State Plane Coordinate System Of 1983
NOAA Manual NOS NGS 5State Plane CoordinateSystem of 1983James E. StemRockville, MDJanuary 1989Reprinted with minor correctionsMarch 1990U.S. DEPARTMENT OF COMMERCENational Oceanic and Atmospheric AdministrationNational Ocean ServiceCharting and Geodetic Services
NOAA Manual NOS NGS 5State Plane CoordinateSystem of 1983James E. StemNational Geodetic SurveyRockville, MDJanuary 1989Reprinted with minor correctionsMarch 1990Reprinted February 1991Reprinted July 1992 eprintedJanuary 1993Reprinted August 1993Reprinted April 1994Re rintedJanuary 1995Re?rinted September 1995U.S. DEPARTMENT OF COMMERCEC. William Verity, SecretaryNational Oceanic and Atmospheric AdministrationWilliam E. Evans, Under SecretaryNational Ocean ServiceThomas J. Maginnis, Assistant AdministratorCharting and Geodetic ServicesR. Adm. Wesley V. HullFor sale by the National Geodetic Information Center, NOAA, Rockville, MD 20852
PREFACEThis manual explains how to perform computations on the State Plane CoordinateSystem of 1983 (SPCS 83). It supplements Coast and Geodetic Survey SpecialPublication No. 235, "The State coordinate systems," and replaces Coast andGeodetic Survey Publication 62-4, "State plane coordinates by automatic dataprocessing.'' These two widely distributed publications provided the surveyingand mapping profession with information on deriving 1927 State plane coordinatesfrom geodetic coordinates based on the North American Datum of 1927 (NAD 27) plusinformation for traverse and other computations with these coordinates. Thismanual serves the similar purpose for users of SPCS 83 derived from the NorthAmerican Datum of 1983 (NAD 83). Emphasis is placed on computations that havechanged as a result of SPCS 83.This publication is neither a textbook on the theory, development, orapplications of general map projections nor a manual on the use of coordinates insurvey computations. Instead it provides the practitioner with the necessaryinformation to work with three conformal map projections: the Lambert conformalconic, the transverse Mercator, and the oblique Mercator. Derivatives of thesethree map projections produce the system which the National Geodetic Survey (NGS)has named the State Plane Coordinate System (SPCS). Referred to NAD 83 or NAD27, this system of plane coordinates is identified as SPCS 83 or SPCS 27,respectively.The equations in chapter 3, Conversion Methodology, form a significant portionof the manual. Chapter 3 is required reading for programmers writing software,but practitioners with software available may skip this chapter. Although amodification of terminology and notation was suggested by some reviewers,consistency with NGS software was deemed more important. Hence, chapter 3documents the SPCS 83 software available from the National Geodetic Survey.iii
ACKNOWLEDGMENTThe mathematics given in this manual were compiled or developed by T. Vincentyprior to his retirement from the National Geodetic Survey (NGS). Hisconsultation was invaluable to the author.Principal reviewer was Joseph F. Dracup, NGS, retired. His many excellentsuggestions were incorporated into the manual. Once again, Joe gave generouslyof his time to assist in the education of the surveying profession.The author appreciates the review and contributions made by Earl F. Burkholder,Oregon Institute of Technology. Earl spent a summer at NGS researching thesubject of map projections and maintains a continuing interest in the subject.In addition, the manual was reviewed by Charles A. Whitten, B. K. Meade, andCharles N. Claire, all retired employees of the former Coast and Geodetic Survey(now NGS). The author was very fortunate to have such experts donate theirservices.Finally, the author appreciates the helpful guidance of John G. Gergen and EdwardJ. McKay, present NGS employees.iv
CONTENTSPreface . . . . . . . . . iiiAcknowledgment.1.1. 11. 21. 31. 41. 51 61. 72.2. 12.22.32.42.52.62.73.3. 13. 113. 1 23. 1 33. 1 43.153.23. 213.223.233.243.253.263.33. 313.323.333.343.353.363.43. 413.42Introduction . . . . . . . Requirement for SPCS 83 . . . . .SPCS 27 background . . . . SPCS 83 design . . . . . . . SPCS 83 local selection . . . . . .SPCS 83 State legislation . . . . . . . .SPCS 83 unit of length . . . . . . . . .The new GRS 80 ellipsoid . .iv24581112Map projections . . . . . . . . . . . 14Fundamentals . . . . . 1 4SPCS 83 grid . . . . . . . . . . . 16Conformal it y . . . . 17Convergence angle . . . . . . . . 1818Grid azimuth "t" and projected geodetic azimuth "T''···················Grid scale factor at a point . . . . . . . 18Universal Transverse Mercator projection . . . . . . . 21Conversion methodology . . . . . . . . .Lambert conformal conic mapping equations (note alternativemethod given in sec. 3. 4) . . . . . .No tat ion and definitions . . . . . .Computation of zone constants . . . . . . .Direct conversion computation . . . .Inverse conversion computation . . . . . . . . . . . .Arc-to-chord correction (t-T) . . . . . .Transverse Mercator mapping equations . . . .Notation and definition . . . . . . . . .Constants for meridional distance . . . . . .Direct conversion computation . . . . . .Inverse conversion computation . . . . . . .Arc-to-chord correction (t-T) . . . . . Grid scale factor of a line . . . . . . . . . . Oblique Mercator mapping equations . . .Notation and definition . . . . . .Computation of GRS 80 ellipsoid constants . .Computation of zone constants . . . . . . Direct conversion computation . . . . . . Inverse conversion computation . . . . .Arc-to-chord correction (t-T) and grid scale factorof a line . . . . . . .Polynomial coefficients for the Lambert projection . . . . .Direct conversion computation . . . . . . . . Inverse conversion computation . . . . . .v242626272828293232333335373838383839404142424445
4.4.14.24.34.4Line conversion methods required to place a survey on SPCS 83.Reduction of observed distances to the ellipsoid .Grid scale factor k 12 of a line.Arc-to-chord correction (t-T).Traverse example . .4646495153Bibliography.62Appendix A.Defining constants for the State Plane Coordinate Systemof 1983. .63Appendix B.Model act for State Plane Coordinate Systems . . . .73Appendix C.Constants for the Lambert projection by the polynomialcoefficient method.76FIGURES1 .4State Plane Coordinate System of 1983 zones.62.1aThe three basic projection surfaces . .152.1bSurfaces used in State Plane Coordinate Systems . . . . .162.5Azimuths . . . . . . . . .192.6Scale factor.202.7Universal Transverse Mercator zones . . . . . . . .223. 4The Lambert grid.434.1aGeoid-ellipsoid-surface relationships . . . . . . .464.1bReduction to the ellipsoid . . . . .474.1cReduction to the ellipsoid (shown with negative geoid height) . 484.2Geodetic vs. grid distances.494.3Projected geodetic vs. grid angles . . . . . . . . .534. 4aSample traverse.544.4bFixed station control information . . . . . . . . . .564.4c(t-T) correction .584. 4dAzimuth adjustment.594.4eTraverse computation by latitudes and departures . .604.4fAdjusted traverse data .60vi
TABLES1.5Status of SPCS 27 and SPCS 83 legislation.93. 0Summary of conversion methods.243.1True values of (t-T) and computational errors in theirdetermination.323.22Intermediate constants for the transverse Mercator. projections.344.3aApproximate size of (t-T) in seconds of arc forLambert or transverse Mercator projection.514.3bSign of (t-T) correction.52vii
THE STATE PLANE COORDINATE SYSTEM OF 1983James E. StemNational Geodetic SurveyCharting and Geodetic ServicesNational Ocean Service, NOAARockville, MD 20852ABSTRACT. This manual provides information and equationsnecessary to perform survey computations on the State PlaneCoordinate System of 1983 (SPCS 83), a map projection systembased on the North American Datum of 1983 (NAD 83). Giventhe geodetic coordinates on NAD 83 (latitude and longitude),the manual provides the necessary equations to compute Stateplane coordinates (northing, easting) using the "forward"mapping equation (cp, . . N, E). "Inverse" mapping equations aregiven to compute the geodetic position of a point defined byState plane coordinates (N,E . cp, .). The manual addressescorrections to angles, azimuths, and distances that arerequired to relate these geodetic quantities between theellipsoid and the grid. The following map projections aredefined within SPCS 83: Lambert conformal conic, transverseMercator, and oblique Mercator. A section on t a UniversalTransverse Mercator (UTM) projection is included. UTM is aderivative of the general transverse Mercator projection aswell as another projection, in addition to SPCS 83, on whichNAD 83 is published by NGS.1.1.1INTRODUCTIONRequirement for SPCS 83The necessity for SPCS 83 arose from the establishment of NAD 83. When NAD 27was readjusted and redefined by the National Geodetic Survey, a project whichbegan in 1975 and finished in 1986, SPCS 27 became obsolete. NAD 83 producednew geodetic coordinates for all horizontal control points in the NationalGeodetic Reference System (NGRS). The project was undertaken because NAD 27values could no longer provide the quality of horizontal control required bysurveyors and engineers without regional recomputations (least squaresadjustments) to repair the existing network. NAD 83 supplied the followingimprovements:oOne hundred and fifty years of geodetic observations(approximately 1.8 million) were adjusted simultaneously,eliminating error propagation which occurs when projectsmust be mathematically assembled on a "piecemeal" basis.oThe precise transcontinental traverse, satellite
triangulation, Doppler positions, baselines established byelectronic distance measurements (EDM), and baselinesestablished by very long baseline interferometry (VLBI),improved the internal consistency of the network.oA new figure of the Earth, the Geodetic Reference System of1980 (GRS 80), which approximates the Earth's true sizeand shape, supplied a better fit than the Clarke 1866spheroid, the reference surface used with NAD 27.oThe origin of the datum was moved from station MEADESRANCH in Kansas to the Earth's center of mass, forcompatibility with satellite systems.Not only will the published geodetic position of each control point change, butthe State plane coordinates will change for the following reasons:oThe plane coordinates are mathematically derived(using mapping equations ) from geodetic coordinates.oThe new figure of the Earth, the GRS 80 ellipsoid, hasdifferent values for the semimajor axis a andflattening "f" (and eccentricity e and semiminor axis"b" ) These ellipsoidal parameters are often embeddedin the mapping equations and their change producesdifferent plane coordinates.oThe mapping equations given in chapter 3 are accurate tothe millimeter, whereas previous equations promulgated byNGS were derivatives of logarithmic calculations withgenerally accepted approximations.oThe defining constants of several zones have beenredefined by the States.oThe numeric grid value of the or1g1n of each zone hasbeen significantly changed to make the coordinatesappear clearly different.oThe State plane coordinates for all points published onNAD 83 by NGS will be in metric units.oThe SPCS 83 uses the Gauss-Kruger form of the transverseMercator projection, whereas the SPCS 27 used the GaussSchreiber form of the equations.1.2SPCS 27 BackgroundThe State Plane Coordinate System of 1927 was designed in the 1930s by the U.S.Coast and Geodetic Survey (predecessor of the National Ocean Service) to enablesurveyors, mappers, and engineers to connect their land or engineering surveys toa common reference system, the North American Datum of 1927. The followingcriteria were applied in the design of the State Plane Coordinate System of 1927:2
oUse of conformal mapping projections.oRestricting the maximum scale distortion (sec. 2.6) to lessthan one part in 10,000.oCovering an entire State with as few zones of a projectionas possible.oDefining boundaries of projection zones as an aggregation ofcounties.It is impossible to map a curved Earth on a flat map using plane coordinateswithout distorting angles, azimuths, distances, or area. It is possible todesign a map such that some of the four remain undistorted by selecting anappropriate "map projection." A map projection in which angles on the curvedEarth are preserved after being projected to a plane is called a "conformal"projection. (See sec. 2.3.) Three conformal map projections were used indesigning the original State plane coordinate systems, the Lambert conformalconic projection, the transverse Mercator projection, and the oblique Mercatorprojection. The Lambert projection was used for States that are long in theeast-west direction (e.g., Kentucky, Tennessee, North Carolina), or for Statesthat prefer to be divided into several zones of east-west extent. The transverseMercator projection was used for States (or zones within States) that are long inthe north-south direction (e.g., Vermont and Indiana), and the oblique Mercatorwas used in one zone of Alaska when neither of these two was appropriate. Thesesame map projections are also often custom designed to provide a coordinatesystem for a local or regional project. For example, the equations of theoblique Mercator projection produced project coordinates for the NortheastCorridor Rail Improvement project when a narrow coordinate system fromWashington, DC, to Boston, MA, was required.Land survey distance measurements in the 1930s were typically made with a steeltape, or something less precise. Accuracy rarely exceeded one part in 10,000.Therefore, the designers of the SPCS 27 concluded that a maximum systematicdistance scale distortion (see sec. 2.6, "Grid scale factor") attributed to theprojection of 1: 10,000 could be absorbed in the computations without adverseimpact on the survey. If distances were more accurate than 1:10,000, or if thesystematic scale distortion could not be tolerated, the effect of scaledistortion could be eliminated by computing and applying an appropriate gridscale factor correction. Admittedly, the one in 10,000 limit was set at anarbitrary level, but it worked well for its intended purpose and was notrestrictive on the quality of the survey when grid scale factor was computed andapplied.To keep the scale distortion at less than one part in 10,000 when designing theSPCS 27, some States required multiple projection "zones." Thus some States haveonly one State plane coordinate zone, some have two or three zones, and the Stateof Alaska has 1 O zones that incorporate all three projections. With theexception of Alaska, the zone boundaries in each State followed countyboundaries. There was usually sufficient overlap from one zone to another toaccommodate projects or surveys that crossed zone boundaries and still limit thescale distortion to 1: 1 o,ooo. In more recent years, survey accuracy usuallyexceeded 1 :10,000. More surveyors became accustomed to correcting distance3
observations for projection scale distortion by applying the grid scale factorcorrection. When the correction is used, zone boundaries become less important,as projects may extend farther into adjacent zones.1.3SPCS 83 DesignIn the mid 1970s NGS considered several alternatives to SPCS 83. Somegeodesists advocated retaining the design of the existing State plane coordinatesystem (projection type, boundaries, and defining constants) and others believedthat a system based on a single projection type should be adopted. The singleprojection proponents contended that. the present SPCS was cumbersome, sincethree projections involving 127 zones were employed.A study was instituted to decide whether a single system would meet theprincipal requirements better than SPCS 27. These requirements included ease ofunderstanding, computation, and implementation. Initially, it appeared thatadoption of the Universal Transverse Mercator (UTM) system (sec. 2.7) would bethe best solution because the grid had long been established, to some extent wasbeing used, and the basic formulas were identical in all situations. However, onfurther examination, it was found that the UTM 6-degree zone widths presentedseveral problems that might impede its overall acceptance by the surveyingprofession. For example, to accommodate the wider zone width, a grid scalefactor of 1 :2,500 exists on the central meridian while a grid scale factor of1:1,250 exists at zone boundaries. As already discussed, similar grid scalefactors on the SPCS rarely exceeded 1:10,000. In addition, the "arc-to-chord"correction term (sec. 2.5) that converts observed geodetic angles to grid anglesis larger, requiring application more frequently. And finally, the UTM zonedefinitions did not coincide with State or county boundaries. These problemswere not viewed as critical, but most surveyors and engineers considered theexisting SPCS 27 the simpler system and the UTM as unacceptable because ofrapidly changing grid scale factors.The study then turned to the transverse Mercator projection with zones of 2 inwidth. This grid met the primary conditions of a single national system. Byreducing zone width, the scale factor and the arc-to-chord correction would be noworse than in the SPCS 27. The major disadvantage of the 2 transverse Mercatorgrid was that the zones, being defined by meridians, rarely fell along State andcounty boundaries. A more detailed review showed that while many States wouldrequire two or more zones, the 2 grid could be
This manual explains how to perform computations on the State Plane Coordinate System of 1983 (SPCS 83). It supplements Coast and Geodetic Survey Special Publication No. 235, "The State coordinate systems," and replaces Coast and Geodetic Survey Publication 62-4, "State plane coordinates by automatic data processing.''
Placing Figures in a Coordinate Plane A coordinate proof involves placing geometric fi gures in a coordinate plane. When you use variables to represent the coordinates of a fi gure in a coordinate proof, the results are true for all fi gures of that type. Placing a Figure in a Coordinate Plane Place each fi gure in a coordinate plane in a way .
Placing Figures in a Coordinate Plane A coordinate proof involves placing geometric fi gures in a coordinate plane. When you use variables to represent the coordinates of a fi gure in a coordinate proof, the results are true for all fi gures of that type. Placing a Figure in a Coordinate Plane Place each fi gure in a coordinate plane in a way .
Jul 03, 2020 · Plotting of Points on the Coordinate Plane If points on the coordinate plane can be named, points can also be plotted or located in the plane using number pairs also called as their coordinates ( x, y ). To locate a point in the coordinate plane, take note of the steps in
EXAMPLE 1 coordinate proof GOAL 1 Place geometric figures in a coordinate plane. Write a coordinate proof. Sometimes a coordinate proof is the most efficient way to prove a statement. Why you should learn it GOAL 2 GOAL 1 What you should learn 4.7 Placing Figures in a Coordinate Plane Draw a right triangle with legs of 3 units and 4 units on a .
1.6 Lesson 36 Chapter 1 Operations with Integers The Coordinate Plane A coordinate plane is formed by the intersection of a horizontal number line and a vertical number line. The number lines intersect at the origin and separate the coordinate plane into four regions called quadrants. An ordered pair is
The Coordinate plane is also called the Cartesian plane. One way to remember how the quadrants are numbered is to write a big "C" on top of the plane. The "C" will begin in quadrant I and end in quadrant IV. Slide the "C" onto the coordinate plane Slide 7 / 52 0 The quadrants are formed by two intersecting number lines called axes.
Four coordinate systems in navigation system are considered: global coordinate system, tractor coordinate system, IMU coordinate system and GPS-receiver coordinate system, as superscripts or subscripts YKJ, t, i and GPS. Tractor coordinate system origin is located at the ground level of the tractor, under the center point of the
coordinate proof, p. 302 prueba de coordenadas VOCABULARY 7-1Polygons in the Coordinate Plane 7-2 Applying Coordinate Geometry 7-3Proofs Using Coordinate Geometry TOPIC OVERVIEW 294 Topic 7 Coordinate Geometry. 3-Act Math 3-Act Math If You Need Help . . . Vocabulary online
