the same point, but corresponding to different horizontal datums, may differ by as much as 900 m. If the hemisphere designation (or latitude-zone designation) is omitted from the UTM coordinate specification, then the point’s position is effectively determined only to within two possible locations – one in each of the northern and southern hemispheres. The one exception to this is points that lie on the equator. The northing coordinates of such points will be either 0 mN (if referenced to the northern hemisphere) or 10 000 000 mN (if referenced to the southern hemisphere). In either case, it would be clear that the point lies on the equator because the minimum UTM northing coordinate in the southern hemisphere is greater than 0 mN and the maximum northing coordinate in the northern hemisphere is less than 10 000 000 mN. Therefore, locations of points on the equator can be specified as either northern hemisphere or southern hemisphere without introducing ambiguity. If the longitude zone is omitted from the UTM coordinate specification, then the point’s longitudinal position is effectively determined only to within 60 possible locations – one in each longitude zone.
24
CHAPTER 2. THE UTM GRID
If the easting coordinate is omitted from the UTM coordinate description, then the point’s easting position is effectively determined only to within a longitude zone. Refer to Table 2.3. Each UTM longitude zone is almost 668 000 m wide at its widest part (i.e., at the equator). In the northern hemisphere, each UTM longitude zone is almost 70 000 m wide at its narrowest part (i.e., at 84◦ N. lat.). In the southern hemisphere each longitude zone is over 116 000 m wide at its narrowest part (i.e., at 80◦ S. lat.). That’s a lot of imprecision.
If the northing coordinate is omitted from the coordinate description, then the point’s northing position is effectively determined only to within a hemisphere. As shown in Table 2.4, UTM longitude zones in the southern hemisphere are over eight million meters long, while those in the northern hemisphere are over nine million meters long (measured from northern to southern boundaries). Again, that’s a great deal of imprecision. The precision can be increased substantially by specifying the latitude zone (see example below).
Example: Increasing the Precision of the Northing Coordinate by Specifying the UTM Latitude Zone Problem: How imprecise (roughly) is the UTM northing coordinate if the UTM latitude zone is included in the coordinate specification, but the numerical value of the northing coordinate is omitted? Assume the earth is spherical. Solution: If the shape of the earth is approximately spherical, then each minute of latitude is approximately equivalent to one nautical mile, or 6076 feet [U.S. Air Force, 2001]. Actually the length of a minute of latitude varies somewhat with latitude, because the earth is more closely approximated by an ellipsoid than by a sphere. However, to model the earth as an ellipsoid requires substantially more mathematical effort than this example requires. Latitude zones C through W are 8◦ wide in the direction of the northing coordinate, so within those zones the imprecision of the northing coordinate is approximately 60’ lat. 6076 ft 0.3048 m ◦ (8 lat.) ≈ 890 000 m 1◦ lat. 1’ lat. ft UTM latitude zone X is 12◦ wide in the direction of the northing coordinate, so within zone X the imprecision of the northing coordinate is approximately 60’ lat. 6076 ft 0.3048 m 12◦ lat. ≈ 1 300 000 m 1◦ lat. 10 lat. ft which is almost 1.5 times that of latitude zones C through W.
2.5. MAP PROJECTIONS, DATUMS, AND THE GRATICULE
25
Example: Preliminary Screening of UTM Coordinate Specifications for Out-of-Bounds Errors Problem: Quickly check each of the following UTM coordinate specifications for out-ofbounds errors: NAD 27, UTM Zone 10, N. hemis., 150300mE, 9951192mN NAD 83, UTM Zone 12, S. hemis., 751334mE, 1116907mN NAD 83, UTM Zone 19, N. hemis., 833980mE, 192mN NAD 27, UTM Zone 17, N. hemis., 150300mE, 34602mN NAD 27, UTM Zone 18, N. hemis., 237811mE, 9328904mN NAD 83, UTM Zone 63, N. hemis., 623300mE, 9328904mN Specify which element or elements, if any, are incorrect and explain why. Solution: For a quick, preliminary screening of UTM coordinate data, use the “0◦ N./S.” section of Table 2.3, and Table 2.4. Beware, however, that this approach will detect only extreme cases of out-of-bounds errors. More refined screening may be required to detect all out-of-bounds errors. “NAD 27, UTM Zone 10, N. hemis., 150300mE, 9951192mN” is incorrect. The easting coordinate is lower than the lower limit given in the “0◦ N./S.” section of Table 2.3, and the northing coordinate exceeds the upper limit listed in Table 2.4. “NAD 83, UTM Zone 12, S. hemis., 751334mE, 1116907mN” is incorrect. The northing coordinate is out of range (too low). “NAD 83, UTM Zone 19, N. hemis., 833980mE, 192mN” is incorrect. The easting coordinate is out of range (too high). “NAD 27, UTM Zone 17, N. hemis., 150300mE, 34602mN” is incorrect. The easting coordinate is out of range (too low). “NAD 27, UTM Zone 18, N. hemis., 237811mE, 9328904mN” is incorrect. The northing coordinate is out of range (too high). “NAD 83, UTM Zone 63, N. hemis., 623300mE, 9328904mN” is incorrect. The zone number is too high (i.e., UTM zone 63 does not exist).
2.5
Map Projections, Datums, and the Graticule
The UTM grid system applies the secant form of the transverse Mercator projection to each UTM longitude zone. The two standard lines (also called lines of secancy) within each longitude zone are approximately 180 000 m east and west of the central meridian; these have coordinates of approximately 320 000 mE and 680 000 mE, respectively [DMA, 1990]. The scale factor of the projection varies with latitude and longitude within each UTM longitude zone; for mathematical details see DMA [1989]. The spatial variation of the scale factor within each UTM longitude zone can be summarized as follows [DMA, 1990]: • The scale factor is 1.000 00 at both of the standard lines. • The scale factor decreases as one moves inward from either of the two standard lines toward
26
CHAPTER 2. THE UTM GRID the central meridian, where it’s equal to 0.9996. • The scale factor increases as one moves outward from either of the two standard lines toward the nearest bounding meridian. Where the bounding meridians intersect the equator, the scale factor is approximately equal to 1.0010.
The projection parameters are based on the particular horizontal datum chosen. The horizontal datums most commonly used in North America are the following: • the North American Datum of 1927 (NAD 27); • the North American Datum of 1983 (NAD 83); and • the World Geodetic System 1984 (WGS 84). All three of these are based on an ellipsoidal, rather than spherical, representation of the globe. For most practical purposes then, the UTM grid system uses the ellipsoidal form of the transverse Mercator projection. DMA [1989] gives mathematical equations for ellipsoid parameters. The UTM grid declination (i.e., convergence of the meridians) varies with both latitude and longitude within each UTM longitude zone; for mathematical details see DMA [1989]. The spatial variation of grid declination within each UTM longitude zone can be summarized as follows [DMA, 1989]: • The grid declination is zero at the central meridian. • The (absolute) grid declination increases with distance from the central meridian. • For those areas on either side of the central meridian, the (absolute) grid declination increases with distance from the equator. Because UTM grid north has a slight easterly or westerly declination, except right at the central meridian of each longitude zone, one should never use map neatlines or the graticule (meridians of longitude and parallels of latitude) as a substitute for UTM gridlines. Doing so introduces error.
2.6
Chapter Summary
Refer to Table 2.5. Each of the 60 regular UTM longitude zones has a separate UTM map projection associated with it, giving a total of 60 separate UTM map projections. Each regular zone has two local Cartesian (xy) coordinate systems associated with it – one for each of the northern and southern hemispheres – giving a total of 120 separate member coordinate systems. Each of the six irregular UTM zones (31V, 32V, 31X, 33X, 35X, 37X) has a separate UTM map projection associated with it, giving a total of six additional map projections. Each irregular zone has one local Cartesian coordinate system associated with it, giving a total of six additional member coordinate systems. Finally, consider the entire set of UTM longitude zones, both regular and irregular, collectively. The UTM grid system uses 66 separate UTM projections and comprises 126 member coordinate systems. The UTM grid system is, in effect, a hybrid system. It combines elements of the geographic coordinate system with numerous local Cartesian coordinate systems. For instance, both the UTM longitude zones and the local easting/northing coordinate system within each longitude zone are defined in terms of the graticule. This is why it’s important to master the principles of geographic and Cartesian (xy) coordinate systems before learning the UTM grid system.
2.6. CHAPTER SUMMARY
27
Table 2.5: Summary of UTM Map Projections and Local Coordinate Systems
Zone Group
Member Zones
Number of Zones
Number of Map Projections
Number of Local Coordinate Systems
Per Zone
Group Total
Per Zone
Group Total
Regular
01 - 60
60
1
60
2
120
Irregular
31V, 32V, 31X, 33X, 35X, 37X
6
1
6
1
6
All
01 - 60, 31V, 32V, 31X, 33X, 35X, 37X
66
1
66
–
126
28
CHAPTER 2. THE UTM GRID
Chapter 3
The UTM Grid and USGS Topographic Maps In this chapter we occasionally refer to the publication date for a particular map product (e.g., paper map, digital map). The publication date is the date on which the map product was released to the public. If a map has been revised, then the publication date of the original edition is earlier than that of the revised edition. If the map has been revised more than once, then each revision will have its own publication date.
3.1
Map Elements Supporting Use of the UTM Grid
The U.S. Geological Survey (USGS) produces topographic quadrangle maps and related products for public distribution. The maps have various elements that facilitate use of the UTM grid system, including the following: • horizontal datum identifier, • UTM longitude zone identifier, • UTM grid tick marks and coordinate labels, • UTM grid declination information, and • UTM gridlines (some maps).
3.1.1
Horizontal Datum Identifier
The horizontal datum is specified in the explanatory text on the map collar, typically on the left-hand side of the lower margin (see Figures 3.1 and 3.2).
3.1.2
UTM Longitude Zone Identifier
The UTM longitude zone is specified in the explanatory text on the map collar, typically on the left-hand side of the lower margin (see Figures 3.1 and 3.2). 29
30
CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
Figure 3.1: UTM Grid Information – Older Map. Source: USGS [1986]
Figure 3.2: UTM Grid Information – Newer Map. Source: USGS [1987]
3.1. MAP ELEMENTS SUPPORTING USE OF THE UTM GRID
3.1.3
31
UTM Grid Tick Marks and Coordinate Labels
In general, a grid tick mark is a short, straight-line segment that marks the location of a gridline. Typically, grid tick marks are placed at points where gridlines intersect other lines, such as orthogonal gridlines (e.g., where an easting gridline crosses a northing gridline) or map neatlines.
UTM grid tick marks are displayed on USGS quadrangle topographic maps published since 1959, and on many quadrangles published before 1959 [Cole, 1977]. The UTM grid tick marks are displayed on the map collar adjacent to the neatline. Each tick mark indicates the point where a UTM gridline intersects the map neatline.
The distance between adjacent grid tick marks is called the grid-tick interval. On each map sheet the grid-tick interval is constant (uniform). All USGS quadrangles use either 1000-meter or 5000-meter grid-tick intervals [USGS, 2001]. In addition, on USGS quadrangles the UTM grid tick mark coordinates are positive integer multiples of 1000 m. The grid-tick interval is specified in the explanatory text displayed on the map collar (see Figures 3.1 and 3.2). On some maps, but not all, the color of the grid tick marks also is specified in the explanatory text on the map collar (e.g., compare Figures 3.1 and 3.2). Typically, UTM grid tick marks are displayed in blue on USGS quadrangle topographic maps.
32
CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
Example: Interpretation of Displayed Explanatory Text Background: The following explanatory text is displayed on the left-hand side of the lower margin of the Nine Mile Falls Quadrangle, Washington, 7.5-minute topographic map [USGS ; 1973b, photorevised 1986]: Mapped, edited, and published by the Geological Survey Control by USGS and NOS/NOAA Topography by photogrammetric methods from aerial photographs taken 1972. Field checked 1973 Underwater contours by Washington Water Power Co. Projection and 10,000-foot grid ticks: Washington coordinate system, north zone (Lambert conformal conic) 1000-meter Universal Transverse Mercator grid ticks, zone 11, shown in blue. 1927 North American datum To place on the North American Datum 1983, move the projection lines 15 meters north and 80 meters east as shown by dashed corner ticks There may be private inholdings within the boundaries of the National or State reservations shown on this map Problem: What is the horizontal datum? What is the UTM longitude zone? What is the UTM grid-tick interval? What color are the UTM grid tick marks? Solution: The horizontal datum is NAD 27. The UTM longitude zone is 11. The UTM grid-tick interval is 1000 m. The UTM grid tick marks are blue.
3.1. MAP ELEMENTS SUPPORTING USE OF THE UTM GRID Example: Explanatory Text Background: The following explanatory text is displayed in the lower margin of the Eagle Cap Quadrangle, Oregon, 15-minute topographic map [USGS, 1954]: Mapped, edited, and published by the Geological Survey Control by USGS and USC&GS Topography from aerial photographs by multiplex methods Aerial photographs taken 1953. Advance field check 1954 Polyconic projection. 1927 North American datum 10,000-foot grid based on Oregon coordinate system, north zone 1000-foot Universal Transverse Mercator grid ticks, zone 11, shown in blue To place on the North American Datum 1983 move the projection lines 17 meters north and 79 meters east There may be private inholdings within the boundaries of the National or State reservations shown on this map Problem: What is the horizontal datum? What is the UTM longitude zone? What UTM grid-tick interval is reported in the explanatory text? Measure the map distance between adjacent UTM grid tick marks, and use the map scale to convert the map distance to the corresponding horizontal ground distance. What is the measured grid-tick interval? What UTM grid-tick interval is reported in the USGS UTM fact sheet [USGS, 2001]? Are the three results obtained above for the UTM grid-tick interval consistent? If not, suggest a possible explanation for any discrepancies. Solution: The horizontal datum is NAD 27. The UTM longitude zone is 11. The grid-tick interval reported in the explanatory text is 1000 feet. The measured UTM grid-tick interval is 1000 meters. The UTM grid-tick interval reported in the fact sheet is 1000 meters [USGS, 2001]. The UTM grid-tick interval reported in the explanatory text differs from both (1) that measured on the map and (2) that reported in the fact sheet. Apparently the grid-tick interval specification “1000-foot Universal Transverse Mercator grid ticks, . . . ” is a misprint.
33
34
CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
A grid tick mark coordinate label is displayed on the map collar adjacent to each corresponding grid tick mark, except in those cases where it would interfere with the display of other map information (e.g., the name of an adjoining map sheet, commonly displayed in the margin adjacent to the neatline). In such cases the coordinate value is clearly implied from the grid-tick interval and the coordinate labels of adjacent grid tick marks. Reading long strings of numerical digits can be tiring and prone to errors. To facilitate their reading, the USGS displays the UTM grid tick mark labels on the map collar using two different font sizes. The units abbreviation (i.e., “m.”), and the three trailing (i.e., rightmost) digits of the numerical string are displayed in a small font. The two adjacent digits on the left of these are displayed in a large font, and the one or two leftmost digits are displayed in the small font.
Examples: Font-Size Convention for UTM Grid Tick Mark Coordinate Labels Problem: Suppose UTM grid tick mark labels for the following UTM grid coordinates are displayed in full on the collars of USGS topographic maps: 5289000 mN 9980000mN 2550000mN 360000 mE 604000mE 499000 mE How would the labels appear? Solution: The labels would appear as follows, respectively: 52 89000m. N 99 80000m. N 25 50000m. N 3 60000m. E 6 04000m. E 4 99000m. E
The difference in font size makes it easier to visually distinguish the various digits of the string. The pair of digits displayed in the larger font are called the principal digits. The left and right principal digits correspond to the ten-thousands’ place and the thousands’ place, respectively. Coordinate labels for the UTM grid tick marks nearest to the northwest and southeast corners are displayed in full as described above [USGS, 2001]. Consequently, on each of the four map margins (upper, lower, left, right), one UTM grid tick mark label is displayed in full. The remaining grid tick mark coordinate labels are abbreviated by truncation. The abbreviated coordinate labels omit the following elements: • the three trailing zero digits (i.e., “000”), • the units designator symbol (i.e., “m” or “m.”), and
3.1. MAP ELEMENTS SUPPORTING USE OF THE UTM GRID
35
• the direction designator (“E” for easting, “N” for northing). Example: Abbreviations for Grid Tick Mark Coordinate Labels Problem: Suppose UTM grid tick mark labels for the following UTM grid coordinates are displayed in abbreviated form on the collars of USGS topographic maps: 5274000 mN 748000mN 2110000mN 355000 mE 625000mE 589000 mE How would the labels appear? Solution: The labels would appear as follows, respectively: 52 74 7 48 21 10 3 55 6 25 5 89
3.1.4
UTM Grid Declination Information
The map’s declination information, which includes information on grid declination and magnetic declination, is displayed in the lower margin. The declination information is conveyed in one of two ways, depending on the map’s publication date. In older edition maps, such as those published prior to 1988, the declination information usually is conveyed via a declination diagram that appears near the lower left-hand corner [Burns and Burns, 2004]. Figure 3.1 shows an example of such a diagram. The declination diagram shows three arrows – to indicate the directions of true north, grid north, and magnetic north – and displayed numerical values of grid declination and magnetic declination. The declination diagram uses the abbreviations “GN” and “MN”, respectively, to denote grid north and magnetic north; the diagram uses a five-pointed star to indicate the direction of true north. A note appears immediately below the diagram, indicating that the declination values indicated in the diagram are for the center of the map; away from the center of the map the declination values vary somewhat from these. In newer edition maps, such as those published in 1988 or later, the UTM grid declination usually is indicated by a statement in the map’s explanatory text rather than a diagram [Burns and Burns, 2004]. Figure 3.2 shows a typical example of such a statement.
3.1.5
UTM Gridlines
Some USGS topographic maps (e.g., newer edition quadrangles), but not all, display UTM gridlines. For instance, USGS topographic maps published in 1988 or later usually display
36
CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
UTM gridlines, while those published prior to 1988 usually do not [Burns and Burns, 2004]. When present, the gridlines are shown in black. Gridlines used for UTM coordinate determination are one of two types – fully displayed or marked-only. A fully displayed gridline is one that’s displayed directly on the map (e.g., either drawn or printed directly on a paper map). A marked-only gridline is one whose position is indicated by the corresponding pairs of grid tick marks displayed on opposite margins of the map sheet; the line itself isn’t explicitly displayed. USGS topographic maps with UTM gridlines displayed on them use the same interval as the grid-tick interval. For example, if the grid-tick interval is 1000 meters, then the UTM easting and northing gridlines are displayed at intervals of 1000 meters. Displaying the UTM gridlines on a map is useful for the following purposes: • Determining the UTM easting and northing coordinates of a known point on a topographic map. See “Determining the UTM Coordinates of a Point on a Map” (Section 3.3). • Locating a point on a topographic map when its UTM coordinates are known. See “Plotting a Point with Known UTM Coordinates on a Map” (Section 3.4). Specifically, displayed UTM gridlines make it possible to quickly and accurately complete the following map-based tasks: • Visually locate those easting and northing gridlines that are nearest to a point of interest (POI). • Measure the projected (perpendicular) distance from the nearest easting and northing gridlines, to the POI, using only a corner ruler (Figure 3.3). Thus, displayed (printed) UTM gridlines are especially convenient for using maps in the field, where a drafting table and drafting instruments may not be available.
3.2
Using Topographic Maps Without Preprinted Gridlines
When one has to work with a paper map that doesn’t have UTM gridlines printed on it, at least three options are available: • Visually estimate gridline positions using the grid tick marks. • Physically overlay a scaled UTM grid transparency on the map. • Physically draw UTM gridlines on the map. Each of these options is discussed below.
3.2.1
Visual Estimation of Gridline Positions
Visually estimating (i.e., “eyeballing”) gridline positions using the corresponding pairs of grid tick marks on opposite margins of the map sheet is the least accurate of the methods considered here. It’s only suitable for obtaining very rough estimates, so this method generally isn’t recommended. However, this method is the least time-consuming, and with practice one can become more accurate.
3.2. USING TOPOGRAPHIC MAPS WITHOUT PREPRINTED GRIDLINES
Figure 3.3: A Basic Corner Ruler
37
38
CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
3.2.2
Overlaying a UTM Grid Transparency
A UTM grid transparency is a sheet of transparent synthetic material (e.g., acetate film, polyethylene film), with sets of easting and northing gridlines printed or drawn on it. One overlays the transparency on the map so that the gridlines on the transparency align with the corresponding pairs of UTM grid tick marks on the map collar. Then one temporarily fastens the transparency to the map sheet so that the two remain fixed in relative position. A separate transparency is required for each map scale. Accurately plotting features on a map while using a transparency is impractical, but one can read existing map features and measure their UTM coordinates. Using a transparency has the advantage that it doesn’t require permanent modification of the map sheet. Grid transparencies can be a variety of sizes. A full-size transparency covers the entire map image, whereas a small transparency might only be 45 square centimeters (7 square inches). For serious map work, where high accuracy is desired, a full-size transparency should be used; the large size allows one to accurately align the transparency grid with the map’s UTM grid tick marks. Using a full-size grid transparency in the field can be awkward, for the following reasons: • Large film sheets are awkward to store and transport. Much more so than paper maps, film sheets are damaged by folding, so transparencies should be stored flat or loosely rolled. • Unsecured large film sheets are easily disturbed by wind gusts, which can easily dislodge or transport them. • Using a large film sheet requires a large level work surface, which may not be available. Thus, using full-size transparencies is recommended for office use only. When using a transparency in the field, take steps to protect it from unnecessary exposure to abrasive debris and degrading UV radiation. UTM grid transparencies can be constructed or purchased. Construction involves drawing or printing UTM gridlines on transparent film. Transparent film can be purchased in sheets and rolls of various widths, thicknesses, and compositions. Constructing a transparency by drawing the gridlines directly on film requires drafting equipment, and is time-consuming and subject to error. Furthermore, constructing additional transparencies (e.g., to replace worn transparencies) requires repeating the drawing effort. Constructing a transparency by printing gridlines on film involves several steps. First the film is acquired, measured, and cut to size. Then an electronic image of the gridline overlay is acquired. Finally, the gridline overlay image is printed onto the film. An electronic image of the gridline overlay can be acquired in various ways, including the following: • Download a grid image file from the worldwide web. For example, New Mexico Bureau of Geology and Mineral Resources [2007] makes some image files for small overlays available to the public in Portable Document Format (PDF). • Draw the gridlines directly on a sheet of paper or film, scan the drawing, and store it as an image file. • Use basic computer drafting/graphing software to draw the grid, and store the drawing as an image file. This is the preferred alternative, because it’s possible to draw the gridlines more accurately and consistently using computer software than by hand.
3.2. USING TOPOGRAPHIC MAPS WITHOUT PREPRINTED GRIDLINES
39
Additional, identical transparencies can be printed at will using the saved overlay image file without repeating the drawing effort. Once a gridline image file is created for one map scale (e.g., 1:24 000), it’s relatively easy to make a similar additional image file suitable for a different map scale (e.g., 1:100 000) by copying the original image file and then editing the copy.
3.2.3
Drawing UTM Gridlines on Maps
UTM gridlines can be drawn on a map in the drafting room prior to entering the field. Once the gridlines are drawn, no transparent overlay is required, nor is one present to interfere with plotting. Thus, using a map with pre-drawn gridlines is a convenient and reliable option for field work. In some cases, permanently modifying a map sheet by drawing UTM gridlines on it might be considered undesirable, for the following reasons: • The gridlines change the map’s overall appearance, possibly degrading its aesthetic appeal. • The gridlines might make it difficult or impossible to read small graphic symbols, alphanumeric labels or other details, thus degrading the map’s usefulness. Drawing UTM gridlines on maps should be completed prior to entering the field, and under controlled conditions, so that the gridlines can be drawn neatly and accurately. Select a work area that has good lighting and good ventilation, with a comfortable, adjustable chair and a large, clean work surface such as a professional drafting table. Wash your hands before you handle the map sheet and drafting equipment. Then fasten the map sheet securely to the drafting table. If you have access to professional drafting equipment, use a good T-square as your straightedge. Otherwise, use a straightedge with a thin cork backing so that (1) the straightedge doesn’t slip during use, and (2) the lower surface of the straightedge is slightly elevated off the map surface so the ink doesn’t wick under the straightedge during drawing. For drawing gridlines on USGS 7.5-minute quadrangle maps, the straightedge should be at least 60 cm (24 in.) long; for 15-minute quadrangles it should be at least 45 cm (18 in.) long. For larger maps, a longer straightedge is required. If the straightedge is too long, it will be awkward to use. A 60-cm (24-in.) straightedge is suitable for use with both 7.5-minute and 15-minute USGS quadrangle maps. Use a technical pen with a fine point so the resulting UTM gridlines are thin and uniform. Excessively thick gridlines reduce a map’s usefulness by (1) needlessly concealing details on the map, and (2) reducing the precision of the gridline locations. Use waterproof ink so the gridlines won’t smear or wash out if the map is inadvertently exposed to moisture. Prior to drawing any gridlines, fill the pen’s ink reservoir so that the ink supply won’t be exhausted while drawing a gridline. If the ink supply is exhausted midway through the drawing of a gridline, it may be difficult to draw the gridline neatly. Before you begin drawing, practice drawing a few gridlines on an old map sheet or scratch paper. Try to determine (1) the optimum amount of downward pressure to apply to the pen, (2) the optimum speed to move the pen tip across the paper, and (3) the best position for holding the pen, for drawing gridlines neatly. For optimum results, practice using the same type of paper as the map sheet. Visually locate all of the UTM grid tick marks on the map before you begin. It helps to systematically plan the order in which you draw the gridlines to minimize the time spent waiting for the ink to dry and to minimize contact with the finished parts of the map. The UTM gridlines
40
CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
that run east-west across the width of the map should be drawn in order from most northerly (i.e., top of map sheet) to most southerly (i.e., bottom of map sheet). This will eliminate the need to lean over the gridlines after they’ve been drawn. Similarly, the UTM gridlines that run north-south across the height of the map should be drawn in order from right to left, or vice versa. Allow each gridline to dry thoroughly before touching it with your hands or any drafting tools (e.g., straightedge), to prevent accidental smearing. After drawing the first set of gridlines (i.e., those that run east-west across the width of the map), allow the ink to dry thoroughly before drawing the second set (i.e., those that run north-south across the height of the map). While waiting for the ink to dry, check the pen’s ink reservoir and refill it if necessary.
3.3
Determining the UTM Coordinates of a Point on a Map
Suppose one knows the plotting position of a point on a topographic map, and wishes to determine its UTM coordinates (all five elements). This section describes the basic procedure for doing so, and gives some tips for measuring the UTM easting and northing coordinates of the point. We’ll assume that UTM gridlines have been drawn or printed on the topographic map. Before proceeding further, some readers may find it helpful to review the basic procedure for accurately measuring the distance between a point and a line (Appendix C).
3.3.1
Basic Procedure
The basic procedure for determining the UTM coordinates of a point on a topographic map is as follows: 1. Locate the point of interest (POI) on a topographic map. For tips on selecting an appropriate USGS topographic map, see Appendix D. 2. Determine the horizontal datum. In North America, typical choices are NAD 27, NAD 83, or WGS 84. For USGS quadrangle maps, see the explanatory text displayed on the map collar, typically on the left-hand side of the lower margin. 3. Determine the UTM longitude zone. The choices are from 01 to 60, inclusive. For USGS quadrangle maps, see the explanatory text displayed on the map collar, typically on the left-hand side of the lower margin. 4. Determine whether the POI is located in the northern hemisphere (N. hemis.) or the southern hemisphere (S. hemis.). All of North America is located in the northern hemisphere. 5. Determine the POI’s UTM easting coordinate. (a) Locate the nearest UTM easting gridline lying east or west of the POI (i.e., the nearest vertically oriented UTM gridline) on the map. (b) Read the gridline’s easting coordinate (xE−gridline ) off the corresponding grid tick mark coordinate label, and record the result.
3.3. DETERMINING THE UTM COORDINATES OF A POINT ON A MAP
41
(c) Determine the projected easting distance (∆x) in meters between the POI and the gridline (see details in following section), and record the result. Remember, easting (x) increases to the east and decreases to the west. If the POI lies east of the gridline, then the easting distance is positive (i.e., ∆x > 0); if the POI lies west of the gridline, then the easting distance is negative (i.e., ∆x < 0). (d) Calculate the POI’s easting coordinate as the sum of the meridian’s easting coordinate and the projected easting distance:
xPOI = xE−gridline + ∆x
(3.3.1)
and record the result.
6. Determine the POI’s UTM northing coordinate.
(a) Locate the nearest UTM gridline lying north or south of the POI (i.e., the nearest horizontally oriented UTM gridline) on the map. (b) Read the gridline’s northing coordinate (yN−gridline ) off the corresponding grid tick mark coordinate label, and record the result. (c) Determine the projected northing distance (∆y) in meters between the POI and the gridline (see details in the following section), and record the result. Remember, northing (y) increases to the north and decreases to the south. If the POI lies north of the gridline, then the northing distance is positive (i.e., ∆y > 0); if the POI lies south of the gridline, then the northing distance is negative (i.e., ∆y < 0). (d) Calculate the POI’s northing coordinate as the sum of the gridline’s northing coordinate and the projected northing distance:
yPOI = yN−gridline + ∆y
and record the result.
7. Record all five components of the UTM coordinates.
(3.3.2)
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CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
Example: Determining UTM Coordinates of a Point on a Topographic Map Problem: Determine the UTM coordinates of the gaging station at Nine Mile Falls, Washington. Use the Nine Mile Falls Quadrangle, Washington (7.5-minute) topographic map [USGS, 1973b]. This example is illustrated graphically in Figure 3.4. On this figure, the UTM easting and northing gridlines nearest to the POI were overlain. These are the heavy, solid black lines that cross the entire image area, one oriented vertically and the other horizontally. Also, an image of a semitransparent corner ruler was overlain. This is the light gray, L-shaped object. Solution: In this example the POI is the gaging station at Nine Mile Falls. Determination of Horizontal Datum and UTM Longitude Zone The explanatory text on the left-hand side of the lower border of the map sheet indicates the horizontal datum is NAD 27, and the area of map coverage is within UTM Zone 11. Determination of Hemisphere All of North America lies in the northern hemisphere, so the POI is in the northern hemisphere. Determination of Easting Coordinate xE−gridline = 459 000 m ∆x =
+288 m (= 12 mm × 24 m/mm)
xPOI = 459 288 m Determination of Northing Coordinate yN−gridline = 5 291 000 m ∆y =
+192 m (= 8 mm × 24 m/mm)
yPOI = 5 291 192 m The conversion factor (24 m ground distance per 1 mm map distance) corresponds to a map scale of 1:24 000. This scale, which is displayed on the collar of the Nine Mile Falls Quadrangle, Washington map, is the standard scale for the USGS 7.5-minute series quadrangle maps. Therefore, the UTM coordinates for the POI are specified as follows: NAD 27, UTM Zone 11, N. hemis., 459288mE, 5291192mN The following is an acceptable alternative specification: NAD 27, Zone 11T, 459288mE, 5291192mN
3.3. DETERMINING THE UTM COORDINATES OF A POINT ON A MAP
43
(See separate image file: use_ruler.pdf)
Figure 3.4: Using a Corner Ruler with a Topographic Map
3.3.2
Measuring Projected Easting and Northing Distances
The method described in this section uses the corner ruler to ensure that the distances from the POI to nearby gridlines are measured along perpendicular segments. Position the corner ruler so that both of the following conditions are satisfied simultaneously: • The entire length of one outside edge is accurately aligned with one of the two nearest gridlines. • The other outside edge intersects the POI. After positioning the corner ruler, hold it steady while you read the distances ∆x and ∆y, in map units, along the outside edges of the corner ruler. One distance (∆x or ∆y, depending on how the ruler is oriented) is measured along the outside edge that’s aligned with a gridline; the distance is measured from the outside corner to the point where the nearby orthogonal gridline crosses this same outside edge. The other distance is measured along the other outside edge, from the outside corner to the POI. Proper positioning of the corner ruler is critical for accurate results. Using the entire length of the outside edge in (1) minimizes alignment error. The two nearest gridlines consist of the nearest easting gridline and the nearest northing gridline. Using the nearest gridlines minimizes
44
CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
the perpendicular distances ∆x and ∆y to be added or subtracted. Also, if the corner ruler is small, it will still be long enough to make the measurements. With this method there will always be at least two ways to measure the easting and northing coordinates of the POI on a topographic quadrangle map – aligning the outside edge of the corner ruler with the nearest easting gridline, or with the nearest northing gridline. The following examples, and the accompanying figures (Figures 3.5 through 3.8), illustrate the method.
Example: Measuring UTM Easting and Northing Coordinates Problem: What are the UTM easting and northing coordinates of the POI shown on Figure 3.5? Solution: The measurements and calculations are performed in the sequence recorded below. Determination of Easting Coordinate xE−gridline = 265 000 m ∆x =
+624 m (= 26 mm × 24 m/mm)
xPOI = 265 624 m Determination of Northing Coordinate yN−gridline = 6 125 000 m ∆y =
−456 m (= −19 mm × 24 m/mm)
yPOI = 6 124 544 m In this example, ∆y is negative because the POI lies south of the northing gridline.
Example: Measuring UTM Easting and Northing Coordinates Problem: What are the UTM easting and northing coordinates of the POI shown on Figure 3.6? Solution: The measurements and calculations are performed in the sequence recorded below. Determination of Easting Coordinate xE−gridline = 265 000 m ∆x =
+384 m (= 16 mm × 24 m/mm)
xPOI = 265 384 m Determination of Northing Coordinate yN−gridline = 6 124 000 m ∆y =
+300 m (= 12.5 mm × 24 m/mm)
yPOI = 6 124 300 m
3.3. DETERMINING THE UTM COORDINATES OF A POINT ON A MAP
45
Figure 3.5: Using a Corner Ruler to Measure the UTM Easting and Northing Coordinates of a Point on a Map – 1st of 4
46
CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
Figure 3.6: Using a Corner Ruler to Measure the UTM Easting and Northing Coordinates of a Point on a Map – 2nd of 4
3.4. PLOTTING A POINT WITH KNOWN UTM COORDINATES ON A MAP
47
Example: Measuring UTM Easting and Northing Coordinates Problem: What are the UTM easting and northing coordinates of the POI shown on Figure 3.7? Solution: The measurements and calculations are performed in the sequence recorded below. Determination of Easting Coordinate xE−gridline = 266 000 m ∆x =
−432 m (= −18 mm × 24 m/mm)
xPOI = 265 568 m Determination of Northing Coordinate yN−gridline = 6 125 000 m ∆y =
−720 m (= −30 mm × 24 m/mm)
yPOI = 6 124 280 m In this example, ∆x is negative because the POI lies west of the easting gridline, and ∆y is negative because the POI lies south of the northing gridline.
Example: Measuring UTM Easting and Northing Coordinates Problem: What are the UTM easting and northing coordinates of the POI shown on Figure 3.8? Solution: The measurements and calculations are performed in the sequence recorded below. Determination of Easting Coordinate xE−gridline = 263 000 m ∆x =
+276 m (= 11.5 mm × 24 m/mm)
xPOI = 263 276 m Determination of Northing Coordinate yN−gridline = 6 125 000 m ∆y =
−792 m (= −33 mm × 24 m/mm)
yPOI = 6 124 208 m In this example, ∆y is negative because the POI lies south of the northing gridline.
3.4
Plotting a Point with Known UTM Coordinates on a Map
Suppose one knows the UTM coordinates (all five elements) of a point of interest (POI), and wishes to plot its position on a topographic map. This section describes a procedure for plotting
48
CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
Figure 3.7: Using a Corner Ruler to Measure the UTM Easting and Northing Coordinates of a Point on a Map – 3rd of 4
3.4. PLOTTING A POINT WITH KNOWN UTM COORDINATES ON A MAP
49
Figure 3.8: Using a Corner Ruler to Measure the UTM Easting and Northing Coordinates of a Point on a Map – 4th of 4
50
CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
the position, subject to the following conditions: • The POI lies within the U.S. • A paper version of the appropriate USGS topographic map has been acquired. At this point some readers may find it helpful to review the procedure for obtaining an appropriate USGS topographic map (Appendix D). The basic procedure for plotting the position of the POI is as follows: 1. Verify that the acquired USGS topographic map is consistent with the following elements of the UTM coordinate specification: • horizontal datum, • UTM longitude zone, and • hemisphere, or UTM latitude zone. 2. On the map, locate the marked-only or fully displayed UTM easting gridline that’s nearest to the POI. Read the gridline’s easting coordinate (xE−gridline ) off the corresponding grid tick mark coordinate label, and record the result. 3. Calculate the projected easting distance (∆x) from the POI to the gridline as the difference between the easting coordinate of the POI (xPOI ) and that of the gridline, i.e., ∆x = xPOI − xE−gridline
(3.4.1)
and record the result. 4. On the map, locate the marked-only or fully displayed UTM northing gridline that’s nearest to the POI. Read the gridlines northing coordinate (yN−gridline ) off the corresponding grid tick mark coordinate label, and record the result. 5. Calculate the projected northing distance (∆y) from the POI to the gridline as the difference between the northing coordinate of the POI (yPOI ) and that of the gridline, i.e.,
∆y = yPOI − yN−gridline
(3.4.2)
and record the result. 6. Measure the projected easting distance (∆x) perpendicularly from the easting gridline, and mark the location by making a small, faint pencil mark. 7. Measure the projected northing distance (∆y) along a line segment that both (a) is perpendicular to the northing gridline, and (b) intersects the pencil mark. Mark the location. This is the location of the POI. 8. Verify that the POI’s location has been plotted accurately. First, determine the UTM coordinate specification of the POI by following the procedure given earlier. Does the coordinate specification you determined match the location you have plotted?
3.5. SOFTWARE FOR USING THE UTM GRID WITH TOPOGRAPHIC MAPS
3.5
51
Software for Using the UTM Grid with Topographic Maps
Some commercially available software products have features for using the UTM grid system with topographic maps. Such features typically include one or more of the following options: • Display the USGS topographic map image, and auxiliary information, on the user’s monitor. • Select a specific horizontal datum for the graticule and UTM grid. • Overlay the corresponding graticule or UTM grid on a USGS topographic map (i.e., display the UTM gridlines). The resulting map image appears on the user’s monitor. • Print the USGS topographic map image, with or without the graticule and UTM gridlines. • Display the UTM easting and northing coordinates, in meters, for any point on the map, by maneuvering the cursor over the POI. For instance, the coordinates might appear in a small window on the monitor, adjacent to the map image. • Plot a point with known UTM coordinates on the map, by entering the numerical values of the easting and northing coordinates.
3.6
Chapter Summary
Chapter 3 discussed the following topics: • those elements of USGS topographic maps that support use of the UTM grid, • basic procedures for using topographic maps without preprinted gridlines, • a procedure for determining the UTM coordinates of a point on a topographic map, • a procedure for plotting a point with known UTM coordinates on a topographic map, and • software for using the UTM grid with topographic maps.
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CHAPTER 3. THE UTM GRID AND USGS TOPOGRAPHIC MAPS
Chapter 4
Horizontal Distance and Bearing Determination Given the UTM coordinates for two points P1 and P2 , how does one determine the horizontal distance, and bearing, between them? The answer to this question depends on whether or not the two points lie within the same UTM longitude zone and the same hemisphere. This is one example of why the full UTM coordinate specification includes not only the easting and northing coordinates, but also the UTM longitude zone and hemisphere.
4.1
Points in the Same UTM Longitude Zone and Hemisphere
If the two points P1 and P2 lie within the same UTM longitude zone and the same hemisphere, then one or both of the following methods can be used to determine the horizontal distance and bearing between them.
4.1.1
Measurement Using a Paper Map
If the two points P1 and P2 plot on the same quadrangle map, then one can plot the two points on the map and measure the horizontal distance and bearing between them. The procedure is as follows: 1. Determine if the two points plot on the same map (e.g., the same USGS topographic quadrangle). 2. If the two points plot on the same map, then identify the particular map. 3. Obtain a paper version of the map. 4. Plot the two points on the paper map. 5. Measure the map distance between the two plotted points (e.g., using a scale). 6. Convert the map distance to a horizontal ground distance, using the map scale. 7. Use a protractor to measure the bearing between the two plotted points. 53
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CHAPTER 4. HORIZONTAL DISTANCE AND BEARING DETERMINATION
4.1.2
Calculation Using Plane Geometry
The two points P1 and P2 share the same local system of easting and northing coordinates because they lie within the same UTM longitude zone and the same hemisphere. In this case, the horizontal distance and bearing between the two points are calculated using plane analytic geometry. The horizontal distance (D) between the points is calculated using the two-point distance formula [Eves, 1984]:
D=
p (x2 − x1 )2 + (y2 − y1 )2
(4.1.1)
where x1 and x2 are the UTM easting coordinates, and y1 and y2 are the UTM northing coordinates, respectively, of points P1 and P2 . The distance formula is based on the Pythagorean theorem. If the easting and northing coordinates are given in meters, then the distance will be in meters. The bearing (β) from point P1 to point P2 is calculated as follows [Langley, 1998]:
β = arctan
x2 − x1 y2 − y1
! (4.1.2)
Equation (4.1.2) is based on the two-point slope formula [see Eves, 1984]. Here the angle β is measured clockwise from the positive y (northing) axis, and is measured in units of radians (rad). To obtain the bearing in units of degrees, multiply by the conversion factor (180◦ /π rad). The UTM easting and northing coordinates of the two points must be expressed relative to a common horizontal datum, or the use of equations (4.1.1) and (4.1.2) will give erroneous results. Equations (4.1.1) and (4.1.2) are convenient to use because of their simplicity. Suppose one has multiple pairs of points for which the horizontal distances and bearings must be calculated. If the number of point pairs is relatively small, then one can easily perform the calculations using a handheld electronic calculator. On the other hand, it’s also possible to write a computer program or to configure a computer spreadsheet to perform the calculations quickly and accurately for thousands of point pairs, if necessary. Calculating the horizontal distance and bearing using equations (4.1.1) and (4.1.2) is more versatile than measuring the distance and bearing on a paper map, in that it doesn’t require that the two points plot on the same quadrangle map. For additional discussion of horizontal distance and bearing calculations, see Langley [1998].
4.1. POINTS IN THE SAME UTM LONGITUDE ZONE AND HEMISPHERE Table 4.1: Data for Distance and Bearing Calculation Example – 1st of 2
Description
Easting x (m)
Northing y (m)
P1
Spring
458 576
5 294 756
P2
Gaging Station
459 288
5 291 192
Point
Notes: (1) Horizontal datum: NAD 27 (2) Both points lie in UTM lon. zone 11, in N. hemis.
Example: Horizontal Distance and Bearing between Points in Same Zone and Hemisphere Problem: Determine the horizontal distance, and bearing, between the unnamed spring near the mouth of Sandy Canyon, Washington and the gaging station at Nine Mile Falls, Washington. Assume that the points have the UTM coordinates listed in Table 4.1. Solution: Note that the UTM coordinates of the two points are expressed relative to the same horizontal datum. Also, both points are known to reside in UTM longitude zone 11, in the northern hemisphere. Therefore, one can use equations (4.1.1) and (4.1.2) to calculate the horizontal distance and bearing, respectively, between the points. Substituting the numerical values tabulated above for the corresponding variables in equation (4.1.1) gives D =
p (9288 − 8576)2 + (1192 − 4756)2 m
= 3634 m Substituting the numerical values tabulated above for the corresponding variables in equation (4.1.2) gives β =
180◦ π rad
! arctan
9288 − 8576 1192 − 4756
!
= 168.7◦ That is, the bearing is approximately 169◦ east of north.
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CHAPTER 4. HORIZONTAL DISTANCE AND BEARING DETERMINATION
Example: Verification by Direct Map Measurement Problem: How can one verify the horizontal distance calculated in the previous example? Solution: In this particular example, both points lie within the coverage area for the Nine Mile Falls Quadrangle, Washington (7.5-minute) topographic map [USGS, 1973b]. Therefore, one can use a paper map to physically measure the map distance between the two points, and then use the map scale to convert the map distance to the equivalent horizontal ground distance. Finally, one can compare the results from the two methods to see if they’re consistent. I obtain the following result for the measured map distance: Dmap = 151 mm Converting to the equivalent horizontal ground distance, D =
Dmap u
= 151 mm
24 m mm
= 3624 m Here u denotes the map scale. This result (3624 m) differs from that obtained in the previous example (3634 m) by only 10 m. Suppose one can directly measure map distance with a precision of about 1 mm. For a map scale of 1:24 000 the equivalent measurement precision of horizontal ground distance is then about 24 m. The difference one obtains using the two methods (i.e., computation versus direct measurement), 10 m, is well within this precision. Therefore, the distance results from the two methods are consistent.
4.1. POINTS IN THE SAME UTM LONGITUDE ZONE AND HEMISPHERE
Table 4.2: Data for Distance and Bearing Calculation Example – 2nd of 2
Description
Easting x (m)
Northing y (m)
P1
Summit
448 496
5 298 673
Four Mound Prairie, WA [USGS, 1973a]
P2
Gaging Station
459 288
5 291 192
Nine Mile Falls, WA [USGS, 1973b]
Point
Quadrangle Name (7.5-minute series)
Notes: (1) Horizontal datum: NAD 27 (2) Both points lie in UTM lon. zone 11, in N. hemis.
Example: Determining Horizontal Distance and Bearing Between Points in Same Zone and Hemisphere Problem: Determine the horizontal distance, and bearing, from the summit of Eagle Rock (Stevens County, Washington) to the gaging station at Nine Mile Falls, Washington. Assume that the points have the UTM coordinates listed in Table 4.2. Solution: The points lie in areas covered by two different USGS quadrangle maps, but are in the same UTM longitude zone and hemisphere, and their UTM coordinates are expressed relative to the same horizontal datum. Therefore one can use equations (4.1.1) and (4.1.2) to calculate the horizontal distance and bearing, respectively, between the points. Substituting the numerical values tabulated above for the corresponding variables in equation (4.1.1) gives D =
p (59 288 − 48 496)2 + (1192 − 8673)2 m
= 13 130 m Substituting the numerical values tabulated above for the corresponding variables in equation (4.1.2) gives β =
180◦ π rad
arctan
59 288 − 48 496 1192 − 8673
= 124.7◦ That is, from the summit, the gaging station is on a bearing of approximately 125◦ east of north.
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CHAPTER 4. HORIZONTAL DISTANCE AND BEARING DETERMINATION
4.2
Points Not in the Same UTM Longitude Zone and Hemisphere
If the two points P1 and P2 lie in different UTM longitude zones or in different hemispheres, or both, then one can use either spherical trigonometry or ellipsoidal trigonometry to calculate the horizontal distance and bearing between them. The following sections briefly discuss the two approaches.
4.2.1
Spherical Earth
Assuming the earth is spherical, then the great-circle distance (D) between the two points P1 and P2 is given by the geographic formula [Snyder, 1987]: D = r arccos [sin φ1 sin φ2 + cos φ1 cos φ2 cos(λ2 − λ1 )]
(4.2.1)
where φ1 and φ2 are the latitudes (in radians) of the points P1 and P2 , respectively; λ1 and λ2 are the longitudes (in radians) of the points P1 and P2 , respectively; and r is the radius of the earth. Although equation (4.2.1) is mathematically exact, as a computational algorithm its practicality is limited. Specifically, if D
