Using state plane coordinate systems

Using state plane coordinate systems

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Authors Noice, Gilbert Vincent,

1943-Publisher The University of Arizona.

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USING STATE PLANE COORDINATE SYSTEMS

by

G ilbert V. Noice

A Thesis Submitted to the Faculty o f the DEPARTMENT OF CIVIL ENGINEERING

AND ENGINEERING MECHANICS

In Partial Fulfillm ent of the Requirements For the Degree of

MASTER OF SCIENCE

WITH A MAJOR IN CIVIL ENGINEERING In the Graduate College THE UNIVERSITY OF ARIZONA

STATEMENT BY AUTHOR

This thesis has been submitted in p a rtia l fu lf illm e n t o f re­ quirements fo r an advanced degree a t The U niversity of Arizona and is deposited in the U niversity L ib rary to be made available to borrowers under rules o f the L ib ra ry .

B rie f quotations from th is thesis are allowable w ithout special permission, provided th a t accurate acknowledgment o f source is made. Requests fo r permission fo r extended quotation from or reproduction of th is manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judg­ ment the proposed use o f the m aterial is in the in te re s ts of scholar­

ship. In a ll other instances, however, permission must be obtained

from the author.

SIGNED:

APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below:

Phi1ip /B . New!in Professor of C iv il Engineering

ACKNOWLEDGMENTS

This paper would not have been possible w ithout the kind assistance o f several people: Professor P h ilip B. New!in fo r his guidance throughout the p ro je c t; Donna Ripley fo r the typ in g ; Susan Angel on fo r the d ra ftin g ; The U n ive rsity Computer Center fo r help in producing Table 14; and my w ife * Maureen fo r moral support.

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS . v

LIST OF TABLES . . v ii ABSTRACT . . . ... v i i i 1. WHY STATE PLANE COORDINATE SYSTEMS ... . . . 1

2. MAP PROJECTIONS... 21

Tangent Plane ... . 23

Secant P l a n e ... 28

Transverse Mercator ... 32

Lambert Conformal Conic P rojection . ... 46

Comparison o f P rojections ... 59

3. DATUM ADJUSTMENTS . . ... . . . 63

4. COMPUTATIONS... 66

T riangulation and T r ila te r a tio n . ... 69

T ri a n g u la tio n ... 69

T r ila te r a tio n . . . . . . 74

Resection . . . 77

Traverse ... . . . 81

Interconversion Between Systems o f Plane Coordinates . . 92

5. SUMMARY... 96

APPENDIX 1: DEFINITIONS . . . . 99

APPENDIX 2: INFORMATIVE MATERIALS ... 100

APPENDIX 3: THE STATE COORDINATE SYSTEMS ... . 106

APPENDIX 4: COMPUTATIONS . . . . 109

APPENDIX 5: INTROCONVERSION BETWEEN GEODETIC AND RECTANGULAR COORDINATES ... . . . 123

REFERENCES CITED ... . . . . 136

LIST OF ILLUSTRATIONS

Figure Page

1. Zones fo r the State o f Arizona . ... 2

2. Control N e tw o r k ... 9

3. Finished Plate ... 10

4. Illu s tr a tio n s o f Factors in Table 1 20

5. E ffe c t o f Linear and Angular Errors (g re a tly exagerated)

Plane View . . ... 25

6. Linear E rror on a Tangent Plane (exagerated cross section) 26

7. Graph o f P rojection E r r o r ... 27

8. Graph o f Elevation E r r o r ... 30

9. Cross Section o f a Secant P rojection . . . 33

10. Transverse Mercator P ro je ctio n : In te rs e c tio n o f C ylinder

and Spheroid ... 34

11. Development o f the Transverse Mercator P rojection . . . . 35

39

12. Arc-to-Chord C orrection: Transverse Mercator P rojection . •

13. Location o f Stations used fo r Table 5 ... 40

14. Graph o f the Arc-Chord Correction ... 4.1

15. Rectangular Coordinates on the Transverse Mercator

P rojection . . . ... . . . 43

1 6 . Universal Transverse Mercator Coordinates from Geodetic .

P o s it io n ... 47

17. Geodetic P osition from Universal Transverse Coordinates . 48

18. Lambert Conformal P ro je c tio n : In te rse ctio n s o f Cone

and S p h e ro id ... 51

v i LIST OF ILLUSTRATIONS--Continued

Figure Page

19. Development o f Lambert P rojection ... 52

20. Arc-to-Cord C orrection: Lambert P rojection ... 54

21. Rectangular Coordinates on the Lambert P rojection . . . . 56

22. Zones o f the State Coordinate System ... 61

23. DIM Zones o f the United S t a t e s ... 62

2,4. Relation o f Adjusted Datums to the I n i t i a l Datum . . . . 64

25. Location o f the Kern T r ia n g u la t io n ... 70

26. Detail o f Kern T ria n g u la tio n , Cochise County, Arizona . . 71

27. In te rs e c tio n (angles tu rn e d )... .. . 72

28. In te rs e c tio n (using bearings) 73 29. In te rs e c tio n (Law o f S in e s ) ... 75

30. T r ila te r a tio n ( d ire c t coordinate s o lu tio n ) ... 76

31. T ri la te r a ti on (Law o f C o s in e s ) ... 78

32. Reduction (verses s o lu tio n ) . . . . ... 82

33. R e s e c tio n ... 83

34. Location o f the Patagonia Traverse . . . 84

35. Detail o f Patagonia Traverse, Santa Cruz County, Arizona. 85 36. Coordinate Conversion . . . 94

LIST OF TABLES

Table Page

1. Factors and Constants Used in Formulas . . . 18

2. Linear and Angular Displacements Away from the Line o f

Tangency . . . . . . . 24

3. Magnitude o f Differences in Distance Due to Elevation . . . 29

4. Convergence Computation . . ... 31

5. Differences Between the Two Arc-to-Chord Formulas ... 37

6. Transverse Mercator Coordinates from Geodetic P osition . . . 44

7. Geodetic P osition from Transverse Mercator Coordinates . . . 45

8. Universal Transverse Mercator Coordinates from Geodetic

P o s it io n ... 49

9. Geodetic P osition from Universal Transverse Coordinates . . 50

10. Lambert Coordinates from Geodetic P osition . . . ... 57

11. Geodetic Positions from Lambert Coordinates ... . . 58

12. Signs fo r Coordinate Differences and Trigonometric Functions

Functions . . . . . . 68

13. Reduction o f Slope Distances ... , 79

14. Table o f R adii, Normals, and Mean Radii from 20 Degrees to

49 Degrees o f Latitude . . . . . . 80

15. Angle Adjustment o f the Patagonia Traverse . . . 87

16. Computation o f the Patagonia Traverse . . . ... 88

17. Computation o f the Patagonia Traverse on an Adjusted Datum . 90

18. Line-up Coordinate and Double Area . . . 91

19. Coordinate Transformation . . . . . . 95

ABSTRACT

Using State Plane Coordinate systems becomes more and more necessary w ith the increased complexity o f modern l i f e and the computer inspired trend o f numbering everything.

There are a few objections to the use o f State Plane Coordinate

systemss but there are many more advantages. The objections center

around two po in ts: education and lack o f c o n tro l,

This paper attempts to aid in the education by discussing the projections involved and presenting computational methods th a t s im p lify surveying processes as w ell as the uses, advantages, and disadvantages o f State Plane Coordinate Systems,

CHAPTER 1

WHY STATE PLANE COORDINATE SYSTEMS

Most surveyors use a local coordinate system f o r computational purposesa While th is is commendable, these surveys would be more b e n e fic ia l i f referenced to a common datum* By indexing i t would be simple fo r another surveyor to determine what control was close to a proposed p ro je ct* Knowing the re la tio n between a ll o f the points th a t have been surveyed would be desirable but the average surveyor does not have the f a c i l i t i e s needed to e sta b lish a control net th a t he could use

to do th is referencing* F ortunately, there are not too many confusing

lo ca l systems in existence* I f there are no precise lo ca l systems a v a il­ able (mainly because o f the expense involved) i t is d i f f i c u l t fo r the local surveyor to t ie his surveys together* The USC&GS (United States Coast and Geodetic Survey) has established reference planes in each o f the states fo r ju s t th is purpose by taking a p ro je c tio n o f a p o rtio n o f " t h e e a rth 's surface th a t is a plane and re la tin g a ll o f th e ir geodetic

s ta tio n s to th is plane in a more e a s ily understood rectangular system o f

X's and Y8s. Each one o f the State Plane Coordinate Zones may cover a ll

or only p a rt o f a state*

Figure 1 shows the zones fo r the State o f Arizona. This fig u re also shows the coordinate axis fo r the East and Central Zones, the Y-axis f o r the West Zone f a lls ju s t o f f the map; and the X-axis f o r a ll three zones is a t 31® la titu d e which is below the state lin e ; thus the State

2 1 14 ° I I 3 ° 1 12 ° 111° | , 0 ° 1 0 9 ° 3 7 M O H A V E C 0 CO N I N O \ A V A J O A P A C H E 3 6 ° 3 6 ° O LU 3 5 ° cn Q _1 LU in ui LU u 3 4 ° 3 4 ° M A R I C O P A LU LU P I N A L 3 3 ° 3 3 ° W E S T Z O N E LU C E N T R A L Z O N E _{E A S T} ZONE 3 2 ° 3 2 LU A R I Z O N A M I L E S 2 0 4 0 Z O N E 0 , 0 F O R T H E C Z O N E 0 , 0 F O R T H E E 11 3 ° 4 5' I I 0 ° I 0 ' X ( C E N T R A L Z O N E ) ( E A S T Z O N E )

Plane Coordinates fo r any one zone are measured from the set o f axes fo r the zone. A l l State Plane Coordinate zones are established so th a t there w i l l be no negative values w ith in the zone. I t is seen from Figure 1 th a t State Plane Coordinates are lik e any other coordinate system w ith the fo llo w in g d iffe re n ce s: the system is regional in nature so th a t the coordinate values are large and the values o f X and Y cannot be con­ sidered a rb itra ry because the o rig in p o in t is d e fin ite ly located.

Geodetic s ta tio n s w ith th e ir p o s itio n expressed as State Plane Coordinates are becoming more and more common w ith attendant ease o f

ty in g in to the National Geodetic Net. There are no State Plane Coordin-

ate sta tio n s p e r sea but instead there are the control s ta tio n s o f the various sta te and federal agencies, as w ell as p riv a te organizations, whose positio n s are expressed in terms o f the State Plane Coordinate System in use in the area. Any survey th a t is executed w ith a t le a s t second order procedures and is tie d in to a s ta tio n th a t is referenced to the State Plane Coordinate System o f the area autom atically becomes a p a rt o f the system, and i f the surveyor w ill permit the re s u lts o f his survey to be used by o th e rs , the system is extended.

Other advantages o f coordinates are summarized by Richardus (1966, p. 1):

The a p p lica tio n o f coordinates in surveying to in d ic a te the re la tiv e p o s itio n o f points in the f ie ld and on maps has many advantages.

In the f i r s t place the use o f coordinates s im p lifie s and standardizes the computational methods, the basic elements w ith which the computations are carried out being defined uniquely. This is so fo r ca lcu la tio n s on the desk c a lc u la tio n machine; i t gains more importance in the case o f e le c tro n ic computers, where i t allows fo r very economical programming.

4 Secondly I t curbs the propagation o f p lo ttin g errors to such a degree th a t the a p p lic a tio n is mandatory fo r th is purpose alone. There is no propagation o f these erro rs in the computa- tions which are necessary to obtain coordinates from the o rig in a l observations 0 . . Therefore the e rro rs which would a ffe c t the ; re la tiv e p o s itio n o f points in a fig u re when p lo tte d g ra p h ica lly

are obviated by p lo ttin g the points by th e ir computed coordinates

separately . 0 "

T h ird ly i t f a c ilit a t e s the u n ific a tio n in to one general coordinate system o f fig u re s given in separate systems.

In the fo u rth place i t s im p lifie s the id e n tific a tio n and adm inistration o f the p o in ts.

For these reasons coordinates are used in control systems o f any major survey p ro je c t by manual conventional and'photogram- m etrical methods.

There are these advantages to regional coordinate systems and

many more. .

Photogrammetric engineering is concerned w ith the a c q u is itio n and in te rp re ta tio n o f both a e ria l and t e r r e s tr ia l photography. The a c q u is itio n o f photography in vo lve s, among other th in g s, the determi­ nation o f the p o s itio n o f a number o f ground control s ta tio n s ; the locations o f these control s ta tio n s are best expressed in reference to a State Plane Coordinate System because o f the area covered by a p ro je c t. With a e ria l photography, these control sta tio n s are required to obtain

the true o rie n ta tio n o f each photograph w ith respect to adjoining photo­ graphs in the p ro je c t area, e ith e r by analog or a n a ly tic a l means.

Analog o rie n ta tio n is the process o f properly o rie n tin g photographs in a mechanical p lo tte r ; c o rre c tly positioned ground control is needed to

control the planim etry o f the re s u ltin g map. Ground control expressed

in State Plane Coordinates is e a s ily p lo tte d and is a l i t t l e more accu­

o rie n ta tio n is a process whereby a few known ground s ta tio n s are used to control the photography and extend the ground control $ w ith the computa­ tions o f coordinates fo r interm ediate photographs being computed by use o f an e le c tro n ic computer. An e le c tro n ic computer can be programmed w ith plane coordinates b e tte r than geodetic, and the output is more com­ prehensible when plane coordinates are used. A n a ly tic a l photogrammetry is suited to State Plane Coordinates because the p ro je c t can cover a large area and the re s u lts are in an e a s ily understood form o f X's and Y 's.

The use o f a n a ly tic a l photogrammetry in cadastral surveying is a tta in in g more and more importance as the accuracy obtained from photo- grammetric processes improves, w ith current accuracies in the th ir d - order range and approaching second-order (depending on the f l i g h t

a ltitu d e ) . The method o f photogrammetric cadastral surveying is f a i r l y

simple: the known ground control sta tio n s and the desired points are

marked before the photography is taken; a fte r the photography is ob­ tained, the photo-coordinates o f a ll points are determined and mathe­ m a tic a lly analized to obtain the coordinates o f the unknown s ta tio n s . In th is process the p o s itio n o f the unknown sta tio n s are generally ex­ pressed in terms o f State Plane Coordinates because the p ositions o f the known sta tio n s are fed in to the computer in terms o f State Plane

Coordinates and i t s im p lifie s fu tu re reference to these points by placing them on a regional reference plane. The accuracies thus fa r a tta in a b le make the method extremely a ttra c tiv e fo r the survey o f large ru ra l tra c ts . Because the accuracy o f the system depend on the fly in g height (c u rre n tly producing X, Y and Z, ground coordinates accurate to

1/10,000 o f the fly in g h e ig h t), i t can also be used on urban tra c ts by varying the fly in g height but a p o in t o f dim inishing returns w ill be

reached. Surveys have been performed in downtown Los Angeles by photo­

graphing, by helicopter, from a height o f 300 fe e t because o f the d i f f i ­

c u lty involved in doing the work on the ground. A n a ly tic a l photogrammetry

w i l l provide, w ith in ten ye a rs, the method o f extending the basic h o ri­ zontal control by e sta b lish in g supplemental second-order control sta tio n s, expressed in terms o f the State Plane Coordinates, a t a fa s te r and

cheaper rate than is now possible.

In t e r r e s tr ia l photogrammetry a ll control sta tio n s and photo sta tio n s should be referenced to the State Plane Coordinate System to f a c ilit a t e p lo ttin g o f d e ta il, map c o n tro l, and planim etry.

By using State Plane C oordinates,for c o n tro llin g the photo- gram m etrically compiled topographic maps, fo r p recisely lo c a tin g the highway and it s structures on the ground and fo r rig h t-o f-w a y determi­ nation and lo c a tio n ; tra n sp o rta tio n engineering has been re vo lutio n ize d . With State Plane Coordinates i t is easy to maintain control on projects

th a t extend over large distances, where the curvature o f the earth could be a problem, and to s im p lify the computations th a t are done manually. By using a n a ly tic a l photogrammetry fo r extending the ground control used to o rie n t the photos th a t are used in map com pilation, highways can be designed w ith a minimum o f f ie ld work. The photos are c o n tro lle d at each end and a t a few interm ediate points by ground control stations whose positions are given in terms o f the State Plane Coordinate System

in use in the area. Rights-of-way can be determined by a n a ly tic a l photo­

are referenced on the State Plane Coordinate System. These points are used to control the analytical.com putations from which fu tu re or

c u rre n tly desired s ta tio n positions are determined, w ith the re su lts

given in terms o f the State Plane Coordinate System o f the area. Earth­

work q u a n titie s can also be determined d ir e c tly from photographs

u t iliz in g d ig ita l readout devices and e le c tro n ic computers w ithout going

as fa r as map com pilation. Most highway departments in the United States

use the State Plane Coordinate System in th e ir work ( i t was at the request o f the North Carolina Highway Department th a t the USC&GS f i r s t developed the State Plane Coordinate Systems fo r the various states in

1932). With highway departments using State Plane Coordinates, the

density o f monumented control along highways is increased.

Other advantages o f State Plane Coordinates are found in road re lo ca tio n where curves are to be elim inated or reduced. The use o f coordinates makes i t easy to tra n s fe r plans made in the o ffic e to

positions on the ground. I f a new road is to be run between points on

other roads, and the coordinates o f these points are known, a d ire c t lin e can be computed and i f no obstacles intervene, run out on the ground w ith o u t the use o f a random lin e . I f obstacles intervene, but a d ire c t lin e is to be located, a traverse is ru n , coordinates o f traverse sta tio n s determined, from which coordinates o f points on the desired lin e can be established on the ground. The s u b s titu tio n o f o ffic e computations fo r some f ie ld surveys is a great convenience because o f the e lim in a tio n o f many expensive random f ie ld traverses.

Control density is increased in the urban areas e ith e r by local governmental surveys or the USC&GS doing the primary control work on a cost sharing basis w ith the local government f i l l i n g in . This is being done in Phoenix, Arizona (Harmon, 1968, p. 512) and in Houston, Texas

(Gale, 1968, p. 415). In Public Land States (those states th a t are sub­

divided by the U. S. Rectangular System o f townships, ranges and

sections) the supplemental work can put coordinates on every section and 1/4 section corner, and eventually most s tre e t in te rse ctio n s would be coordinated. In the non-public land s ta te s , a g rid work o f sta tio n s

every h a lf m ile could be established. Figure 2 is the control network

established by the C ity o f Tucson as the s ta r t o f a p ro je c t to coordinate

a ll section and 1/4 section corners in the C ity o f Tucson. Figure 3 is

a fin ish e d p la t o f a p o rtio n o f th is p ro je c t; to maintain c o rre la tio n between the record dimensions and the surveyed q u a n titie s as w ell as the reduction o f confusion, bearings and ground distances are shown as w ell as the g rid data.

The a v a ila b ilit y o f, and method o f obtaining control data from the USC&GS is given in Appendix 2. The control established by the U. S. Geological Survey to control it s topographic mapping is available from the regional o ffic e fo r the area o f in te re s t.

The d e scrip tio n o f property corners using the State Plane Coordi­ nate System is the most permanent means o f referencing a vailable because the sta tio n s are tie d in to the National Geodetic Control Net, which is almost in d e s tru c tib le , and thus a ll sta tio n s th a t have State Plane Coordinate values have im plied geodetic p o s itio n .

>

10 < ? Z - £ - A /A / <S7~ <?^4/VZ- XPO / V •9 9 '’ o -* CXS"£T * S £ : C 7 - / 0 / V 3 3 o x - Z - / 3 3 v > e / < t £T. ^ & / A *.

Coordinates are a calculated q u a n tity and consequently they do not carry the legal weight o f a. monument in place (or th a t was set) or a bearing and distance. Monuments in place must control over everything, but i t is unfortunate th a t bearing and distance control over coordinates because the distances and bearings th a t are recorded on a p la t were probably computed from coordinates derived a fte r the adjustment o f the

traverse. Coordinates are also used to compute area (which is sub­

ordinate to coordinates) as w ell as fo r determining the true bearing and distance o f a random traverse along a property boundary (th is computed

lin e carries more legal weight than the coordinates from which i t was

d e riv e d ). A ll deeds th a t use coordinates to reference the property

corners must sta te the State Plane Coordinate System th a t is used as the basis o f the coordinates w ith in the deed.

Besides s im p lify in g map construction and property surveying, there are other ways o f using a State Plane Coordinate System. They improve the record keeping a b ilit y o f t i t l e companies and the county recording o ffic e s by givin g each piece o f property an id e n t if ie r th a t is unique (the State Plane Coordinates o f, say, the southeast corner as the id e n t if ie r ) . This system would be e sp e cia lly useful in the non-public land states where most o f the descriptions are on a metes-and-bounds

system. This type o f system is best illu s tr a te d by the Massachusetts

Land Court where a sta te agency is attempting to place a ll property corners on the State Plane Coordinate System in a jo in t p u b lic -p riv a te e f f o r t , to s im p lify t i t l e s and perpetuate corners. I f the U. S. Bureau o f Land Management were to update th e ir procedures fo r the subdivision o f Public Lands, State Plane Coordinates would have to be considered as

12

the best and fa s te s t way o f subdividing these lands, because the south­ east corner o f each township would not have to be used as the s ta rtin g p o in t, but any p o in t th a t had acceptable State Plane Coordinates could be used fo r the o rig in . In dependent resurveys a ll points th a t are re ­ established should be tie d to the State Plane Coordinate System in the

p ro je c t area. State Plane Coordinates would give tax assessors a b e tte r

way o f lo c a tin g tax parcels and improving the accuracies o f tax maps (which are q u ite often very poor and should never be used fo r property surveys). An in te re s tin g a p p lica tio n o f State Plane Coordinates is in the area o f ru ra l f i r e and p o lice p ro te ctio n . Instead o f givin g the p o s itio n o f a burning barn as: north on Route 10, fiv e m ile s , then l e f t on Route 15, 1.7 miles and then r ig h t on a d i r t road 1.3 m ile s; one could give the lo c a tio n as 0797-1351, which would be the coordinates in un its o f 1,000 fe e t, the dispatcher could give the inform ation to the f i r e tru c k s , which would have gridded maps o f the f i r e d i s t r i c t , and they could fin d the best way to the f i r e ; the same p rin c ip le also applies to p o lice work.

Probably the most in te re s tin g a p p lic a tio n o f State Plane Coordinates is th e ir use in conjunction w ith e le c tro n ic computers in the mineral in d u strie s (g e o lo g ic a l, petroleum, mining engineering and geology).

The area o f regional c o rre la tio n and study o f geological f o r ­ mations is one where the work can e a s ily be reduced and the output

increased. The X, Y, and Z coordinates o f points on the top o f the bed under study are obtained by f ie ld methods (mainly subsurface d r illin g ) and by using photogeological methods to obtain the inform ation regarding

13 the lo ca tio n o f outcrops ( th is reduces f ie ld work considerably). The inform ation thus obtained is fed in to a computer which runs a coqrdi- natograph th a t would p lo t the s tru c tu re contours on the top o f the bed. I f the X, Y, and Z coordinates are also obtained fo r the base o f the bed, an isopach map could be produced q u ite rapidly, which would speed up the paleogeographic studies needed in the search fo r bedded mineral

deposits and petroleum. Because o f the regional nature o f these studies,

a reference system is needed th a t w i l l not be grossly affected by the curvature o f the earth and can be e a s ily understood by everyone; State Plane Coordinates are the answer.

Other a p plications in mineral in d u s trie s are in p lo ttin g zones o f a lte ra tio n in a mining d i s t r i c t or around an igneous body, th is work can also be speeded up by using co lo r a e ria l photographs to obtain

a lte ra tio n values as w ell as the coordinates o f the points where a value is determined; in th is case the computer would not run a coordinatograph but instead a p rin te r would be used to show the various in te n s ity le v e ls . This type o f computer output can also be used fo r ore control w ith in a mine or fo r determining the ore concentrations in a prospect area, since

the maps could be p rin te d as fa s t as inform ation became a va ilable as to

the d ire c tio n th a t mining or exploration should proceed. In th is case,

only local conditions p re v a il, but to c o rre la te between mines and prospect areas State Plane Coordinates should be used to obtain the

maximum usefulness from a system o f regional c o rre la tio n . Using e ith e r

a coordinatograph or a p r in te r , a computer can also draw maps o f anomol.ies (magnetic, geochemical, g r a v ita tio n a l, and radiom etric) as

14 w e ll as s tru c tu ra l fe a tu re s . Because these features are often obtained

on a regional scale, a regional coordinate system is required and State Plane Coordinates are again the best answer.

W ithin mine workings, connections have to be made between

d iffe re n t levels and between various d r if t s on a le v e l, th is is done by coordinates. These coordinates are generally on a local system which is acceptable i f a ll o f the work is to be done w ith in the one mine. I f the workings are to be extended beyond the end lin e s o f the claim in to an

area where there are workings o f other mines, then a ll the workings o f the various mines in a d i s t r i c t should be correlated and referenced on

the State Plane Coordinate System. Then there would not be an accidental

holing through because o f confusion over d iffe r e n t coordinate systems and i t w ill be easier to t e l l who owns what and where. Another problem

encountered in mine operations is depth. Although the surface is

referenced to the State Plane Coordinate System on an adjusted datum (see Chapter 3 ), there w i l l be a disagreement when the depth o f the workings go below the 2,000 fo o t li m i t fo r the 1/10,000 ground to g rid discrepancies established fo r the surface. The only way to handle the problem is to esta b lish a series o f adjusted datums a t 4,000 fo o t in t e r ­ vals (2,000 fe e t down from one and 2,000 fe e t above the next one down) w ith the ju n c tio n between datums in an unused or barren pa rt o f the mine

to elim inate confusion over what datum is in use. The f i r s t datum down would be established about 2,000 fe e t below the mean elevation o f the surface structures and operations; each datum would be designated by a le t t e r and the combination fa c to r required to place everything on the State Plane Coordinate Reference System. As an example, the mean

elevation o f a mine is 4,000 fe e t and the scale fa c to r a t the mine is 0.99993 and the elevation fa c to r is 0.99981 or a combination fa c to r (0.99993) (0.99981) = 0.99974. The fa c to r o f 1.00026 which is

(1/0.99974) is used to elevate a ll e x is tin g control based on the State Plane Coordinate System; the fa c to r, 0.99974, is used to convert the adjusted coordinates back to the State Plane Coordinate System; the surface datum would be designated as A .99974 and the next datum down would be designated as B .99979 and so on down the mine.

Another use o f State Plane Coordinates is in conjunction w ith

a n a ly tic a l photogrammetry. A ll mining claim corners in an area would

be premarked w ith old t ir e s and lime fo r good photographic imagery. A fte r the photographs were obtained, they would be mathematically

analized to obtain the coordinates o f the claim corners. Once the

State Plane Coordinates fo r the corners are known, i t is easy to p lo t them and determine the true re la tio n s between the various claims in a d i s t r i c t .

Since State Plane Coordinates have been in existence fo r more than t h ir t y years, there must be reasons why they have not been more

widely accepted. Brown and Eldridge (1962, p. 378) give the fo llo w in g

major disadvantages:

1. Many surveyors, attorneys, and t i t l e men do not understand

the g rid p ro je c tio n s .

2. Many states have no permissive le g is la tio n .

3. Extra cost is involved in a d d itio n a l control surveys.

4. Many areas do not have s u ffic ie n t control fo r tie s to the system.

16 5. Many e x is tin g surveys are not o f s u ffic ie n t q u a lity or

accuracy to contribute positions to the system.

The f i r s t problem can only be overcome by education, through short courses, seminars, and conferences sponsored by the surveying associations in conjunction w ith lawyers and t i t l e men.

Once the major users o f State Plane Coordinates have been educated on the need and usefulness o f the system, i t is easier to get the sta te le g is la tu re s to pass enabling le g is la tio n fo r the use o f the system in the s ta te . A sample b i l l is given in M itc h e ll (1948). At le a s t twenty-nine states have passed enabling le g is la tio n thus fa r (see Appendix 3 ); these states are generally where the surveyor's associations

have been most active in the ro le o f educators in the use o f the system. The th ir d and fo u rth problems can be overcome by using modern equipment and methods, e sp e cia lly e le c tro n ic distance measuring equipment

and photogrammetry. For example, a lin e fifte e n miles long can be

measured in a day or two e le c tro n ic a lly , whereas the conventional methods would take ten days or more. Moreover, there would be a decrease in

costs (an e le c tro n ic u n it can be rented fo r what one would pay a fo u r man crew per day, but i t does the work much fa s te r and more a ccurately).

Photogrammetry can reduce o ve ra ll costs o f supplemental control by 50%. The la s t problem is the biggest one, because the only way to up­ grade e x is tin g surveys is to redo them using more accurate methods and placing the re s u lts on the State Plane Coordinate System. The objection to redoing surveys is th a t many people th in k th a t the resurveys would have to be done immediately; th is is not necessary since most surveys

17 w ill eventually be redone any way and they can be placed on the State

Plane Coordinate System as they are executed. Not only are older surveys

a problem, but many recent surveys are being done in a substandard manner and the idea o f having to redo one's surveys to get them up to standard (p re fe ra b ly second-order but no worse than th ird -o rd e r) causes some surveyors to oppose the State Plane Coordinate System; education can be a help in th is s itu a tio n .

This paper w i l l examine plane coordinates in some d e t a il, e sp e cia lly computations using coordinates, as a co n trib u tio n to the education o f surveyors in the use o f State Plane Coordinates.

Table 1 defines the fa cto rs and constants used in th is paper; terms not defined here w ill be defined when f i r s t used.

18 TABLE 1„

1 sin

Factors and Constants Used in Formulas (Figure 4 illu s tr a te s terms marked *)

1 = ca* = the e a rth 's semi-major axis ~ 6 378 206.4 m

) = cb* = the e a rth 's semi-minor axis = 6 356 583.8 m

2 a2 - b2

i = ——^—— = the e c c e n tric ity r a tio o f the earth squared a

= 0.00676 86579 97 (i = 3.80833 3333 f t .

I" = tan 1" = arc 1" = 0.00000 48481 37 > = 20,906,000 fe e t

Factors

I = r p* = radius o f curvature o f the earth at any p o in t on

the e llip s o id - 6,335 034 .^ m (1 - e sin $ ) 3/ Z

j = nm* = radius o f curvature o f the earth in the prime v e rtic a l = . 6 .378 M W m.

(1 - e sin <f>) z

j - cn* = mean radius o f curvature o f the earth =

\ = arc o f p a ra lle l fo r 1" o f la titu d e = (Ncos <j> arc 1")

> = na* = meridional distance between two p a ra lle ls

6 335 034.502(A cj> - B (sin 2*2 - s in 2*) + C(sin^*2 " s in % ^))

19

TABLE 1. Factors and Constants Used in Formulas—Continued

where <j> is in radians

A = 1,005 10093 B = 0,005 1202 C = 0,000 0108 Rb = n t* Nq cot (j)

<j> is the geodetic latitude and X is the geodetic longitude,

A<j> = difference in latitude in seconds = <f>? “ 4>-j

AX = difference in longitude in seconds = Xg - x^

Aa = the convergence o f the meridian fo r both geodetic coordinates and the Transverse Mercator p ro je c tio n . <j> » the mean l a t i tude =■ ^ 2 * *1^

e - the mapping angle or convergence in the Lambert p ro je c tio n . R, = the radius o f curvature o f an arc o f la titu d e in the

Lambert p ro je c tio n .

k = p ro je c tio n or scale fa c to r

The sub script £ indicates the o rig in lin e or p o in t o f a system. Values fo r JR and N_ are found in Table 14.

20

o

CHAPTER 2

MAP PROJECTIONS

The re la tiv e p o s itio n o f points on the e a rth 's surface can be indicated by a three dimensional coordinate system, Bomford (1962* p. 83) discusses fiv e methods o f doing th is . The only ones th a t w i l l be

considered'here are plane and geodetic coordinates. The height* Z* is

measured above the mean s e a -le v e l. The other two coordinates * X and Y* are derived by a p ro je c tio n o f the geodetic coordinates ( la titu d e and longitude) onto a surface which can be developed in to a plane. The methods o f p ro je c tio n and the re la tio n s h ip between plane and geodetic coordinates are the subject o f th is chapter.

The surface o f the earth cannot be fla tte n e d w ith o u t some d is to r tio n ; hence* i f a considerable p o rtio n o f the e a rth 's surface is to be shown on a map* the dimensions must be d is to rte d in some way. The character o f the d is to rtio n can be co n tro lle d i f the points on the

e a rth 's surface are mathematically projected upon a surface th a t can be developed in to a plane. A fte r such p ro je c tio n and development* the points w i l l represent, w ith a minimum o f scale d is to r tio n , the co rre ct re la tiv e p ositions o f the corresponding points on the e a rth 's surface.

D is to rtio n s on a map are unimportant only i f they are too small to be p lo tte d a t the scale o f the map. However, d is to rtio n s entering in to the use o f the plane coordinates are unimportant only i f they are

22 so small as to f a l l w ith in the usual lim its o f accidental errors

incurred in f ie ld operations (Rayner, 1957$ p. 475).

The fo llo w in g l i s t o f c r it e r ia (adapted from Colvocoresses, 1965$ p. 12) is useful in deciding what type o f p ro je ctio n is to be used fo r a plane coordinate system:

1. I t must be conformal to minimize angular e rro rs ,

2. I t must be continuous over a large area w ith the fewest zone

boundaries possible.

3, I t should not develop scale errors in excess o f 1:10*000.

4. I t must be s u ita b le f o r extension over a comparatively large area o f the e arth, .

5, I t must be s u ita b le fo r unique referencing in a rectangular system o f Northings (Y) and Eastings (X).

6, The convergence o f the meridians should not exceed fiv e degrees.

The p ro je c tio n methods th a t w i l l s a tis fy most o f the above

requirements are: a tangent plane and a secant plane. The former does

not completely s a tis fy a ll o f the requirements because o f i t / s lim ite d width when the scale e rro r equals 1:10$000 (a t 56 miles r a d ia l, when

tangent a t a p o in t, or 56 miles on e ith e r side o f the lin e o f tangency fo r developed p ro je c tio n s ). The la t t e r s a tis fie s more o f the c r it e r ia fo r a p ro je c tio n systern and permits the use o f wider zones as w i l l be shown. The tangent plane w ill be discussed f i r s t because o f it s wide­ spread use in local systems as w ell as it s a p p lic a b ility to a ll systems o f plane coordinates, developed or not. A fte r discussing the tangent piane systems, the two secant p rojections th a t are used in the State

23 Plane Coordinate Systems ( the Transverse Mercator and the Lambert

Conformal Conic) w i l l be examined.

Tangent Plane

A plane can be tangent to the earth a t e ith e r a p o in t or a lin e , as is the case o f developed p ro je c tio n s . With the lo ca l systems in use, the plane is tangent a t a p o in t; th is p o in t should be a USC&GS tria n g u ­ la tio n s ta tio n located near the center o f the p ro je c t ( th is permits easy

conversion to the State Plane Coordinate System whenever d e s ire d ). Most

o f the tim e , however, i t is an a rb itra ry p o in t o f unknown la titu d e and lo n g itu d e ; the conversion to State Plane Coordinates can s t i l l be done w ith some ad d itio n a l f ie ld work and the formulas given on page 92, The e levation o f the plane should be the mean ele va tio n o f the p ro je c t area.

Choosing a p o in t near the center o f the p ro je c t area is

necessary because the scale varies r a d ia lly from the p o in t o f tangency ( lin e a r ly away from a lin e o f tangency f o r developed p ro je c tio n s ). Suf­ f ic ie n t ly large X and Y coordinate values should be assigned to th is p o in t to avoid negative values. The e ffe c t o f scale change on the r e l­ a tiv e accuracy o f both lin e a r and angular u n its is apparent in Table 2 and Figures 5-7.

Using the mean e levation o f the p ro je c t area saves computation time and elim inates an a d d itio n a l source o f e rro r th a t would arise i f the distances were reduced to a plane, tangent at s e a -le v e l. Another reason fo r using the mean elevation is to maintain a close s im ila r ity

between ground and g rid distances. I f the datum plane is to be tangent

TABLE 2. Linear and Angular Displacements Away from the Line o f Tangency (C la rk , 1963, p. 440)

24

Distance from Central Meridian (M iles) 50 100 150 200 R elative E rror (Distance) 1:12,510 1:3,142 1:1,396 1:785 Angular E rror and Displacement 8" .2 1:25,154 32" *8 1:6,289 1" 13" .9 1:2,791 2 6 11" e3 1:1,571

25 L I N E A R ERROR - P L I N E A R ERROR L I N E A R DI SP L A C E ME N T " P , G E O D E T I C NE P L A N E L I NE LINE T A N G E N T TO T H E G E O D E T I C L I N E A N G U L A R ERROR POI NT OF T A N G E N C Y

Figure 5. E ffe c t o f Linear and Angular Errors (g re a tly exagerated)

F ig u re 6. L in e a r E rr o r on a T a n g e n t P la ne (e x a g e ra te d cr o ss s e c tio n ) LINEAR ERROR PLANE DISTANCE P O I N T OF TANGENCY--► G E O D E T I C DI STANCE GEOID / C E N T E R OF THE E A R T H £

S C A L E F A C T O R

DISTANCE ( IN MI LES FROM C E N TR AL MERIDAN) 95 7 6 5 7 3 8 . 0 0 0 4 2 5 0 0 U T M 0 0 0 3 3 3 3 3 1.0002 5 0 0 0 A P P R O X I M A T E L I M I T OF S T A T E P L A N E P R O J E C T I O N E RROR .0001 ₁₀₀₀₀ 1.0000 DI STANCE IN 1 0 0 , 0 0 0 FOOT U N I T S

Figure 7. Graph o f Projection Error

ro ' -j R E L A T IV E ERROR

. 28 fa c to r fo r the reduction o f distances measured on the ground to sea- level is computed w ith the fo llo w in g formula: SL = - -P-y (use the approximate value o f p ) . The value is 0,00000004783 fe e t per fo o t o f e le va tio n , thus the co rre ctio n at 5 90Q0 fe e t would be 0,00024 fe e t per fo o t or .24 fe e t per thousand fe e t o f ground distance. A ll ground distances must be reduced to e ith e r sea-level or the datum fo r the

area, only one reduction fa c to r is used. Table 3 and Figure 8 show

the e ffe c t o f e levation in producing a ground to g rid discrepancy. A ll

geodetic distances are sea-level distances.

As one gets fa rth e r and fa rth e r away from the tangent p o in t o r the lin e o f tangency, the e ffe c ts o f the convergence o f the meridians on the azimuth become more pronounced. The amount o f th is e rro r is given by the fo llo w in g formula (Reynolds* 1932* p. 8 ): Aa" = AX"sin<j>m, Table 4 gives examples o f the computation fo r convergence* fig u re s illu s t r a t in g convergence are shown on pages 35 and 52,

Secant Plane

The d iffe re n ce between a secant plane and a tangent plane is th a t the former "cu ts" the earth w hile the la t t e r does not. To make a tangent plane secant to the earth* a negative scale fa c to r is introduced along the lin e o f tangency. This w ill increase the width o f the zone to about 158 miles before the scale e rro r exceeds 1:10,000. The use o f a secant p ro je c tio n balances the scale throughout the p ro je c tio n area.

The two p rojectio n s th a t w i l l be considered are the Transverse Mercator and the Lambert Conformal Conic p ro je c tio n s , because they meet the requirements lis te d on page 22. The Transverse Mercator p ro je ctio n

TABLE 3. Magnitude o f Differences in Distance Due to Elevation (P ryo r9 1967, p« 51) Elevation (fe e t) 1,000 2,000 3.000 4.000 5.000 6.000 8,000 10,000 12,000 14,000 R elative E rro r 1:20,910 1:10,450 1:6,960 5,220 4,180 3,480 2,610 2,090 1,740 1:1,490

IQ C 1 fD CO £u *o O -+) fD < Oi c+ O Q m -s o -s 0.9992 ₁₂₅₀ 0 . 9 9 9 4 ₁₆₆₇ o: g % u_ FACTOR = R + Z O 0 . 9 9 9 6 _{2 5 0 0} O Q Q UJ cr Z 0 . 9 9 9 8 O 5 0 0 0 UJ UJ 1.0000 10000 2000 1 4 0 0 0 8 0 0 0 ) 6 0 0 0 ELEVATI ON IN FEET 4 0 0 0 2 0 0 0 -2000 w o R E L A T IV E E R R O R

31

TABLE 4. Convergence Computation

Geographic:

Act" = AX sin <j> AX11 = 3784.79

= (3784.79)(.53349560) <j> = 32° 14' 31" Aa" = 2019.169

o r Aa = 331 39.17"

Transverse Mercator (Arizona Central Zone):

Aa " = MX' X » 872793.81

= (.006171)(327206.19)X' = 327206.19

A a " = 2019.169 Y ~ 390000.00

or Aa = 336 39.17" M - .006171

Lambert (Colorado Central Zone):

6" = arctan (X '/R ^-y) X = 879637.91

= arctan (1120362.09/25845230.9)

Xs = 1120362.09

6" « .0433488907 Y * 397821.83

32 is a " c y lin d r ic a l" development o f the e a rth 8s surface onto a plane and the Lambert p ro je ctio n is a "conic" development o f the e a rth 's surface, A p ro je c tio n o f th is type should not be in te rp re te d as a p ro je ctio n from

the center o f the spheroid (defined in Appendix 1) upon e ith e r the cone or c y lin d e r (in Figure 4 the radius o f p ro je c tio n is shown as r p ) „

Figure 9 is a cross-section o f a ty p ic a l secant p ro je c tio n .

Transverse Mercator

The Transverse Mercator p ro je ctio n is a conformal p ro je ctio n th a t may be v isu a lize d as a c y lin d e r on which a p o rtio n o f the e a rth 's surface has been developed (Figure 10). The p a ra lle ls and meridians in the p ro je c tio n (except the Central Meridian) are curves (Figure 11), The Central Meridian is a s tra ig h t lin e w ith a constant scale and rec­ tangular coordinates are measured p a ra lle l and perpendicular to i t . The axis o f the c y lin d e r is normal to the plane o f the Central Meridian. Because o f th is la s t fe a tu re , the p ro je c tio n is s u ita b le fo r use in areas w ith a large north-south extent or where an area can be subdivided in to

the fewest zones by use o f the long north-south dimension.

The lin e s o f exact scale are e q u id is ta n t from the Central

Meridian and form minor c irc le s on the sphere (Figure 10). The area

between the lin e s o f exact scale covers 2/3 o f the zone width and the area beyond the lin e s covers 1/6 o f the remaining area on each side

because the scale e rro r formula is parabolic (see Figure 7 ), The approxi­ mate scale fa c to r is given by M itch e ll and Simmons (1945$ p. 60) as:

x ! 2

k (fe e t per thousand) = R - 0.0001144 ("ffffiffg’) where R is the reduction in fe e t per thousand along the Central Meridian and is found in Appendix

( G R I D D I S T A N C E S AR E G R E A T E R ( GRI D D I S T A N C E S ARE G R E A T E R ( P R O J E C T I O N , OR GRI D, D I S T A N C E S A R E S H O R T E R T H A N S E A - L E V E L D I S T A N C E S . ^ ^ M A P P R O J E C T I O N S U R F A C E 4 SCA L E FACTOR E Q U A L S 1 . 0 0 0 0 0 ( S E A - L E V E L D I S T A N C E S = G R I D D I S T A N C E S ) G R I D , O R C O M B I N A T I O N , F A C T O R E Q U A L S 1 . 0 0 0 0 0 ( M E A S U R E D DI S T ANC E S = GRI D D I S T A N C E S ) L I M I T OF Z O N E ( U S U A L L Y AT A S T A T E OR C O U N T Y L I N E ) S E A - L E V E L F A C T O R E Q U A L S 1 . 0 0 0 0 0 ( M E A S U R E D D I S T A N C E S = S E A - L E V E L D I S T A N C E S )

34

ARC AT L E V E L OF SURVEY AREA

ARC AT S E A - L E V E L , A R C P R O J E C T E D

ONTO T H E CYL I NDER

LU LU LU o LU O o _o LU O (/) _LU S E A L E V E L S P H E R O I D OF E A R T H

Figure 10. Transverse Mercator P ro je c tio n : In te rse ctio n s o f

35 G R A T I C L E GR I D R E L A T I V E O F S C A L E M A G N I T U D E E R R O R

Figure 11. Development o f the Transverse Mercator P rojection

S C A L E G R E A T E R T H A N U N IT Y

36

3, fo r each zone. The precise formula is given in Appendix 5, The

scale fa c to r projects the geodetic (s e a -le ve l) distance onto the re fe r­ ence plane to obtain the g rid distance.

Since the meridians converge on the g rid * there w i l l be a discrepancy between the geodetic and g rid azimuths, which are measured

from south, except along the Central Meridian. The convergence, which

can be fig u re d using e ith e r geodetic or rectangular coordinates, is dependent on the X and Y coordinates or the la titu d e and longitude o f

the p o in t (see Figure 11). Using rectangular coordinates, the formula

is = MX’ - e; w ith e_ a tabulated correction fo r the second term o f

the precise formula (Appendix 5 ), and X' = X - X value o f the Central

Meridian (u su a lly 500,000); the sign is im portant, M is tabulated in

each o f the USC&GS Special P ublications fo r the d iffe re n t Transverse Mercator p ro je c tio n zones th a t are used by the several states based on

th a t p ro je c tio n . Using geodetic coordinates the formula is

Aa = AA"sin<f> - g, w ith £ a tabulated co rre ctio n from the precise formula

(Appendix 5 ). Examples are shown in Table 5. From these re la tio n s the

g rid azimuth equals the geodetic azimuth minus Aa minus the arc-to-chord c o rre c tio n .

The arc-to-chord correction must be applied when the length o f a tria n g le side or a transverse leg exceeds about three miles depending on the Y value; f o r the Transverse Mercator p ro je ctio n the formula is

approximation: 2.36 AXAy; where ax is the average X8 values and Ay is

the d iffe re n ce in the Y values o f the two p o in ts ; the ax and Ay values

(6Pq sin I " )

TABLE 5» Differences Between the Two Arc-to-Chord Formulas

S tation Distance Correction (Seconds)

(M iles) (Exact) (Approximate)

From CE to : Pusch 9.5 3.26 3.54 Sahuaro 6.4 1.15 1.26 Black H ills 11.9 "3.39 -3.60 Warner 2,8 "0.58 "0.62 Cat Mountain 7.2 "1.17 -1.24 ,x: Wasson 11.6 0.82 0.86.-'OS' •

38 are in un its o f 100,000 fe e t. The curvature o f the geodesic (defined in Appendix 1) lin e is always concave toward the Central Meridians. An example from Simmons (1968) is shown in Figure 12. This formula is _ accurate to the nearest 1/2 second, but beyond th a t problems can a ris e , as shown in Table 5 and Figure 13$ where the co rre ctio n was computed fo r a series o f d ire c tio n s turned on the ro o f o f the C iv il Engineering

b u ild in g a t The U n ive rsity o f Arizona fo r both the exact and approximate formulas. Table 4 shows th a t the arc-chord co rre ctio n is dependent on Y. The e ffe c t o f th is correction is shown in Figure 14. This graph was

computed using the approximate formula.

For Plane Coordinate Systerns based on the Transverse Mercator p ro je ctio n the fo llo w in g formulas (from Special P u b lication 257, Plane Coordinate P rojection Tables f o r Arizona) are used fo r the in te r -

conversion between geodetic and rectangular coordinates. This procedure s im p lifie s the 1ocation o f engineering structures th a t require geodetic p o s itio n s , such as radio towers. A ll o f the functions required fo r the computations are found Yn the USC&GS Special P ublications f o r the states

th a t use the Transverse Mercator P ro je ctio n . The conversion o f geodetic

coordinates to plane coordinates uses the form ulas: Xs = Hax” + a*b

X = X' + False Easting

Y = v o +

v

2tc

39 Central Lavender Meridian P ro je c te d ge odetic tine Indian Carnes

Geodetic Angles Corr.

Lavender 56' 11 36.21 -0.57

Carnes 48 59 21.81 +7.39

Indian 74 49 05.24 -10.06

180 00 03.26 -3.24

All coordinates in units of 100 thousand feet.

y Lavender 1.61 15.' Carnes 2.43 14.: Indian 1.19 14.1 dx Ay Corr. = 2.36( A x ) ( Ay) (1) 3.60 1.06 +9.01 (2) 3.19 0.14 -1.05 (3) 3.19 0.14 + 1.05 (4) 2.98 1.20 +8.44 (5) 2.98 1.20 -8.44 (6) 3.60 1.06 -9.01 G rid A ngles 35. M 29.20 55.18 00.02 Corr. to Angles -<1H<2) -10.06 -(3)+(4) +7.39 -(5)+(6) -0.57 -3.24 3.39 2.57 3.81

40 ."V PUSCH SAHUARO WASSON WARNER CE CAT ' M I N BLACK H I L L S NOTE: GEODETI C L I N E S GREATLY E X A G G E R A T E D ON CURVES. 5 0 5 10 S T A T U T E M I L E S 1___________I__________I________ J

A Y (IN U N IT S OF 1 0 0 ,0 0 0 F E E T ) 20 . 7 0 9 -0 2 _L_l 3. 0 i_L j i _i i i_j i i i i__ 2.0 A X ( IN UNI TS OF 1 0 0 , 0 0 0 ) 4. 0

42 The reverse process is given by the fo llo w in g formulas:

P ( X V I O O O O ) 2 + d = V ( A X ' V I O O ) 2 + c

YQ = Y - P(Xl /10000)2 - d

The la titu d e is obtained from the table f o r Using the la titu d e ju s t obtained8 H is obtained from the ta b le s:

X' = X - False Easting

approximate a x" = X'/H

The arc-sine co rre ctio n fa cto rs are £ from the la titu d e and b^ from the approximate a x" ; then a x" = (X8 + a»b)/H. Then a x" is added algebra­ ic a lly to the longitude o f the Central Meridian to obtain the longitude o f the p o in t. The re la tio n s h ip s between the fa cto rs and examples o f computations are shown in Figure 15 and Tables 6-7. Tabular values used fo r the computations are found in Appendix 4.

The m ilita r y has established a world-wide system o f Transverse Mercator p ro je ctio n zones known as the Universal Transverse Mercator

(UTM) system. The differences between the UTM and the Transverse

Mercator p ro je c tio n used in the sta te coordinate systems are: (1) a

wider zone width $ 6° versus 2° o f longitude maximum f o r the state systems; (2) consequently, a la rg e r scale fa c to r (1 /2 ,5 0 0 ); and (3) a d iffe re n t set o f tables which are computed from the basic formulas in Appendix 5. Since a UTM zone is the same everywhere on a p a rtic u la r spheroid, only one set o f tables are needed fo r the interconversion computations. These tables include a b u i l t in scale fa c to r on the

43 LU LU U A X |Ao P ( X' /I O. OOO) 100 X = 5 0 0 , 0 0 0 F E E T Y=0 0

Figure 15. Rectangular Coordinates on the Transverse Mercator

S T A T E : A R I Z O N A Z O N E : ^ . C E N T R A L C E N T R A L M E R I D I A N : I M 0 5 5 ' S T A T I O N P U S C H ( u s e s ) _x' = H A X ± a b 3 0 2 0 1 5 . 6 8 <P 32 ° 2 2 ' 18'.' 960 _{1 3 8 0 . 5 2} X 1 1 0 ° 5 6 ' 18'.'4 13 T A B U L A R Y 4 9 9 0 1 9 . 9 3 A X = C E N T R A L MER.-X 5 8 ' 4 l" 587 X 8 0 2 0 1 5 . 6 8 A X" 3 5 2 1 . 5 8 7 Y 5 0 0 4 0 0 . 4 5 r A X ' ' i 2 . 1 0 0

J

V 1 .11 3 2 9 1 GEOD. AZ. TO AZ. MK. 0 3 ° 5 7 ' 01:' 6

a b - 0 . 8 9 6 6 . 0 1 6 G R I D AZ. TO AZ. MK. 0 3 ° 2 5 ' 3 6" 0 C ab - 0 . 1 3 3 - 5 . 3 9 0 X = x' + 5 0 0 , 0 0 0 A a " = A X " S i n + d Y = T A B. Y + V A X

-r±

S T A T E : _ A R I Z O N A Z O N E : C E N T R A L C E N T R A L M E R I D I A N : 1 1 1 * 5 5 ' S T A T I O N P U S C H ( U S G S ) X 8 0 2 0 1 5 . 6 8 Y 5 0 0 4 0 0 . 4 5 c — 5 0 0 0 0 0 . 0 0 P ( X / 1 0 0 0 0 ) 2 + d — 1 3 8 0 . 5 2 X* 3 0 2 0 1 5 . 6 8 Y o 4 9 9 0 1 9 . 9 3 p 1.5 13 3 9 _{A P P R O X A X"} = X' v H 3 5 2 1 . 5 3 d + 0 . 1 0 A X = ( X 1 a b ) — H 3 5 2 1 . 5 8 7 H 8 5 . 7 6 2 7 7 3 A X 5 8' 4 l'.' 5 8 7 o b - 0 . 8 9 6 6 . 0 1 6 - 5 . 3 9 0 C E N T R A L M E R I D I A N II 1° 5 5 ' 00".0 0 0 $ 3 2 2 2 1 8 . 9 6 X = C M — A X 1 1 0 ° 5 6 1 8 : 4 1 3 W H E N a b IS r p r - X' N U M E R I C A L L Y G R I D A Z I M U T H = G E O D E T I C A Z I M U T H — A a " I IN v r> ^ M o L.

46 the basic equations f o r the UTM are given in the U„ S. Army Technical

Manuals 5-241-2 through 5-241-31. These tabulations are designed fo r

the various spheroids th a t are in use throughout the world today.

In s tru c tio n s on using the UTM is given in the U. S. Amy Technical Manual 5-237; Surveying Computer's Manuals, (1964). The re la tio n s h ip between these parts and a sample computation are shown in Figures 16-17 and Tables 8-9. Tabular values used fo r the computations are found in Appendix 4.

More w ill be said about the Transverse Mercator p ro je ctio n la te r.

The Lambert Conformal Conic P rojection (h e re in a fte r calle d the Lambert p ro je c tio n ) is a conformal p ro je ctio n th a t may be visu a lize d as

a cone on which a po rtio n o f the earth has been projected. Being a

secant p ro je c tio n , the cone 'in te rs e c ts * the earth along two p a ra lle ls o f 1:1 scale, known as Standard P a ra lle ls (Figure 18). The axis o f the cone is normal to the plane o f the Central P a ra lle l and coincident w ith the pola r a xis. The meridians are converging s tra ig h t lin e s and the p a ra lle ls are a series o f concentric arcs (Figure 19). The meridian a t the center o f the p ro je c tio n is re fe rre d to as the Central M eridian, but i t is not a lin e o f constant scale as was the case w ith the Transverse Mercator p ro je c tio n ; the p a ra lle ls are arcs o f uniform scale.

Because the scale d eteriorates in a north-south d ire c tio n , the Lambert p ro je ctio n is used fo r states having a large east-west dimension, or th a t can be subdivided in to the fewest possible zones th a t w ill cover the desired area.

47

5 00,00 0 M

A a

3 < -< E'

EQUAT OR

Figure 15. Universal Transverse Mercator Coordinates from Geodetic

C E N T R A L M E R ID IA N 48 E Q U A T O R

U N I V E R S A L T R A N S V E R S E M E R C A T O R G R I D C O O R D I N A T E S S T A T I O N C E 1 o r AT 1 ON : P I M A C O U N T Y , A R I Z O N A 7 0 N F : *2 S P H F R O I D : C L A R K E 1 8 6 6 U N I T : M E T E R ua t iT u o ? , <p 3 2 ° 1 4 ' 04 " 7 7 P _{0. 0 1 7 0 9 2} L O N G I T U D E , \ _{H O ° 5 7 ‘ 0 9" 0 8} T A B U L AR ( I V ) (-;v en mi nut e o f <t> ) 261 7 1 4 . 9 4 2 P 2 0 . 0 0 0 2 9 2 1 4 C E N T R A L M E R . D i A N . \ _ "o o o I N T E R P O L A T I O N F O R S E C O N D S OF - 3 . 7 9 9 p 3 0 . 0 0 0 0 0 5 A X 0 2' 5 0'.' 92 A 2( I V ) F R O M g r a p h - 0 . 0 0 1 p 4

( I V) 2 6 1 71 1 . 1 4 2 TABULAR i 1 _{I EVEN M.NUTE OF} ) < p ) 3 5 6 6 1 1 0 . 9 4 6

tabular (V ) t E, EN MIN JTE OF ( p ) 4 4 . 5 4 9 INTERPOLATION FOR SECONDS OF <p 1 4 6 . 8 6 7 INTERPOLATION FOR SECONDS OF <£ - 0 . 0 0 5 _{( i ) 3 5 6 6 2 5 7 . 8 1 3} ( V ) 4 4 . 5 4 4 TABULAR (II) (E » EN M.NUTE OF <£) 3 3 8 3 . 7 6 9

(IV) P 4 4 7 3 . 1 6 6 INTERPOLATION SECONDS OFFOR 0 . 0 7 5

( V ' P 2 0. 0 ! 3 ( ID 3 3 8 3 . 8 4 4 ( i i ) p * 0 . 9 8 8

FROM GRAPH _{B 5} ( I I I ) 2 . 2 0 3 ( ii i ) p 4

WE S ^ ° F CENTRA L MER.DIAN E* 4 4 7 3 . 1 7 9 FROM GRAPH _As

FA^SE EASTING F E 5 0 0 0 0 0 . 0 0 0

E 5 0 4 4 7 3 . 1 7 9 N 3 5 6 6 2 5 8 . 8 0