LOAN COPY: RETURN TO AFWL (WLIL-2) KCRTLAND AFB, N MEX

AIRCRAFT NAVIGATION

by SI SI FedcbipzTransport PressMoscow, 1966

i

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7.

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N A T I O N A L A E R O N A U T I C S A N D SPACE A D M I N I S T R A T I O N

W A S H I N G T O N , D. C.

FEBRUARY 1 9 6 9

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k

TECH LIBRARY KAFB,

NM

I111 111l0068952iltl1 llIWI 111H I1 ll l

AIRCRAFT NAVIGATIONBy S. S. Fedchin

Translation of: "Samoletovozhdeniye . I t "Transport" Press, Moscow, 1966

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION . -- -_ - For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 CFSTI price $3.00

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TABLE OF CONTENTS

..................................................... INTRODUCTION ................................................. CHAPTER O N E . C O O R D I N A T E S Y S T E M S A N D ELEMENTS OF AIRCRAFT NAVIGATION ........................................ 1. E l e m e n t s o f A i r c r a f t M o v e m e n t i n S p a c e ............... 2 . C o n c e p t s o f S t a b l e and U n s t a b l e F l i g h t C o n d i t i o n s ........................................... 3 . Form a n d D i m e n s i o n s o f t h e E a r t h ..................... 4 . E l e m e n t s Which C o n n e c t t h e E a r t h ' s S u r f a c e w i t h T h r e e - D i m e n s i o n a l S p a c e ......................... 5 . C h a r t s . Maps. a n d C a r t o g r a p h i c P r o j e c t i o n s ........... D i s t o r t i o n s o f C a r t o g r a p h i c P r o j e c t i o n s ............ E Z Z i p s e o f D i s t o r t i o n s ........................... D i s t o r t i o n o f L e n g t h s ............................ D i s t o r t i o n of D i r e c t i o n s ......................... D i s t o r t i o n of A r e a s ..............................ABSTRACTClassification of Cartographic Projections

xi

xiii

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4 7 9 12 14 14 15

. 2. 3. 4.1

D i v i s i o n o f P r o j e c t i o n s by t h e Nature of the DistortionsIsogonal or

Azimuthal

........................... ................ C e n t r a l p o l a r ( g n o m o n i c p r o j e c t i o n ) ................ E q u a l l y s p a c e d a z i m u t h a l ( c e n t . r a 1 ) p r o j e c t i o n ...... S t e r e o g r a p h i c p o l a r p r o j e c t i o n ..................... N o m e n c l a t u r e o f M a p s ...............................International projection(Perspective) Projections

................... ............................ N o r m a l ( e q u i v a l e n t ) c y l i n d r i c a l p r o j e c t i o n ......... S i m p l e e q u a l l y s p a c e d c y l i n d r i c a l p r o j e c t i o n ....... I s o g o n a l c y l i n d r i c a l p r o j e c t i o n .................... I s o g o n a l o b l i q u e c y l i n d r i c a l p r o j e c t i o n s ........... I s o g o n a l t r a n s v e r s e and c y l i n d r i c a l G a u s s i a n p r o j e c t i o n ......................................... C o n i c P r o j e c t i o n s .................................. S i m p l e n o r m a l c o n i c p r o j e c t i o n ..................... I s o g o n a l c o n i c p r o j e c t i o n .......................... C o n v e r g e n c e a n g l e o f t h e m e r i d i a n s ................. P o l y c o n i c p r o j e c t i o n s ..............................Cylindrical Projections

E q u a l l y spaced o r e q u i d i s t a n t p r o j e c t i o n s Equally large or equivalent projections Arbitrary projections D i v i s i o n o f P r o j e c t i o n s According t o t h e Method o f C o n s t r u c t i o n ( A c c o r d i n g t o t h e A p p e a r a n c e o f t h e Normal G r i d )

......... ............................... c o n f o r m a l p r o j e c t i o n s ..............

16 17

1.8 18 18 19 19 20 20 20 20 22 23 24

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25 27 28 29 30 31 32 34 36 38 38 41

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.................. .............................................. O r t h o d r o m e o n t h e E a r t h ' s S u r f a c e .................. O r t h o d r o m e o n T o p o g r a p h i c a l Maps o f D i f f e r e n t P r o j e c t i o n s ........................................ L o x o d r o m e o n t h e E a r t h ' s S u r f a c e ................... General Recommendations f o r M e a s u r i n g D i r e c t i o n s a n d D i s t a n c e s ...................................... 7 . S p e c i a l C o o r d i n a t e S y s t e m s o n t h e E a r t h ' s S u r f a c e .... O r t h o d r o m i c C o o r d i n a t e S y s t e m ...................... A r b i t r a r y (Ob1 i q u e a n d T r a n s v e r s e ) S p h e r i c a l a n d P o l a r C o o r d i n a t e S y s t e m s ........................... P o s i t i o n L i n e s o f an A i r c r a f t on t h e E a r t h ' s S u r f a c e ............................................ B i p o l a r A z i m u t h a l C o o r d i n a t e S y s t e m ................ G o n i o m e t r i c Range-Finding C o o r d i n a t e System ........ B i p o l a r Range-F ind ing ( C i r c u 1 a r ) C o o r d i n a t e S y s t e m ............................................. L i n e s o f E q u a l A z i m u t h s ............................6

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Maps U s e d f o r A i r c r a f t N a v i g a t i o n

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M e a s u r i n g D i r e c t i o n s and D i s t a n c e s on t h e E a r t h ' s Surface

45 45

55 60 65 66 67

71 73

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78 80 81

85 88 88

.................................. ............................................. 8 . E l e m e n t s o f A i r c r a f t N a v i g a t i o n ...................... E l e m e n t s w h i c h d e t e r m i n e F l i g h t D i r e c t i o n .......... 1 . Assymetry o f the Engine T h r u s t o r A i r c r a f t D r a g ( F i g . 1 . 5 9 ) ...............................

Difference-Range-Finding (Hyperbolic) C o o r d i n a t e System Overall-Range-Finding ( E l l i p t i c a l ) Coordinate System

.............................. ................................. . ......................................... . ............. ........................................ ......................... ........... Calculating F l i g h t A l t i t u d e i n Determining D i s t a n c e s o n t h e E a r t h ' s S u r f a c e ................... E l e m e n t s o f A i r c r a f t R o l l .......................... 1 . C o m b i n a t i o n o f R o l l w i t h ' a S t r a i g h t L i n e ....... 2 . C o m b i n a t i o n o f t w o r o l l s ....................... 3 . L i n e a r p r e d i c t i o n o f r o l l ( L P R ) ................ C H A P T E R TWO . A I R C R A F T N A V I G A T I O N U S I N G MISCELLANEOUS D E V I C E S .................................................... 1. G e o t e c h n i c a 7 Means o f A i r c r a f t N a v i g a t i o n ............ 2 . C o u r s e I n s t r u m e n t s a n d S y s t e m s .......................iv

A l l o w a b l e L a t e r a l B a n k i n g o f an A i r c r a f t i n Horizontal Flight C o r i o l i s Force 4 Two-dimensional F l u c t u a t i o n s i n the A i r c r a f t Course 5 G l i d i n g D u r i n g Changes i n t h e L a t e r a l Wind Speed Component a t F l i g h t A l t i t u d e E l e m e n t s Which C h a r a c t e r i z e t h e F l i g h t Speed o f an A i r c r a f t N a v i g a t i o n a l Speed T r i a n g l e Elements Which D e t e r m i n e F l i g h t A l t i t u d e

. 3.2

94

94 95

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96 98 101 103 107 110 110 111 113

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Methods o f U s i n g t h e M a g n e t i c F i e l d o f t h e Earth t o Determine D i r e c t i o n V a r i a t i o n s and O s c i l l a t i o n s i n t h e E a r t h ' s Magnetic F i e l d M a g n e t i c Compasses D e v i a t i o n o f M a g n e t i c Compasses a n d i t s Compensation

.................................. ........................................ G y r o s c o p i c C o u r s e D e v i c e s .......................... P r 5 n c i p Ze of O p e r a t i o n of G y r o s c o p i c I n s t r u m e n t s ...................................... Degree of Freedom of t h e G y r o s c o p e . . . . . . . . . . . . . . . D i r e c t i o n of P r e c e s s i o n of t h e G y r o s c o p e A x i s . . . .............................. ...................... G y r o i n d u c t i o n Compass .............................. D e t a i l s o f D e v i a t i o n O p e r a t i o n s on D i s t a n c e G y r o m a g n e t i c a n d G y r o i n d u c t i o n Compasses . . . . . . . . . . . Methods o f U s i n g Course Devices f o r Purposes o f A i r c r a f t N a v i g a t i o n ............................. Methods of U s i n g C o u r s e D e v i c e s Under C o n d i t i o n s I n c l u d e d i n t h e F i r s t Group ...................... Methods of U s i n g C o u r s e D e v i c e s Under C o n d i t i o n s of t h e S e c o n d Group .............................. Methods of U s i n g C o u r s e D e v i c e s Under t h e C o n d i t i o n s of t h e T h i r d Group .................... 3 . B a r o m e t r i c A l t i m e t e r s ................................ Description o f a Barometric A l t i m e t e r .............. Errors i n Measuring A l t i t u d e w i t h a Barometric A l t i m e t e r .......................................... 4 . A i r s p e e d I n d i c a t o r s .................................. E r r o r s i n M e a s u r i n g A i r s p e e d ....................... R e l a t i o n s h i p B e t w e e n E r r o r s i n Speed I n d i c a t o r s a n d F l i g h t A l t i t u d e ................................G y r o s c o p i c Semicompass D i s t a n c e G y r o m a g n e t i c Compass

Change i n D e v i a t i o n of M a g n e t i c Compasses a s a F u n c t i o n o f t h e M a g n e t i c L a t i t u d e of t h e L o c u s of t h e A i r c r a f t E Z i m i n a t i o n of D e v i a t i o n i n t h e M a g n e t i c Compasses

....................... ..................................... ................................. ....................................... E q u a l i z i n g t h e M a g n e t i c F i e l d of t h e A i r c r a f t .... D e v i a t i o n FormuZas ............................... C a Z c u Z a t i o n of A p p r o x i m a t e D e v i a t i o n C o e f f i c i e n t s .....................................

114

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126 128 131 133 134 141

142 144

146

A p p a r e n t R o t a t i o n of G y r o s c o p e A x i s on t h e E a r t h ' s S u r f a c e ..................................

146 149 152 158

162 165 166

168

172 175 180

183 186 193

. . 7. 8. 9.5 6

Measurement o f t h e T e m p e r a t u r e o f t h e O u t s i d e Air Aviation Clocks

A u t o m a t i c N a v i g a t i o n I n s t r u m e n t s ..................... P r a c t i c a l Methods o f A i r c r a f t N a v i g a t i o n U s i n g Geotechnical Devices

.... ...................................... S p e c i a l R e q u i r e m e n t s f o r A v i a t i o n C l o c k s ........... N a v i g a t i o n a l S i g h t s ...................................................................V

196 199 201 202 204 210 214

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T a k e o f f of the A i r c r a f t at the S t a r t i n g Point of the Route....................................... S e l e c t i n g t h e C o u r s e ' t o be F o l l o w e d for the F l i g h t Route............ .Change i n Navigational Elements D u r i n g F l i g h t M e a s u r i n g the W i n d a t F l i g h t A l t i t u d e a n d C a l c u l a t i n g Navigational Elements a t S u c c e s s i v e Stages C a l c u l a t i o n o f the P a t h of the A i r c r a f t and Monitoring Aircraft Navigation i n Terms of D i s t a n c e s and Direction.. Use of A u t o m a t i c Navigational Devices for C a l c u l a t i n g the A i r c r a f t P a t h a n d M e a s u r i n g the W i n d P a r a m e t e r s Details of A i r c r a f t N a v i g a t i o n U s i n g Geotechnical M e t h o d s i n Various F l i g h t Conditions.....

215 218 221

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224

227 230

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10. Calculating and Measuring Pilotage Instruments..

.... Purpose of Calculating a n d Measuring Pilotage ......................... Instruments............... Navigational S l i d e Rule N L - l O M . . . . .................

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233 234

234 235

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250 251

CHAPTER THREE. AIRCRAFT NAVIGATION USING RADIO-ENGINEERING DEVICES ....................................................

1.

2.

............................. ................................. ................................... ..................................... ..................................... Goniometric and G o n i o m e t r i c - R a n g e f i n d i n g Systems .... A i r c r a f t N a v i g a t i o n U s i n g G r o u n - B a s e d Radio Direction-Finders.................... ............. S e Z e c t i o n of t h e C o u r s e t o b e FoZZowed and C o n t r o Z of F Z i g h t D i r e c t i o n ..................... P a t h C o n t r o l . i n Terms of D i s t a n c e and D e t e r m i n a t i o n of t h e A i r c r a f t ' s L o c a t i o n ............. D e t e r m i n a t i o n of t h e Ground S p e e d , D r i f t AngZe, and Wind ........................................ A u t o m a t i c A i r c r a f t Radio D i s t a n c e - F i n d e r s ( R a d i o c o m p a s s e s ) .................................. R a d i o c o m p a s s D e v i a t i o n . ......................... A i r c r a f t N a v i g a t i o n U s i n g R a d i o c o m p a s s e s on Board t h e A i r c r a f t ..............................W a v e Polarization.... P r o p a g a t i o n of E l e c t r o m a g n e t i c O s c i l l a t i o n s i n H o m o g e n e o u s Medi'a P r i n c i p l e s of S u p e r p o s i t i o n a n d I n t e r f e r e n c e of Radio W a v e s . P r i n c i p l e C h a r a c t e r i s t i c s of R a d i o n a v i g a t i o n a l Instruments.. O p e r a t i n g P r i n c i p l e s of R a d i o n a v i g a t i o n a l Instruments..

Principles o f the Theory o f Radionavigational Instruments .........................................

253

257 257

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265 269

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S p e c i a l F e a t u r e s o f U s i n g Radiocompasses on Board A i r c r a f t a t High A Z t i t u d e s and F Z i g h t Speedsvi

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292

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U l t r a - S h o r t w a v e G o n i o m e t r i c and G o n i o m e t r i c Range F i n d i n g S y s t e m s

3.

................ ............................. D e t a i 2s of U s i n g G o n i o m e t r i c - R a n g e F i n d i n g S y s t e m s a t D i f f e r e n t F Z i g h t A Z t i t u d e d ........... F a n - S h a p e d G o n i o m e t r i c R a d i o B e a c o n s ..............

D e t a i Z s of U s i n g R a d i o c o m p a s s e s i n Makhysics, aerodynamics, r a d i o engineering, r a d i o e l e c t r o n i c s , etc.

Navigation technology is developing at a rapid pace; a i r c r a f t a n d g r o u n d f a c i l i t i e s for a i r c r a f t n a v i g a t i o n a r e ' c o n t i n u a l l y b e i n g p e r f e c t e d and t h e p r o f e s s i o n a l t r a i n i n g and n a v i g a t i o n a l p r e p a r a t i o n o f f l i g h t and ground p e r s o n n e l h a s improved. A l l t h i s has r a d i c a l l y r a i s e d t h e r e l i a b i l i t y o f aircraft n a v i g a t i o n , i t s accu r a c y , and i t s c h i e f c r i t e r i o n , s a f e t y . Modern t e c h n i c a l means o f a i r c r a f t n a v i g a t i o n a r e d i v i d e d i n t o f o u r b a s i c groups according t o t h e p r i n c i p l e o f o p e r a t i o n . 1. G e o t e c h n i c a Z means of a i r c r a f t n a v i g a t i o n , w h i c h a r e b a s e d on t h e p r i n c i p l e o f m e a s u r i n g d i f f e r e n t p a r a m e t e r s o f t h e E a r t h ' s fields. They i n c l u d e : magnetic compasses, gyroscopic n a v i g a t i o n and p i l o t i n g d e v i c e s , gyromagnetic and g y r o i n d u c t i o n t e l e c o m p a s s e s , course systems, airspeed indicators, barometric altimeters, exter n a l a i r thermometers, navigation i n d i c a t o r s , i n e r t i a l i n d i c a t o r s , mechanical clocks, e t c . 2. R a d i o - e n g i n e e r i n g means of a i r c r a f t n a v i g a t i o n , w h i c h a r e b a s e d on t h e o p e r a t i n g p r i n c i p l e o f r a d i o - e l e c t r o n i c t e c h n o l o g y . These i n c l u d e goniometer r a d i o - e n g i n e e r i n g systems ( r a d i o compasses w i t h ground t r a n s m i t t i n g r a d i o s t a t i o n s , ground radiogoniometers w i t h a i r c r a f t r e c e i v i n g - t r a n s m i t t i n g r a d i o s t a t i o n s , and r a d i o bea cons w i t h a i r c r a f t r e c e i v i n g r a d i o equipment), r a n g e f i n d i n g systems, g o n i o m e t e r - r a n g e f i n d i n g s y s t e m s , ground and a i r c r a f t r a d a r , Doppler m e t e r s a n d s y s t e m s , r a d i o a l t i m e t e r s , c o u r s e - l a n d i n g beam s y s t e m s w i t h t h e i r ground and a i r c r a f t e q u i p m e n t , e t c .3. A s t r o n o m i c a l ( r a d i o a s t r o n o m i c a l ) means of a i r c r a f t n a v i g a t i o n , w h i c h a r e b a s e d on t h e p r i n c i p l e o f m e a s u r i n g t h e m o t i o n '

/6

parameters of heavenly bodies. These i n c l u d e a v i a t i o n s e x t a n t s , astrocompasses, astronomical o r i e n t a t o r s , e t c .4. L i g h t e n g i n e e r i n g means of a i r c r a f t n a v i g a t i o n , w h i c h a r e b a s e d on t h e p r i n c i p l e o f u s i n g l i g h t e n e r g y r a d i a t i o n . These i n c l u d e ground l i g h t beams, l i g h t and p u l s e - l i g h t equipment f o r t a k e o f f and l a n d i n g s t r i p s as w e l l as a i r c r a f t , e n c l o s u r e s f o r t h e l i g h t i n g equipment of t h e r o u t e s and a i r p o r t s (housings f o r ground i n s t a l l a t i o n s ) , various pyrotechnic devices, etc.

A t t h e h e a r t o f a s a f e and a c c u r a t e f l i g h t a c c o r d i n g t o a s e t r o u t e , i n t h e v i c i n i t y o f t h e a i r p o r t , or d u r i n g t a k e - o f f a n d l a n d ing, lies t h e p r i n c i p l e of t h e o v e r a l l usage of a l l t h e available t e c h n i c a l means o f a i r c r a f t n a v i g a t i o n , b o t h g r o u n d f a c i l i t i e s and those aboard t h e a i r c r a f t .

NASA T T F-524

C H A P T E R O.NE COORDINATE SYSTEMS AND ELEMENTS O F A I R C R A F T N A V I G A T I O N 1. Elements o f Aircraft Movement

in Space

The f u n d a m e n t a l p r o b l e m o f a i r c r a f t n a v i g a t i o n i n a l l s t a g e s o f f l i g h t i s m a i n t a i n i n g a g i v e n t r a j e c t o r y o f a i r c r a f t movement i n a l t i t u d e , d i r e c t i o n a n d t i m e by means o f a complex u t i l i z a t i o n o f n a v i g a t i o n a l means a n d m e t h o d s . A successful solution t o these p r o b l e m s d e p e n d s on c o n s t a n t a n d a c c u r a t e i n f o r m a t i o n c o n c e r n i n g t h e position of t h e craft r e l a t i v e t o a given f l i g h t t r a j e c t o r y , t h e n a t u r e of t h e a i r c r a f t movement, and t h e a c t i o n s of t h e c r e w .A s a r e s u l t of t h e c u r v a t u r e of t h e E a r t h ' s s u r f a c e , f l i g h t t r a j e c t o r y of an aircraft i s c u r v i l i n e a r . However, i n t o account t h e l a r g e r a d i u s of curvature of t h e Earth's a s m a l l a r e a c a n a l w a y s b e d e l i n e a t e d on i t whose s u r f a c e a s s u m e d t o b e p l a n e ( F i g . 1.1).

any given by t a k i n g surface, can be

L e t us e r e c t a perpendicular 01Y from t h e c e n t e r o f t h e s m a l l area which w e have chosen and c o n t i n u e it u n t i l it i n t e r s e c t s the center of the Earth. Obviously, t h i s w i l l be a p e r p e n d i c u l a r l i n e , which w e can c a l l t h e v e r t i c a z of t h e Z o c u s .

I n t h e p l a n e o f t h e s m a l l area which w e have c h o s e n , l e t u s draw a s t r a i g h t l i n e t h r o u g h t h e p o i n t 01 a n d t a k e i t a s t h e X a x i s ; t h e n l e t u s draw a n o t h e r s t r a i g h t l i n e t h r o u g h t h e p o i n t 01 i n t h e plane of t h e area, perpendicular t o t h e f i r s t , and c a l l it t h e Z a x i s . F i g . 1.1. R e c t a n g u l a r Coordinate System on t h e Earth's Surface.~-

T h u s , a t p o i n t 0 1 o n t h e E a r t h ' s surface, w e w i l l o b t a i n a r e c t a n g u l a r system of space coordinates X, Y, Z.__..

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;t.

Numbers i n t h e m a r g i n i n d i c a t e p a g i n a t i o n i n t h e f o r e i g n t e x t .

1

The t r a v e l of a n a i r c r a f t o v e r t h e E a r t h ' s s u r f a c e w i l l i n v o l v e b o t h a s h i f t i n t h e p o i n t 01 ( o r i g i n o f t h e c o o r d i n a t e s ) a n d t h e r o t a t i o n of t h e axes o f t h e c o o r d i n a t e system around t h e c e n t e r of t h e E a r t h ( p o i n t 0 ) . However, t h e s y s t e m o f c o o r d i n a t e s which w e have o b t a i n e d can b e u s e d for d e t e r m i n i n g t h e d i r e c t i o n s o f t h e a i r c r a f t a x e s a n d t h e component f l i g h t s p e e d v e c t o r s . Since t h e o r i g i n of t h i s system is b e i n g c o n t i n u o u s l y s h i f t e d , l e t u s d e s i g n a t e it as a g l i d i n g r e c

/8

t a n g u l a r s y s t e m of c o o r d i n a t e s .In t h i s coordinate system, t h e f o l l o w i n g elements can be d i s t i n guished:

44

F i g . 1 . 2 . Dip Angle of t h e T r a j e c t o r y of A l t i t u d e Gain.

;(a) P o s i t i o n o f t h e l o n g i t u d i n a l axis of t h e a i r c r a f t i n t h e horizontal plane (aircraft course),(b) Position of the longitu d i n a l a x i s of t h e a i r c r a f t i n t h e v e r t i c a l p l a n e ( a n g l e of p i t c h o f

the a i r c r a f t ) ,

( c > Position of t h e l a t e r a l a x i s of t h e a i r c r a f t i n t h e ver t i c a l plane (lateral banking), ( d ) D i s t a n c e a l o n g t h e v e r t i c a l from t h e E a r t h ' s s u r f a c e ( t h e a r e a w h i c h we h a v e c h o s e n ) t o t h e a i r c r a f t ( f l i g h t a l t i t u d e ) , ( e ) V e r t i c a l s p e e d ( a l t i t u d e g a i n and

loss),

( f ) C o m p o n e n t f l i g h t s p e e d a l o n g t h e X a n d Z a x e s or t h e v e c t o r o f g r o u n d s p e e d a n d i t s d i r e c t i o n ( g r o u n d s p e e d and f l i g h t a n g l e ) , ( g ) A n g u l a r v e l o c i t y o f a i r c r a f t roll, ( h ) Component wind speed along t h e

X and

Z axes of

t h e system,

or t h e w i n d v e c t o r a n d i t s d i r e c t i o n ( w i n d s p e e d and d i r e c t i o n ) .U s u a l l y t h e p o s i t i o n o f t h e c r a f t on t h e E a r t h ' s s u r f a c e i s t r e a t e d i n s u r f a c e - c o o r d i n a t e s y s t e m s , t h e most widely used of which a r e t h e g e o g r a p h i c s y s t e m a n d t h e r e f e r e n c e s y s t e m whose m a j o r a x i s c o i n c i d e s w i t h a g i v e n f l i g h t t r a j e c t o r y on t h e E a r t h ' s s u r f a c e . The p o s i t i o n o f t h e a i r c r a f t i n s u r f a c e - c o o r d i n a t e s y s t e m s i s a s s u m e d t o b e t h e p o s i t i o n of t h e o r i g i n o f t h e g l i d i n g s y s t e m . To a n a l y z e t h e e l e m e n t s o f a i r c r a f t n a v i g a t i o n , l e t u s combine t h e X a x i s of t h e gliding-coordinate system with a given f l i g h t t r a j e c t o r y of the aircraft. I n o r d e r t o keep t h e a i r c r a f t i n t h e r e c t i l i n e a r h o r i z o n t a l2

segment o f t h i s t r a j e c t o r y , t h e crew must m a i n t a i n a f l i g h t c o n d i t i o n i n which t h e a i r c r a f t w i l l n o t be s h i f t e d a l o n g t h e v e r t i c a l ( a l t i t u d e g a i n and l o s s ) , t h e r e w i l l be no l a t e r a l d e v i a t i o n ( t o t h e r i g h t or l e f t ) , i . e . , t h e v e r t i c a l v e l o c i t y Vy a n d t h e l a t e r a l component o f t h e v e l o c i t y V z , w i l l b e equal t o z e r o , and t h e l o n g i t u d i n a l f l i g h t v e l o c i t y Vx ( a l o n g t h e X a x i s ) w i l l b e a s g i v e n . If t h e f l i g h t t r a j e c t o r y i s i n c l i n e d ( s e g m e n t s o f a l t i t u d e g a i n a n d loss), t h e c r e w m u s t h o l d t h i s t r a j e c t o r y b y m a i n t a i n i n g t h e /9 v e r t i c a l and l o n g i t u d i n a l f l i g h t v e l o c i t i e s ( V y and Vx>, i . e . , m a i n t a i n a given d i p a n g l e of t h e t r a j e c t o r y 0 ( F i g . 1 . 2 ) . Obviously, a t a constant dip angle of t h e f l i g h t t r a j e c t o r y , t h e l a t t e r w i l l have a c u r v a t u r e i n t h e v e r t i c a l p l a n e j u s t as i n horizontal flight. T h e r e f o r e , i f we n e g l e c t t h e c u r v a t u r e o f t h e h o r i z o n t a l f l i g h t t r a j e c t o r y , we may a s s u m e

-

(1.1)

where 0 i s t h e d i p a n g l e of t h e f l i g h t t r a j e c t o r y ; X I , X 2 a r e t h e c o o r d i n a t e s of t h e i n i t i a l and f i n a l p o i n t s o f t h e s l o p i n g segment o f t h e t r a j e c t o r y ; H I , H2 r e p r e s e n t a g i v e n a l t i t u d e a t t h e i n i t i a l and f i n a l p o i n t s . When t h e a i r c r a f t t r a v e l s f r o m t h e i n i t i a l p o i n t X I t o t h e mov i n g p o i n t X , t h e f l i g h t a l t i t u d e i s changed by t h e v a l u eAH= (. XI) tg 8, X-

(1.2)

and t h e v a l u e o f t h e moving f l i g h t a l t i t u d e i s

or i f we t a k e F o r m u l a (1.1) i n t o a c c o u n t ,

Since the a l t i t u d e during a sloping trajectory i s a variable value, a given f l i g h t t r a j e c t o r y is maintained a t a constant value of t h e v e r t i c a l velocity

vy*v,tgeo'r

v -vy-

H2-Hl X,-X,

,

(1.5)

C h e c k i n g of t h e p o s i t i o n o f t h e a i r c r a f t a t g i v e n v a l u e s o f the varying f l i g h t a l t i t u d e is carried out only at specific points on t h e s l o p i n g t r a j e c t o r y .

Translator's note:

tg = tan.3

2.

Concepts o f S t a b l e and U n s t a b l e F l i g h t C o n d i t i o n s

A n a v i g a t i o n a l f l i g h t c o n d i t i o n i s d e t e r m i n e d by t h e m o t i o n p a r a m e t e r s o f a n a i r c r a f t a l o n g a t r a j e c t o r y or b y n a v i g a t i o n a l elements of f l i g h t : course, speed, and a l t i t u d e .The m o t i o n p a r a m e t e r s o f a n a i r c r a f t a r e u s u a l l y m e a s u r e d r e l ative t o airspace. However, c o n s i d e r i n g t h a t t h e a i r s p a c e a l s o s h i f t s , t h e y a r e s e l e c t e d i n s u c h a way a s t o e n s u r e r e t a i n i n g t h e given f l i g h t t r a j e c t o r y r e l a t i v e t o t h e E a r t h l s surface. B a s e d on t h e n a t u r e o f t h e t r a j e c t o r y a n d t h e c o n d i t i o n s o f a i r c r a f t n a v i g a t i o n , f o u r main f l i g h t c o n d i t i o n s a r e d i s t i n g u i s h e d : h o r i z o n t a l r e c t i l i n e a r f l i g h t , a l t i t u d e g a i n , a l t i t u d e loss, a n d

/10

~~~

roll.H o r i z o n t a l r e c t i l i n e a r f l i g h t i s c h a r a c t e r i z e d by t w o c o n s t a n t parameters: h e i g h t and f l i g h t d i r e c t i o n . A l t i t u d e g a i n and l o s s c o n d i t i o n s e a c h have two c o n s t a n t param eters: f l i g h t d i r e c t i o n , a n d v e r t i c a l v e l o c i t y or d i p a n g l e o f t h e t r aj e c to r y

.

The c o n d i t i o n o f roll i s a l w a y s c o m b i n e d w i t h o n e o f t h e f i r s t t h r e e f l i g h t c o n d i t i o n s , s o t h a t t h e f l i g h t d i r e c t i o n becomes v a r i a b l e and can be r e p l a c e d by a p a r a m e t e r which c h a r a c t e r i z e s t h e c u r v a t u r e o f t h e roll t r a j e c t o r y t h r o u g h t h e r a d i u s o f roll or t h e angular velocity.

A f l i g h t condition is s t a b l e i f i t s p a r a m e t e r s a c q u i r e c o n s t a n t v a l u e s , and unstable i f i t s parameters a r e v a r i a b l e .F l i g h t p r a c t i c e shows t h a t f l i g h t c o n d i t i o n s , s t r i c t l y s p e a k i n g , a r e n e v e r f i x e d f o r any p r o l o n g e d t i m e , s i n c e t h e r e a r e always f a c t o r s changing t h e a i r c r a f t ' s motion parameters. The m a i n s i g n o f a s t a b l e f l i g h t c o n d i t i o n i s t h e e q u a l i t y t o zero of t h e first derivative of t h e given parameter with time d2S or o f t h e s e c o n d d e r i v a t i v e p a t h w i t h t i m e -

() g

d.t

.

For e x a m p l e , f o r t h e v e l o c i t y p a r a m e t e r V = c o n s t , i f

-dvdf

-0

dzS or - - 0.dt2

Analogously, f o r t h e f l i g h t d i r e c t i o n parameter a l t i t u d e parameter (HI:+=consf,

( 9 ) and t h e

i f -- 0 , dd,l dt

H=const,

dH i f --- - 0.

dt

4

I f f o r c e s a r i s e d u r i n g f l i g h t w h i c h c h a n g e t h e a i r c r a f t ' s mo t i o n parameters, t h e extreme v a l u e s o f t h e motion parameters ( ? . e . , t h e p o i n t s o f t h e maxima a n d m i n i m a o n t h e c u r v e w h i c h c h a r a c t e r i z e s t h e change o f t h e g i v e n p a r a m e t e r w i t h t i m e ) i n d i c a t e e q u i l i b r i u m of these forces.A stable f l i g h t condition based on a given parameter e x i s t s o n l y a t t h e extreme p o i n t s , s i n c e t h e first d e r i v a t i v e p a r a m e t e r s b a s e d on t i m e a t these points are equal t o zero while t h e d i s t u r b i n g f o r c e s are absent.

F i g . 1 . 3 . Graph o f t h e Changes o f a N a v i g a t i o n a l P a r a m e t e r and P o i n t s w i t h a Stable Flight Condition.

The d i s t u r b i n g f o r c e s a c q u i r e a maximum v a l u e a t p o i n t s o f i n f l e c t i o n , i . e . , when t h e s e c o n d d e r i v a t i v e p a r a m e t e r s b a s e d on t i m e a r e e q u a l t o On a c u r v e c o n zero (Fig. 1.3). structed f o r the velocity parameter, t h e p o i n t s of a s t a b l e condition are d e s i g n a t e d b y o n e l i n e , w h i l e p o i n t s of maximum d i s t u r b i n g f o r c e s a r e d e s i g n a t e d by two l i n e s . From a e r o d y n a m i c s , w e know t h a t i n h o r i z o n t a l f l i g h t a t a v e - l o c i t y s i g n i f i c a n t l y less than t h e speed of sound, t h e drag of an aircraft i n a counterflow is

/11 -

w h e r e ex i s t h e c o e f f i c i e n t o f d r a g o f t h e a i r c r a f t , S i s t h e c r o s s s e c t i o n a l a r e a o f t h e m i d s h i p s e c t i o n , and p i s t h e a i r d e n s i t y a t flight altitude.I t i s obvious t h a t t h e a i r s p e e d w i l l be s t a b l e i f t h e t h r u s t o f t h e e n g i n e s ( P ) i s e q u a l t o t h e d r a g o f t h e a i r c r a f t P = Qx.

With a d i s t u r b a n c e o f t h i s e q u i l i b r i u m , t h e r e a r i s e s a d i s t u r b i n g f o r c e which changes t h e f l i g h t v e l o c i t y . For example, w i t h an i n c r e a s e i n t h e t h r u s t of t h e engines t h e d i s t u r b i n g f o r c e w i l l be equal t o :AP=P'-c~

p, "2 '

which c a u s e s an i n i t i a l a c c e l e r a t i o n o f t h e a i r c r a f tdV -=dt

AP m '

where m i s t h e m a s s of t h e a i r c r a f t i n kg.

5

L a t e r , with an increase i n v e l o c i t y , t h e drag of t h e aircraft w i l l also increase. T h e v a l u e of t h i s d r a g w i l l a p p r o a c h t h e v a l u e of t h e t h r u s t o f t h e e n g i n e s , i . e . , t h e v e l o c i t y v e r y s l o w l y ap proaches a s t a b l e value logarithmically. Changes i n a i r s p e e d which a r e a n a l o g o u s i n n a t u r e a r i s e d u r i n g c h a n g e s i n t h e v e l o c i t y o f t h e h e a d w i n d or t h e i n c i d e n t a i r f l o w a t flight altitude. For e x a m p l e , w i t h a n i n c r e a s e i n t h e v e l o c i t y o f t h e incident airflow, t h e airspeed diminishes. This provides a surplus of engine t h r u s t . Subsequently, an increase i n airspeed occurs logarithmically.

If t h e l a t e r a l component o f t h e wind s p e e d c h a n g e s , a l a t e r a l p r e s s u r e on t h e s u r f a c e o f t h e a i r c r a f t a r i s e s :

w h e r e c z i s t h e c o e f f i c i e n t o f l a t e r a l d r a g o f t h e a i r c r a f t ; S, i s t h e c r o s s - s e c t i o n a l a r e a o f t h e a i r c r a f t i n t h e XY p l a n e ; Y , i s t h e l a t e r a l v e l o c i t y c o m p o n e n t e q u a l t o uz. The i n i t i a l l a t e r a l a c c e l e r a t i o n o f t h e a i r c r a f t i s :

Subsequently, the lateral velocity of t h e aircraft w i l l log a r i t h m i c a l l y a p p r o a c h t h e l a t e r a l component o f t h e wind v e l o c i t y , i . e . , t h e f l i g h t c o n d i t i o n w i l l approach a c o n d i t i o n which i s s t a b l e in direction. Usually, during navigational c a l c u l a t i o n s f o r each parameter, i t s mean v a l u e f o r a d e f i n i t e l e n g t h o f t i m e i s c a l l e d a s t a b l e flight condition: mean v e l o c i t y , mean v e r t i c a l v e l o c i t y , mean d i rection, etc. From t h e p o i n t o f v i e w o f m a i n t a i n i n g f l i g h t d i r e c t i o n , a i r craft r o l l is an u n s t a b l e condition. If a g i v e n t r a j e c t o r y i s c u r v i l i n e a r , t h e r o l l c o n d i t i o n i s a l s o e x a m i n e d a s s t a b l e or u n s t a b l e . T h e e n t r a n c e or e x i t o f a n a i r c r a f t f r o m roll, a s w e l l a s roll w i t h v a r i a b l e b a n k i n g , c a n s e r v e a s e x a m p l e s o f u n s t a b l e roll c o n d i t i o n s . The r o l l i n g o f a n a i r c r a f t i s c o n s i d e r e d t o b e c o o r d i n a t e d i f t h e l o n g i t u d i n a l a x i s o f t h e a i r c r a f t c o n s t a n t l y c o i n c i d e s w i t h t.he t a n g e n t t o t h e t r a j e c t o r y o f i t s m o v e m e n t , ? . e . , e x t e r n a l or i n t e r T h i s i s a c h i e v e d by t i l t i n g t h e r u d n a l a i r c r a f t glide is absent. d e r o f t h e a i r c r a f t f o r b a n k i n g i n a roll. D u r i n g b a n k i n g o f a n a i r c r a f t , i t s l i f t (Y) i s d i r e c t e d n o t a l o n g t h e v e r t i c a l p l a n e b u t a l o n g t-he a x i s o f t h e a i r c r a f t , which i s d e f l e c t e d from it ( F i g . 1 . 4 ) .

/12

6

R o l l i n g o f an a i r c r a f t w i t h o u t d e s c e n t o r w i t h s t a b l e v e r t i c a l v e l o c i t y i s p o s s i b l e o n l y when t h e v e r t i c a l c o m p o n e n t o f t h e l i f t ( Y 1 ) i s e q u a l t o t h e w e i g h t of t h e a i r c r a f t G. I n t h i s case, t h e h o r i z o n t a l ( c e n t r i p e t a l ) component o f t h e l i f t i s :Yr=Gtg8.

w h e r e f3 craft.

is t h e banking angle of t h e air

Since w e a r e examining a c o o r d i n a t e r o l l (without gliding of the aircraft), t h e c e n t r i f u g a l f o r c e i n t h e roll Fig. 1.4. Resolution of Forces During R o l l i n g of an A i r craft.F,=R

m V2

w i l l be e q u a l t o t h e c e n t r i p e t a l f o r c e , i.e.,

where m i s t h e m a s s o f t h e a i r c r a f t ; and R i s t h e r a d i u s o f t h e co o r d i n a t e d roll.

G Transforming t h i s equation, taking i n t o account t h a t m = -

9 ,

w e w i l l o b t a i n formulas f o r d e t e r m i n i n g b o t h t h e r a d i u s and p a t h o f t h e a i r c r a f t w i t h c o o r d i n a t e d roll:

F o r m u l a s ( 1 . 6 ) r e l a t e t h e r a d i u s o f s t a b l e c o o r d i n a t e d roll o f t h e a i r c r a f t w i t h t h e a i r s p e e d and a l s o w i t h banking i n r o l l i n g , and t h e y a r e used i n c a l c u l a t i o n s o f t h e r a d i u s and p a t h o f t h e a i r craft along a curvilinear f l i g h t trajectory.

3.

Form and D i m e n s i o n s o f t h e E a r t h

/3 1

I n t h e p r a c t i c e of aircraft navigation, it i s necessary first of a l l t o d e a l w i t h d i s t a n c e s a n d d i r e c t i o n s on t h e E a r t h ' s s u r f a c e which are t h e r e s u l t o f t h e mutual d i s t r i b u t i o n o f o b j e c t s through which t h e f l i g h t p a t h p a s s e s . The E a r t h ' s s u r f a c e , i t s r e l i e f a n d m u t u a l d i s t r i b u t i o n o f o b j e c t s c a n b e m o s t a c c u r a t e l y e x p r e s s e d on a m o d e l o f t h e E a r t h ( a globe). H o w e v e r , a g l o b e w i t h a r e p r e s e n t a t i o n o f t h e E a r t h ' s sur f a c e t h a t s a t i s f i e s t h e demands o f a i r c r a f t n a v i g a t i o n would b e s o l a r g e t h a t i t s u s e i n f l i g h t would be i m p o s s i b l e . Therefore, dif f e r e n t means o f r e p r e s e n t i n g t h e s u r f a c e o f t h e E a r t h , which i s curved i n a l l d i r e c t i o n s , on a p l a n e ( s h e e t s of p a p e r ) are used.7

The E a r t h h a s a c o m p l e x f o r m c a l l e d a g e o i d ( w i t h o u t c o n s i d e r i n g t h e l o c a l r e l i e f , i f w e imagine t h a t i t s e n t i r e s u r f a c e i s cov The s u r f a c e o f a g e o i d a t a n y p o i n t e r e d w i t h w a t e r a t sea l e v e l ) . i s p e r p e n d i c u l a r t o t h e d i r e c t i o n o f t h e a c t i o n of g r a v i t y . A de s c r i p t i o n of a g e o i d by m a t h e m a t i c a l e x p r e s s i o n s i s v e r y c o m p l e x , and i f w e c o n s i d e r t h e f o l d s i n t h e r e l i e f o f t h e E a r t h ' s s u r f a c e , t h e n it i s p r a c t i c a l l y impossible t o e x p r e s s i t s form mathematically, T h e r e f o r e , i n c a l c u l a t i o n s t h e form o f t h e E a r t h i s t a k e n as an e Z Z i p s o i d of revoZu+.ion, t h e f o r m c l o s e s t t o a g e o i d .

Fig.

1.5. G r e a t a n d S m a l l C i r c l e s on t h e E a r t h ' s S u r f a c e . a x i s o f t h e E a r t h a n d Great C i r c l e ; b ) S m a l l C i r c l e .

a ) Semi-

A c c o r d i n g t o m e a s u r e m e n t s made b y S o v i e t s c i e n t i s t s u n d e r t h e s u p e r v i s i o n of F . N . K r a s o v s k i y , t h e m a j o r s e m i a x i s o f t h i s e l l i p s o i d ( a ) , which c o i n c i d e s w i t h t h e r a d i u s of t h e e q u a t o r , i s e q u a l t o 6 , 3 7 8 , 2 4 5 km. The m i n o r s e m i a x i s o f t h e e l l i p s o i d ( b ) , w h i c h coincides with t h e a x i s of t h e E a r t h ' s r o t a t i o n , is equal t o 6 , 3 5 6 , 8 6 3 km ( F i g . 1 . 5 , a ) . The f l a t t e n i n g o f t h e E a r t h a t t h e p o l e s i se = - -a-- b

/14

a

1 298.3

'

T h e s e d i m e n s i o n s show t h a t t h e E a r t h ' s e l l i p s o i d o f r e v o l u t i o n i s p r a c t i c a l l y c l o s e t o a s p h e r e ; t o s i m p l i f y t h e s o l u t i o n of t h e m a j o r i t y o f p r o b l e m s i n a i r c r a f t n a v i g a t i o n , i t i s t a k e n as a t r u e s p h e r e , e q u i v a l e n t i n volume t o t h e E a r t h ' s e l l i p s o i d . The r a d i u s o f s u c h a s p h e r e i s e q u a l t o 6 3 7 1 km. T h e maximum d i s t o r t i o n o f d i s t a n c e s c a u s e d b y t h e r e p l a c e m e n t

8

o f t h e E a r t h ' s e l l i p s o i d b y a s p h e r e d o e s n o t e x c e e d 0.5%, a n d t h e d i s t o r t i o n o f d i r e c t i o n s i s n o t more t h a n 1 2 m i n u t e s o f a n g l e .

I n g e o d e s y a n d c a r t o g r a p h y , t h e p l o t t i n g o f maps, as w e l l as i n o t h e r b r a n c h e s o f s c i e n c e w h e r e more a c c u r a t e c a l c u l a t i o n s o f d i s t a n c e s and d i r e c t i o n s a r e n e c e s s a r y , t h e E a r t h ' s s u r f a c e i s t a k e n a s a n e l l i p s o i d of r e v o l u t i o n .

4. E l e m e n t s W h i c h C o n n e c t t h e E a r t h ' s S u r f a c e with Three-Dimensional Space

T a k i n g t h e E a r t h as a t r u e s p h e r e , w e w i l l l o c a t e a p e r p e n d i c u l a r ( a r e s t i n g pendulum) a t any p o i n t above t h e E a r t h ' s s u r f a c e . Then, d i s r e g a r d i n g t h e p o s s i b l e i n s i g n i f i c a n t d e v i a t i o n s c a u s e d by t h e varying r e l i e f , t h e i r r e g u l a r i t y of d i s t r i b u t i o n of t h e densest masses i n t h e E a r t h ' s c r u s t , and t h e t a n g e n t i a l a c c e l e r a t i o n s con nected with t h e E a r t h ' s r o t a t i o n , it i s p o s s i b l e t o consider t h a t t h e l i n e of t h e perpendicular runs i n t h e d i r e c t i o n of t h e c e n t e r of the Earth. The p e r p e n d i c u l a r l i n e ( s e e F i g . 1 . 5 , a ) j o i n i n g t h e c e n t e r o f t h e E a r t h w i t h t h e p o i n t o f t h e o b s e r v e r ' s p o s i t i o n , and c o n t i n u e d i n t h e d i r e c t i o n o f t h e c e l e s t i a l s p h e r e (Y), i s c a l l e d t h e geo

c e n t r i c v e r t i c a l of t h e l o c u s .The p l a n e on t h e E a r t h ' s s u r f a c e , t a n g e n t t o t h e s p h e r e a t t h e p o i n t of t h e o b s e r v e r and p e r p e n d i c u l a r t o t h e t r u e v e r t i c a l o f t h e l o c u s , i s c a l l e d t h e p l a n e of t h e t r u e h o r i z o n . The d i r e c t i o n a n d v e l o c i t y o f a i r c r a f t movement a t e v e r y p o i n t on t h e E a r t h ' s s u r f a c e a r e examined i n t h e p l a n e o f t h e t r u e h o r i zon, w h i l e t h e a l t i t u d e change i s examined i n t h e d i r e c t i o n o f t h e true vertical.If another Earth), Earth's which w

we c u t t h e p l a n e o f t h i s t r u e h o r i z o n i n a n y d i r e c t i o n b y plane along the t r u e v e r t i c a l (through t h e center of t h e t h e l i n e formed by t h e i n t e r s e c t i o n o f t h i s p l a n e w i t h t h e s u r f a c e f o r m s a c l o s e d g r e a t c i r c l e , t h e mean r a d i u s o f i l l be equal t o t h e r a d i u s o f t h e Earth.

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The s h o r t e s t d i s t a n c e b e t w e e n two p o i n t s A B on t h e E a r t h ' s s u r f a c e or p a r t o f t h e a r c o f a g r e a t c i r c l e i s c a l l e d t h e o r t h o d r o m e ( s e e Fig. 1.5, a ) . T h e mean r a d i u s o f a g r e a t c i r c l e i s a s s u m e d t o b e e q u a l t o 6 3 7 1 km. The l e n g t h o f t h e c i r c u m f e r e n c e o f s u c h a r a d i u s i s e q u a l t o 4 0 , 0 0 0 km. One d e g r e e o f a r c o f a g r e a t c i r c l e i s e q u a l t o 111.1 km, w h i l e o n e m i n u t e o f a r c i s e q u a l t o 1 , 8 5 2 km. The l e n g t h o f a segment of t h e arc o f a g r e a t c i r c l e a t one minute o f a n g l e i s c a l l e d a nautical mile. With an i n t e r s e c t i o n o f t h e E a r t h ' s s p h e r e by a p l a n e which9

d o e s n o t p a s s t h r o u g h t h e c e n t e r of t h e E a r t h , t h e l i n e o f i n t e r s e c t i o n of t h i s p l a n e w i t h t h e E a r t h ' s s u r f a c e forms a c l o s e d s m a l l c i r c l e , t h e r a d i u s o f w h i c h w i l l a l w a y s b e l e s s t h a n t h e mean r a d i u s of t h e E a r t h . The s m a l l c i r c l e s p a r a l l e l t o t h e p l a n e o f t h e e q u a t o r are c a l l e d parallels ( s e e Fig. 1 . 5 , b ) . For t h e hurposes o f a i r c r a f t navigation, a coordinate system which unequivocally determines t h e p o s i t i o n of an a i r c r a f t and o b j e c t s on t h e E a r t h ' s s u r f a c e is necessary. Obviously, a s p h e r i c a l coordinate system w i l l be t h e most convenient ( F i g . 1 . 6 ) . A spherical coordinate sys tem i s d i s t i n g u i s h e d from a r e c t a n g u l a r s y s t e m ( C a r t e s i a n ) by the fact t h a t instead of deterF i g . 1 . 6 . R e l a t i o n s h i p Between a S p h e r i c a l System of Coordimining t h r e e distances t o a n a t e s and a R e c t a n g u l a r System. p o i n t i n t h e d i r e c t i o n s of t h e X , Y , a n d Z a x e s , we d e t e r m i n e the length of the radius-vector R from t h e c e n t e r o f t h e c o o r d i n a t e s y s t e m t o a p o i n t , and two a n gles: a n g l e X b e t w e e n t h e XY p l a n e and t h e p r o j e c t i o n o f t h e 'ra d i u s - v e c t o r (R) t o t h e p l a n e X Z , a n d a n g l e 4 b e t w e e n t h e X Z p l a n e and t h e d i r e c t i o n o f t h e r a d i u s - v e c t o r ( R ) . T h e r e i s an o b v i o u s r e l a t i o n b e t w e e n s p h e r i c a l a n d r e c t a n g u l a r . coordinate systems:

With a c o n s t a n t l e n g t h o f t h e r a d i u s - v e c t o r R , i f a n g l e s X and 4 a s s u m e all p o s s i b l e v a l u e s , t h e g e o m e t r i c l o c a t i o n o f t h e p o i n t s of t h e end of t h e v e c t o r r a d i u s w i l l be a s p h e r e .

To d e t e r m i n e c o o r d i n a t e s on t h e E a r t h ' s s u r f a c e , t h e r e i s n o This coorneed t o i n d i c a t e t h e r a d i u s of t h e E a r t h ( R ) each time. d i n a t e i s c o n s i d e r e d , o n c e and f o r a l l , c o n s t a n t .Thus, t h e s p h e r i c a l coordinate system is transformed i n t o a two-dimensional s u r f a c e system which i s c a l l e d a geographic system of coordinates.

/6 1

The p l a n e o f t h e e q u a t o r a n d t h e p l a n e o f t h e p r i m e ( G r e e n w i c h ) meridian a r e taken as t h e i n i t i a l r e f e r e n c e p l a n e s i n a geographic coordinate system. The p o i n t c o o r d i n a t e s o n t h e E a r t h ' s s u r f a c e b e a r t h e name " l o n g i t u d e o f t h e l o c u s " a n d " l a t i t u d e of t h e l o c u s " (Fig. 1.7).

10

The d i h e d r a l a n g l e b e t w e e n t h e p l a n e o f t h e p r i m e m e r i d i a n a n d t h e p l a n e of t h e m e r i d i a n o f a g i v e n p o i n t i s c a l l e d t h e l o n g i t u d e of t h e p o i n t ( A ) . Determination of t h e longitude can be given i n arc values: t h e length of the a r c o f t h e e q u a t o r (or t h e p a r a l l e l ) , expressed i n degrees, be PN tween t h e prime m e r i d i a n and t h e meridian of a given point is called the longitude of the point. Reading o f t h e l o n g i t u d e i s c a r r i e d o u t from 0 t o 180 e a s t o f t h e rpime m e r i d i a n ( e a s t long i t u d e ) a n d f r o m 0 t o 180 w e s t o f t h e rpime m e r i d i a n (west long i t u d e ) . I n navigational calcu l a t i o n s , east longitude is taken as p o s i t i v e and i s d e s i g n a t e d by a plus sign, while w e s t longitude i s n e g a t i v e a n d i s d e s i g n a t e d by a minus s i g n . However, i n c a r r y ing out navigational calculations, i t i s more c o n v e n i e n t t o c a r r y o u t a reading of longitude i n t h e e a s t e r l y d i r e c t i o n from z e r o t o360O.

ps

Fig. 1.7. Spherical Coordinate S y s t e m on t h e E a r t h ' s S u r f a c e .

The a n g l e b e t w e e n t h e p l a n e of t h e e q u a t o r a n d t h e t r u e v e r t i c a l o f a g i v e n p o i n t (or t h e l e n g t h o f t h e m e r i d i a n a r c , e x p r e s s e d i n d e g r e e s , from t h e p l a n e of t h e e q u a t o r t o t h e p a r a l l e l o f a g i v e n p o i n t ) i s c a l l e d t h e Z a t i t u d e of t h e p o i n t ( 9 ) . Since a set of t r u e v e r t i c a l s a t a c o n s t a n t l a t i t u d e forms a cone w i t h t h e v e r t e x i n t h e c e n t e r of t h e E a r t h and an a n g l e a t t h e v e r t e x e q u a l t o 90-4, then i n c o n t r a s t t o t h e d i h e d r a l angle between t h e p l a n e s of t h e meridians, we s h a l l call a similar angle i n other spherical systems, t h e conic

angle.Reading of t h e l a t i t u d e i s c a r r i e d o u t from t h e plane of t h e e q u a t o r t o t h e n o r t h a n d s o u t h f r o m 0 t o 9 0 ( n o r t h and s o u t h Z a t i tude). In navigational calculations, north l a t i t u d e is considered p o s i t i v e and s o u t h , n e g a t i v e .A geographic coordinate system i s a surface c u r v i l i n e a r system, ;.e., t h e m e r i d i a n s of t h e c o o r d i n a t e g r i d on t h e E a r t h a r e n o t parallel. However, i f w e e x a m i n e t h e m e r i d i a n s a n d p a r a l l e l s on any u n i t area of t h e E a r t h ' s s u r f a c e , t h e y t u r n o u t t o be o r t h o g o n a l ( p e r p e n d i c u l a r i n one p l a n e ) . Two s p e c i a l p o i n t s o n t h e E a r t h ' s surface ( t h e geographic poles) are an exception.

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A geographic coordinate system is used n o t only t o determine t h e l o c a t i o n o f a p o i n t ( o b j e c t ) on t h e E a r t h , b u t t o d e t e r m i n e d i r e c t i o n from one p o i n t t o a n o t h e r .

11

The a n g l e i n c l u d e d b e t w e e n t h e n o r t h e r n d i r e c t i o n o f t h e m e r i d i a n which p a s s e s t h r o u g h a g i v e n p o i n t and t h e orthodrome d i r e c t i o n Read t o a p o i n t s e t t i n g a c o u r s e i s c a l l e d t h e b e a r i n g or a z i m u t h . i n g o f t h e a n g l e s o f b e a r i n g or a z i m u t h i s d o n e c l o c k w i s e f r o m 0 t o 360O. S i n c e t h e m e r i d i a n s on t h e E a r t h ' s s u r f a c e a r e g e n e r a l l y n o t p a r a l l e l , t h e v a l u e o f t h e a z i m u t h c h a n g e s w i t h a c h a n g e i n t h e mov i n g l o n g i t u d e a l o n g t h e l i n e which j o i n s t h e two p o i n t s ; t h e g r e a t e r Therefore, f o r t h e orthodrome t h e l a t i t u d e , t h e more i t c h a n g e s . d i r e c t i o n t o g e t h e r w i t h an i n d i c a t i o n o f t h e a z i m u t h , it i s n e c e s s a r y t o mention from which meridian t h i a d i r e c t i o n i s measured. The c h a n g e i n a z i m u t h w i t h a c h a n g e i n t h e moving l o n g i t u d e d o e s n o t make i t p o s s i b l e t o u s e m a g n e t i c c o m p a s s e s f o r moving a l o n g t h e orthodrome without i n t r o d u c i n g corresponding c o r r e c t i o n s , espe c i a l l y when t h e t w o p o i n t s a r e f a r a p a r t .If t h e m a g n e t i c d e c l i n a t i o n d o e s n o t c h a n g e , f o l l o w i n g a con s t a n t magnetic course w i l l cause t h e meridians t o i n t e r s e c t a t iden tical angles. The l i n e w h i c h i n t e r s e c t s t h e m e r i d i a n s a t a c o n s t a n t a n g l e i s c a l l e d t h e loxodrome.

I n o r d e r t o p r o c e e d t o a more d e t a i l e d e x a m i n a t i o n o f t h e e l e ments o f a i r c r a f t n a v i g a t i o n and t h e i r measurement, it i s n e c e s s a r y t o become a c q u a i n t e d w i t h t h e m a k i n g o f m a p s , t h e i r s c a l e s , a n d some f e a t u r e s o f c a r t o g r a p h i c p r o j e c t i o n s .

5.

C h a r t s , Maps

,

and C a r t o g r a p h i c P r o j e c t i o n s

The r e p r e s e n t a t i o n o f a s m a l l p a r t o f t h e E a r t h ' s s u r f a c e on a plane is called a chart. D i s t o r t i o n as a r e s u l t o f t h e c u r v a t u r e o f t h e E a r t h ' s s u r f a c e i s p r a c t i c a l l y a b s e n t on a c h a r t . The c o n v e n t i o n a l r e p r e s e n t a t i o n o f t h e E a r t h ' s p l a n e i s c a l l e d a map. surface in a

A map i s a c o n t i n u o u s r e p r e s e n t a t i o n o f t h e s u r f a c e o f t h e E a r t h or a p a r t o f i t w i t h o u t d i s c o n t i n u i t i e s a n d f o l d s , made w i t h a variable scale according t o a d e f i n i t e r u l e . The s p h e r i c i t y o f t h e E a r t h ' s s u r f a c e d o e s n o t a l l o w i t t o b e r e p r e s e n t e d w i t h com p l e t e a c c u r a c y on a p l a n e s u r f a c e . T h e r e f o r e , t h e r e a r e many w a y s o f p r o j e c t i n g t h e E a r t h ' s s u r f a c e o n t o a p l a n e w h i c h make i t p o s s i b l e t o r e p r e s e n t m o s t a c c u r a t e l y on t h e map o n l y t h o s e p a r a m e t e r s ( e l e m e n t s ) which a r e most n e c e s s a r y under t h e g i v e n c o n d i t i o n s of application. M e t h o d s or l a w s o f r e p r e s e n t i n g t h e E a r t h ' s are called cartographic projections. s u r f a c e on a p l a n e

/8 1

A common g e o m e t r i c a l p r o j e c t i o n i s t h e p o i n t o f i n t e r s e c t i o n o f t h e l i n e of s i g h t ( w h i c h p a s s e s t h r o u g h t h e e y e o f t h e o b s e r v e r

12

and t h e p r o j e c t e d p o i n t ) w i t h t h e p l a n e o n t o which t h e g i v e n p o i n t is projected. I t i s a s p e c i a l case o f c a r t o g r a p h i c p r o j e c t i o n .A c a r t o g r a p h i c p r o j e c t i o n i s set a n a l y t i c a l l y as a f u n c t i o n o f g e o g r a p h i c a l c o o r d i n a t e s on t h e E a r t h ( s p h e r e ) b e t w e e n t h e c o o r d i n a t e s o f a p o i n t on a p l a n e .

If w e c a l l o n e o f t h e m a i n d i r e c t i o n s o n a map t h e X a x i s a n d t h e perpendicular t o it t h e Z a x i s , t h e n X = F l ( r p ; A) a n d z=&('9; 1 ; )

or

P=

1) an d ii = F, (~p; A),

w h e r e p a n d 6 a r e t h e m a i n d i r e c t i o n s o n maps o f c o n i c a n d a z i m u t h a l p r o j e c t i o n s , a n d I$ a n d X a r e t h e g e o g r a p h i c a l c o o r d i n a t e s o f a p o i n t on t h e E a r t h ( s p h e r e ) . The p r o p e r t i e s o f t h e p r o j e c t i o n s w i l l d e p e n d on t h e p r o p e r t i e s of t h e s e f u n c t i o n s (F1, F2, F 3 , and F 4 ) , which must be c o n t i n u o u s a n d w e l l - d e f i n e d , s i n c e t h e map i s made w i t h o u t d i s c o n t i n u i t i e s s o t h a t a s i n g l e p o i n t o n t h e map c o r r e s p o n d s t o e v e r y p o i n t i n t h e location. Map S c a l e s The map-making a) globe. b) p r o c e s s i s d i v i d e d i n t o two s t a g e s .

The E a r t h i s d e c r e a s e d t o t h e d e f i n i t e d i m e n s i o n s o f a

The g l o b e i s u n r o l l e d t o f o r m a p l a n e .

The e x t e n t of t h e o v e r a l l d e c r e a s e i n t h e E a r t h ' s d i m e n s i o n s t o t h e f i x e d dimensions of a globe i s c a l l e d a principaz scale. A p r i n c i p a l s c a l e i s a l w a y s i n d i c a t e d o n t h e e d g e o f a map a n d makes it p o s s i b l e t o j u d g e t h e d e c r e a s e o f t h e l e n g t h o f a s e g m e n t i n t r a n s f e r r i n g i t from t h e E a r t h ' s s u r f a c e t o t h e g l o b e . A principal scale is numerically equal t o t h e r a t i o of t h e d i s t a n c e on t h e g l o b e t o t h e a c t u a l d i s t a n c e a t a l o c a t i o n :

A. s a M= ASe.s.4

w h e r e M i s t h e p r i n c i p a l s c a l e , A S g i s a s e g m e n t on t h e g l o b e , a n d i s a s e g m e n t on t h e E a r t h ' s s u r f a c e w h i c h c o r r e s p o n d s t o t h e segment on t h e g l o b e . On m a p s , t h e p r i n c i p a l s c a l e i s u s u a l l y s h o w n a s a f r a c t i o n ( n u m e r i c a l s c a l e ) a n d by means o f a s p e c i a l s c a l e ( l i n e a r s c a l e ) .13

The n u m e r i c a l s c a l e . i s a f r a c t i o n , t h e n u m e r a t o r o f w h i c h i s o n e , w h i l e t h e d e n o m i n a t o r s h o w s how many s u c h u n i t s o f m e a s u r e m e n t fit i n t o t h e location./19 F o r e x a m p l e , 1 : 1 , 0 0 0 , 0 0 0 means t h a t i f w e t a k e 1 c m on a maps t h e n 1 , 0 0 0 , 0 0 0 c m a t a l o c a t i o n ( i . e . , 1 0 km) w i l $ c o r r e s p o n d t o i t .

-

A l i n e a r s c a l e i s a s c a l e on a map i n w h i c h a d e f i n i t e n u m b e r o f k i l o m e t e r s a t a l o c a t i o n correspond t o s p e c i a l segments of t h e scale. However, a p r i n c i p a l s c a l e ( n u m e r i c a l and l i n e a r ) i s i n s u f f i c i e n t f o r a c c u r a t e l y m e a s u r i n g d i s t a n c e s on t h e e n t i r e f i e l d o f a map. I t i s n e c e s s a r y t o know t h e l a w s o f d i s t o r t i o n o f d i s t a n c e s and d i r e c t i o n s . The l a w s o f c h a n g e i n t h e p r i n c i p a l s c a l e a l o n g t h e map f i e l d a r e d e t e r m i n e d by a s p e c i a l s c a l e .

A special scale i s t h e r a t i o of an i n f i n i t e l y small segment i n a g i v e n p l a c e on t h e map i n a g i v e n d i r e c t i o n , t o a n a n a l o g o u s s e g A t e a c h p o i n t on t h e m a p , t h e s p e c i a l ment i n a l o c a t i o n ( g l o b e ) . scale is different. I t i s e i t h e r s o m e w h a t l a r g e r or s o m e w h a t s m a l l e r than the principal scale.Distort ions o f Cartographic Project ions

E l l i p s e of D i s t o P t i o n sL e t u s draw on a s p h e r e ( g l o b e ) , a n i n f i n i t e l y s m a l l c i r c l e with radius r ; l e t us a l s o designate a rectangular coordinate system' Then on t h e s p h e r e by x a n d z ( F i g . 1 . 8 , a ) .f2

= x2

+ P.

(1.8)

Fig.

1.8.

D i s t o r t i o n o f S c a l e s on a P l a n e : ( b ) S c a l e on a P l a n e .

( a ) S c a l e on a G l o b e ;

14

I n t h e t r a n s f e r o f t h e coordinate system from t h e sphere (globe) t o the plane, the direction of the coordinate,axes is dis torted (Fig. 1.8, b). Having d e s i g n a t e d t h e s p e c i a l s c a l e s on a p l a n e (map) b y m i n t h e d i r e c t i o n X and n i n t h e d i r e c t i o n z , w e o b t a i n : XI = mx; 21 = nz

S u b s t i t u t i n g t h e l a t t e r i n (1.81,

a n d t h e n d i v i d i n g b o t h s i d e s o f t h e e q u a t i o n b y r2, w e o b t a i n

mr

+ (")

nr

=

,

(1.9)

From m a t h e m a t i c s , i t i s known t h a t t h i s i s t h e f o r m u l a o f a n e l l i p s e with conjugate diameters; therefore: a) Any i n f i n i t e l y s m a l l c i r c l e o n t h e s u r f a c e o f t h e E a r t h ' s s p h e r e i n any p r o j e c t i o n i s r e p r e s e n t e d by a n i n f i n i t e l y s m a l l e l lipse. b) On t h e s u r f a c e o f t h e E a r t h ' s s p h e r e ( g l o b e ) , i t i s p o s s i b l e t o choose two m u t u a l l y p e r p e n d i c u l a r d i r e c t i o n s which w i l l be t r a n s f e r r e d t o a map w i t h o u t a n y d i s t o r t i o n s . These d i r e c t i o n s are c a l l e d p r i n c i p a l d i r e c t i o n s . Knowing t h e s p e c i a l s c a l e s ( m a n d n ) i n t h e p r i n c i p a l d i r e c t i o n s , it i s always p o s s i b l e t o c o n s t r u c t an e l l i p s e o f d i s t o r t i o n s which w i l l make it p o s s i b l e t o judge t h e n a t u r e o f t h e d i s t o r t i o n s o f t h e p r o j e c t i o n as a whole. In the majority of projections, the d i r e c t i o n s along t h e m e r i d i a n s and p a r a l l e l s are t a k e n as t h e p r i n cipal directions.

Distortion of L e n g t h sI f a n i n f i n i t e l y s m a l l c i r c l e on t h e E a r t h i s r e p r e s e n t e d by an e l l i p s e (Fig. 1 . 9 , b) with i t s t r a n s f e r t o a p l a n e , t h e d i s t o r t i o n o f t h e s p e c i a l scale i n any d i r e c t i o n ( A S , ) can be expressed as f o l l o w s : ASa=-- 0 1 4 - V ( W + n i p ( D (1.10) OM r

/21

15

b u t from t h e c i r c l e i n F i g u r e 1 . 9 ,

a:

~ = s ! n a r , ~ i 1 ett=cosar, wh thenASa=

m sinza + n2 cos2 a , z

(1.11)

Fig.

1.9.

Distortion

i n a Plane: ( a ) L e n g t h on a G l o b e ; on a P l a n e .

(b) L e n g t h

Xl

Fig.

1.10.

D i s t o r t i o n of D i r e c t i o n s o n a Map. ( a ) D i r e c t i o n on a G l o b e ; ( b ) D i r e c t i o n o n a Map

i . e . , k n o w i n g t h e s p e c i a l s c a l e s for t h e p r i n c i p a l d i r e c t i o n s , we can always judge t h e value o f t h e d i s t o r t i o n o f t h e s p e c i a l s c a l e i n any d i r e c t i o n ( a n d t h e r e f o r e , t h e d i s t o r t i o n of t h e l e n g t h o f t h e segment as a w h o l e ) .

D i s t o r t i o n of D i r e c t i o n sLet us take t h e radius r = 1 (Fig. c i r c l e on t h e E a r t h ; t h e ntga=%-,X

1.10)nz

o f an i n f i n i t e l y small

I

w h i le t g e = - . mr

(1.12)

16

Dividing Equations

(1.12)

i n t o one a n o t h e r , w e o b t a i n :tgB

=mtg a.II

(1.13)

Obviously, knowing t h e s p e c i a l s c a l e s f o r t h e p r i n c i p a l d i r e c t i o n s , i t i s a l w a y s p o s s i b l e t o f i n d a n a n g l e f3 o n a map f o r a n a n g i e a i n a l o c a t i o n , and v i c e v e r s a .

D i s t o r t i o n of A r e a sT h e d i s t o r t i o n o f a r e a s AP c a n b e d e t e r m i n e d b y a c o m p a r i s o n o r d i v i s i o n o f t h e a r e a of t h e e l l i p s e ( S e l l by t h e a r e a o f a c i r c l e (sei); s e e F i g u r e 1.11: (1.14) b u t i f we t a k e t h e r a d i u s o f t h e c i r c l e o n t h e E a r t h a s e q u a l t o 1 , then AP = ab

/22

or, i f w e e x p r e s s a a n d b b y s p e c i a l s c a l e s f o r t h e p r i n c i p a l d i r e c t i o n s , we o b t a i n : LIP = mn,, (1.15)

xFig.1.11.

D i s t o r t i o n o f A r e a s o n a Map. ( a ) Area on a G l o b e ; ( b ) A r e a o n a Map.

The d i s t o r t i o n o f a r e a s i s e q u a l t o t h e p r o d u c t o f t h e s p e c i a l scales f o r t h e p r i n c i p a l d i r e c t i o n s . H e n c e , w e s e e t h a t i f w e know t h e s p e c i a l s c a l e s for t h e p r i n c i p a l d i r e c t i o n s , w e can give t h e complete c h a r a c t e r i s t i c s of any map p r o j e c t i o n ., .

17

C 1 a s s i f i c a t i o n o f C a r t o g r a p h i c P r o j e c t i,on s

T h e r e a r e many c a r t o g r a p h i c p r o j e c t i o n s . a c c o r d i n g t o two b a s i c c h a r a c t e r i s t i c s :

They c a n b e d i v i d e d

(a) a c c o r d i n g t o t h e n a t u r e o f t h e d i s t o r t i o n s , and (b) a c c o r d i n g t o t h e means o f c o n s t r u c t i o n ' o r t h e a p p e a r a n c e o f t h e normal g r i d . By norma2 grid w e mean t h e c o o r d i n a t e s y s t e m o n a g l o b e w h i c h i s m o s t s i m p l y r e p r e s e n t e d o n a map. Obviously, t h i s i s a system o f m e r i d i a n s and p a r a l l e l s .

Division of Projections b y the Nature of the Distortions

The c h o i c e o f c a r t o g r a p h i c p r o j e c t i o n s d e p e n d s on t h e p r o b l e m s f o r whose s o l u t i o n t h e y a r e i n t e n d e d . According t o t h e n a t u r e of t h e e l e m e n t s w h i c h h a v e t h e l e a s t d i s t o r t i o n on a map, c a r t o g r a p h i c projections are divided i n t o t h e following groups:

1.

Isogonal o r conformal p r o j e c t i o n s

These p r o j e c t i o n s must s a t i s f y t h e r e q u i r e m e n t o f e q u a l i t y o f a n g l e s and s i m i l a r i t y o f f i g u r e s ( c o n f o r m a b i l i t y ) w i t h i n t h e l i m i t s o f u n i t areas of t h e E a r t h ' s s u r f a c e , i . e . , s o t h a t i n p r o j e c t i n g a s u r f a c e o f a g l o b e o n t o a p l a n e (map)., t h e a n g l e s a n d s i m i l a r f i g u r e s do n o t change. b)

CIc

I 41

I xf

IX

z

Fig. 1.12. C o n f o r m a b i l i t y o f F i g u r e s o n Maps. ( a ) Preserving the C o n f o r m a b i l i t y of a U n i t A r e a ; ( b ) D e s t r o y i n g t h e C o n f o r m a b i l i t y o a Long S t r i p . A c c o r d i n g t o t h e s t i p u l a t i o n , t h e a n g l e o n a map m u s t b e e q u a l t o t h e a n g l e a t t h e l o c a t i o n : L 6 = L a , b u t from ( 1 . 1 3 ) it i s ob v i o u s t h a t i n t h i s c a s e m = n. Therefore, t h e equation of s p e c i a l scales f o r p r i n c i p a l d i r e c tions is a condition for isogonality.

18

On l a r g e p a r t s o f t h e s u r f a c e , w i t h i n t h e l i m i t s o f w h i c h i t i s impossible t o d i s r e g a r d t h e change i n scale, t h e c o n f o r m a b i l i t y (and therefore t h e isogonality) are not preserved. Figure 1.12 g i v e s a n example of p r e s e r v i n g t h e c o n f o r m a b i l i t y o f a u n i t area and d e s t r o y i n g t h e c o n f o r m a b i l i t y o f a l o n g s t r i p . T h e u n i t a r e a ( F i g . 1 . 1 2 , a ) i s t r a n s f e r r e d t o t h e map o n a d e f i n i t e scale without d i s t o r t i o n s . The l o n g s t r i p ( F i g . 1 . 1 2 , b ) c a n b e d i v i d e d i n t o a number o f u n i t a r e a s , e a c h o f w h i c h w i l l b e t r a n s f e r r e d t o t h e map o n a s o m e w h a t c h a n g e d s c a l e . Since t h e scales mx,and nz a r e i n c r e a s e d p r o p o r t i o n a l l y i n t h e d i r e c t i o n o f t h e s t r i p , e a c h o f t h e s m a l l a r e a s i s r e p r e s e n t e d o n t h e map w i t h t h e c o n f o r m a By e q u a t i n g t h e b i l i t y b e i n g p r e s e r v e d , o n l y on a d i f f e r e n t s c a l e . /24 l a t e r a l l i m i t s o f t h e s m a l l a r e a s , w e do n o t o b t a i n a c o n f o r m a l figure, i.e., the similarity of small figures i n isogonal projec tions is preserved, while the similarity of large figures (large l a k e s , seas, e t c . ) is destroyed.

2.

Equally spaced o r equidistant projections

The e q u i v a l e n c e t o u n i t y o f t h e s p e c i a l s c a l e s f o r a p r i n c i p a l d i r e c t i o n ( m = 1 o r n = 1) i s a n e c e s s a r y c o n d i t i o n o f t h i s g r o u p of projections.

z

4X

$ ? ,

T h i s m e a n s t h a t t h e map s c a l e w i l l be p r e s e r v e d i n one o f t h e p r i n c i p a l directions. T h e r e f o r e , when u s i n g s u c h a map w e c a n m e a s u r e t h e d i s t a n c e i n o n e o f t h e d i r e c t i o n s by means o f a scale. The n a t u r e o f t h e d i s t o r t i o n of c o n f o r m a b i l i t y i n t h e s e projections i s shown i n F i g . 1 . 1 3 . Here m = c o n s t , while n is a function of Z.

J,

3.

Fig. 1.13. Distortion of Conformability i n Equally Spaced P r o j e c t i o n s : ( a ) Appearance o f a Figu r e i n a L o c a t i o n ; (b) Appearance of t h e F i g u r e o n a Map.

Equally large or equivalent projections

T h i s group o f p r o j e c t i o n s must s a t i s f y the condition of equivalence of a r e a s i.e. , the product of t h e s p e c i a l scales f o r t h e p r i n c i p a l d i r e c t i o n s m u s t e q u a l u n i t y ( m n = 1); t h e r e f o r e , t h e r e l a t i o n between t h e s p e c i a l scales f o r t h e p r i n c i p a l d i r e c t i o n s w i l l be i n v e r s e l y p r o p o r t i o n a l :

m=-;

1

n

n=-.

1 m

These p r o j e c t i o n s do n o t have an e q u i v a l e n c e o f a n g l e s and a similarity of figures.

19

4.

Arbitrary projections

P r o j e c t i o n s o f t h i s g r o u p d o n o t s a t i s f y a n y of t h e c o n d i t i o n s mentioned above. H o w e v e r , t h e y a r e a l s o u s e d when c o m p a r a t i v e l y s m a l l p o r t i o n s o f t h e E a r t h ' s surface are projected onto a plane where t h e d i s t o r t i o n s o f t h e a n g l e s and t h e s c a l e s f o r t h e p r i n c i p a l d i r e c t i o n s a n d a l o n g t h e e n t i r e map f i e l d a r e i n s i g n i f i c a n t a n d t h e s i m i l a r i t y o f f i g u r e s and a r e a s which s a t i s f y t h e n e e d s o f t h e i r This group o f p r o j e c t i o n s i n practical application is preserved. c l u d e s a b a s i c f l i g h t map o n a s c a l e o f 1:1,000,000, w h i c h i s c o n s t r u c t e d a c c o r d i n g t o a s p e c i a l l a w a n d w h i c h h a s b e e n a c c e p t e d by i n t e r n a t i o n a l agreement.

For t h e p u r p o s e s o f a i r c r a f t n a v i g a t i o n , t h e m o s t n e c e s s a r y c o n d i t i o n s a r e ( o b v i o u s l y ) i s o g o n a l i t y and e q u a l s c a l e o f t h e maps. E q u a l l y l a r g e a n d e q u a l l y s p a c e d p r o j e c t i o n s o f maps a r e u s e d i n a i r c r a f t n a v i g a t i o n o n l y a s s u r v e y maps f o r s p e c i a l a p p l i c a t i o n s .They i n c l u d e maps o f h o u r z o n e s , m a g n e t i c d e c l i n a t i o n s , c o m p o s i t e d i a g r a m s o f t o p o g r a p h i c a l map s h e e t s , c l i m a t o l o g i c a l a n d m e t e o r o l o g i c a l maps, e t c ./25 -

D i v i s i o n of P r o j e c t i o n s A c c o r d i n g t o t h e Method of C o n s t r u c t i o n ( A c c o r d i n g t o t h e A p p e a r a n c e of t h e Normal G r i d )Depending on t h e method o f c o n s t r u c t i o n , c a r t o g r a p h i c p r o j e c t i o n s a r e d i v i d e d i n t o s e v e r a l g r o u p s , t h e b a s e s o f which a r e t h e following: (a) group o f c y l i n d r i c a l p r o j e c t i o n s ; (b) group o f c o n i c p r o j e c t i o n s and t h e i r v a r i a n t s , projections; ( c ) group of azimuthal p r o j e c t i o n s ; (d) group of s p e c i a l p r o j e c t i o n s .

polyconic

Each of t h e s e p r o j e c t i o n s i s d i v i d e d i n t u r n i n t o t h e f o l l o w i n g categories: normal, i f t h e E a r t h ' s a x i s c o n c i d e s w i t h t h e a x i s o f t h e f i g u r e o n t o which t h e E a r t h ' s s u r f a c e i s p r o j e c t e d ; t r a n s v e r s e , if t h e E a r t h ' s a x i s f o r m s a n a n g l e o f 90 w i t h t h e a x i s o f t h e f i g u r e , and o b l i q u e , i f t h e a x i s of t h e E a r t h d o e s n o t c o i n c i d e w i t h t h e a x i s of t h e f i g u r e and i n t e r s e c t s it a t an a n g l e which i s n o t e q u a l t o 90.Cy1 i n d r i c a l

Projections

Normal ( e q u i v a l e n t ) cy1 i n d r i c a l p r o j e c t i o nA l l c y l i n d r i c a l p r o j e c t i o n s a r e f o r m e d by means o f t h e i m a g i n a r y t r a n s f e r o f t h e E a r t h ' s s u r f a c e ( g l o b e ) t o a t a n g e n t i a l or i n tersecting cylinder, with subsequent unrolling. I n F i g u r e 1 . 1 4 , a simple normal c y l i n d r i c a l p r o j e c t i o n i s g i v e n , i . e . , a p r o j e c t i o n o f t h e E a r t h on a t a n g e n t i a l c y l i n d e r ,

t h e a x i s of which c o i n c i d e s w i t h t h e a x i s * o f t h e E a r t h ( g l o b e ) , while the height of t h e cylinder is proportional t o the length of the axis.

PS

Fig.

1.14.

Normal

(Equivalent) Cylindrical Projection

Fig.

1.15.

Simple E q u a l l y S p a c e d C y l i n d r i c a l P r o j e c t i o n

I n t h i s p r o j e c t i o n , t h e meridians are compressed while t h e p a r a l l e l s are extended t o a d e g r e e which i n c r e a s e s w i t h l a t i t u d e . The p r o j e c t i o n i n c l u d e s a c a t e g o r y o f e q u a l l y l a r g e a n d e q u i v a l e n t p r o j e c t i o n s , s i n c e it satisfies t h e condition o f an equivalence of areas.I t s equation can be w r i t t e n i n t h e following form:X = R sin v; 2 = RS,

(1.16)

where X r e p r e s e n t s t h e c o o r d i n a t e s o f a p o i n t a l o n g t h e m e r i d i a n ; Z r e p r e s e n t s t h e coordinates of a point along t h e equator; and R is the Earth's radius.L e t u s d e t e r m i n e what t h e s p e c i a l s c a l e s f o r t h e d i r e c t i o n s are equal t o i n t h i s p r o j e c t i o n :

21

(1.17)

(1.18)

where m i s a p a r t i a l scale a l o n g a m e r i d i a n ; n i s a p a r t i a l scale along a parallel; i s a n i n c r e a s e i n d i s t a n c e on t h e map; d S g l o b e i s a n i n c r e a s e i n d i s t a n c e on t h e g l o b e .

asmap

The p r o d u c t o f t h e s p e c i a l s c a l e s i s1

=1

mn -cos

or m = d , w h i l e n=; 1 1n

'p

cos 'p

Therefore, the given projection is equal.

Since

m # n ; m # 1 and

n # 1 i n t h e p r i n c i p a l d i r e c t i o n s ( m e r i d i a n s and p a r a l l e l s ) it i s

n o t i s o g o n a l and n o t e q u a l l y spaced. Only i n t h e e q u a t o r i a l band, i n t h e l i m i t s f r o m 0 t o f 5 O a l o n g i t s l a t i t u d e , i s it p r a c t i c a l l y p o s s i b l e t o c o n s i d e r i t i s o g o n a l and e q u a l l y spaced.

/27 -

S i m p l e equally spaced c y l i n d r i c a l p r o j e c t i o nIf w e t a k e t h e h e i g h t of a c y l i n d e r t o be p r o p o r t i o n a l n o t t o the length of the Earth's axis, but t o t h e length of a meridian, and i n s t e a d o f simply p r o j e c t i n g w e u n f o l d t h e m e r i d i a n s t o t h e c y l i n d e r w a l l s , a s shown i n F i g . 1 . 1 5 , t h e n a s i m p l e , e q u a l l y s p a c e d I t i s r e g a r d e d as normal s i n c e cylindrical projection is obtained. t h e axis of t h e globe coincides with t h e a x i s o f t h e cylinder.I n t h i s p r o j e c t i o n , t h e meridians w i l l be transformed t o t h e i r f u l l s i z e d u r i n g t h e i r t r a n s f e r f r o m t h e g l o b e ' s s u r f a c e t o a map ( i . e . , m = l), a n d t h e e q u a t o r a l s o w i l l b e t r a n s f o r m e d t o full s i z e ( a t t h e e q u a t o r , n = 11, w h i l e t h e p a r a l l e l s w i l l b e e x t e n d e d The m a g n i t u d e o f t h e j u s t as i n a normal ( e q u i v a l e n t ) p r o j e c t i o n . effect increases with l a t i t u d e . T h e c o o r d i n a t e g r i d o f t h e map o f t h i s p r o j e c t i o n h a s t h e a p pearance of a uniform r e c t a n g u l a r r u l i n g . I t s e q u a t i o n s have t h e form : X=R'p; 2 - R A . The s p e c i a l s c a l e s a r e e q u a l t o :

(1.19)

1 - sec = cos'p

along the p a r a l l e l

n=

'p.

(1.20)

22

S i n c e m = 1, t h e p r o j e c t i o n i s e q u a l l y s p a c e d a l o n g t h e m e r i d i a n s and a l s o a l o n g t h e e q u a t o r . S i n c e m # n a n d mn # 1, t h e p r o j e c t i o n i s n o t i s o g o n a l and n o t e q u a l l y l a r g e , e x c e p t f o r t h e equa t o r i a l band i n t h e l i m i t s from 0 t o + 5 O a l o n g t h e l a t i t u d e , where it i s p r a c t i c a l l y p o s s i b l e t o c o n s i d e r i t i s o g o n a l and e q u a l l y l a r g e . Maps i n n o r m a l ( e q u i v a l e n t ) a n d s i m p l e , e q u a l l y d i s t a n t c y l i n maps d r i c a l p r o j e c t i o n s are u s e d i n a v i a t i o n o n l y as r e f e r e n c e s : o f h o u r z o n e s , maps o f n a t u r a l l i g h t , e t c .

I s o g o n a l cy1 i n d r i c a l p r o j e c t i o nAn i s o g o n a l c y l i n d r i c a l p r o j e c t i o n ( M e r c a t o r p r o j e c t i o n ) i s t h e most v a l u a b l e of a l l t h e c y l i n d r i c a l p r o j e c t i o n s f o r n a v i g a t i o n . I t i s o b t a i n e d from a s i m p l e , e q u a l l y spaced c y l i n d r i c a l p r o j e c t i o n by a r t i f i c i a l l y e x t e n d i n g t h e s c a l e a l o n g t h e l a t i t u d e ( l e n g t h e n i n g t h e m e r i d i a n s ) , p r o p o r t i o n a l t o t h e change i n s c a l e a l o n g t h e l o n g i tude. T h e c o o r d i n a t e g r i d o f t h e map o f t h i s p r o j e c t i o n i s s h o w n i n Figure 1.16. The r e a s o n f o r i t s u s e i s t h e f a c t t h a t t h e a n g l e s m e a s u r e d o n t h e map a r e e q u a l t o t h e c o r r e s p o n d i n g a n g l e s a t t h e l o c a t i o n , % . e . , m = n = s e c 6.Le t i o n of along a dinate) f i n d m: t us w r i t e an equa t h i s map p r o j e c t i o n meridian (X-coor f o r which w e can

/28

m=

dS dS

--d X globe

Rdv '

w h e r e dS i s a n i n c r e a s e o f distance along the meridian on t h e map; a n d R d + i s a n increase i n distance along t h e meridian at the loca tion.W must have e m = sec 'p,W s h a l l then equate the e r i g h t - h a n d s i d e s of t h e s e equations:

Fig. 1.16. Coordinate Grid of an Isogonal Cylindrical Projection.

- .sec +,'w h e n c e R 4dXe :

dX-

Rd'p -a(l.21) cos'p

23

After i n t e g r a t i n g (1.211, along the meridian:

we w i l l obtain t h e X-coordinate

X = R I n t g 45( O

+-

i ),

(1.22)

while the Z-coordinate ple equation:

a l o n g t h e p a r a l l e l i s d e t e r m i n e d by t h e s i m

z= Rh.

(1.22a)

(m # 1

Since m = n , the projection is isogonal but not equally spaced a n d n # 1) and n o t e q u a l l y l a r g e ( m n # 1 ) .

T h e b a s i c a d v a n t a g e o f maps i n a n i s o g o n a l c y l i n d r i c a l p r o j e c t i o n i s t h e s i m p l i c i t y o f t h e i r u s e w i t h m a g n e t i c c o m p a s s e s for moving f r o m o n e p o i n t on t h e E a r t h t o a n o t h e r , s i n c e t h e loxodrome Therefore, i n t h i s p r o j e c t i o n has t h e appearance of a s t r a i g h t l i n e . t h e isogonal c y l i n d r i c a l projection has been used widely, primarily i n m a r i n e n a v i g a t i o n d u r i n g t h e c o m p i l a t i o n o f n a v a l maps. The c h a n g e i n s c a l e w i t h l a t i t u d e i s a d i s a d v a n t a g e o f n o r m a l cylindrical projections. Here, i n normal ( e q u i v a l e n t ) and s i m p l e , e q u a l l y s p a c e d c y l i n d r i c a l p r o j e c t i o n s , t h e map s c a l e i s n o t i d e n t i c a l i n t h e p r i n c i p a l d i r e c t i o n s ( n o r t h - s o u t h and e a s t - w e s t ) , s o t h a t t h e d i s t a n c e b e t w e e n two p o i n t s i n d i r e c t i o n s n o t p a r a l l e l t o t h e l i n e s o f t h e g r a t i n g c a n b e d e t e r m i n e d o n l y by c a l c u l a t i o n . I n a n i s o g o n a l c y l i n d r i c a l p r o j e c t i o n , t h e map s c a l e a l o n g t h e l a t i t u d e i s a l s o v a r i a b l e , b u t a t a n y p o i n t o n t h e map i t i s i d e n t i cal in the principal directions. T h i s makes it p o s s i b l e t o m e a s u r e d i s t a n c e s b y m e a n s of c o m p a s s e s , for w h i c h a s c a l e ( v a r y i n g w i t h t h e l a t i t u d e ) i s drawn on t h e w e s t e r n an d e a s t e r n e d g e s o f t h e map. Means for m e a s u r i n g d i s t a n c e s o n maps w i t h s u c h a p r o j e c t i o n a r e i n d i c a t e d i n m a n u a l s for m a r i n e n a v i g a t i o n .

Isogonal oblique cylindrical projections

T h e b a s i s f o r c r e a t i n g maps i n a n i s o g o n a l c y l i n d r i c a l p r o j e c its isogonality. t i o n i s a property of t h e Mercator projection: S u c h p r o j e c t i o n s a r e u s e d i n t h e p r e p a r a t i o n o f s p e c i a l f l i g h t maps and 1:4,000,000 which a r e o n s c a l e s o f 1:1,000,000, 1 : 2 , 0 0 0 , 0 0 0 , used i n c i v i l a v i a t i o n . T h e t a n g e n t i a l ( F i g . 1.17) or i n t e r s e c t i n g ( F i g . 1.18) c y l i n d e r is s i t u a t e d a t such an angle t o t h e axis of t h e globe t h a t t h e tan g e n t o f t h e c y l i n d e r ' s s u r f a c e t o t h e g l o b e or t h e i n t e r s e c t i o n r u n s along t h e f l i g h t path. Usually the s t r i p along t h e tangent does not e x t e n d m o r e t h a n 5 0 0 - 6 0 0 km t o e i t h e r s i d e o f t h e r o u t e (or t h e m i d d l e l i n e o f t h e r o u t e , i f i t h a s d i s c o n t i n u i t i e s ) , w h i l e on t h e i n t e r s e c t i n g c o n e i t d o e s n o t e x t e n d m o r e t h a n 1 0 0 0 - 1 4 0 0 km t o e i t h e r s i d e of t h e given middle l i n e of t h e r o u t e s .24

I n p r a c t i c e , s u c h f l i g h t maps a r e i s o g o n a l , e q u a l l y s p a c e d , and e q u a l l y l a r g e ; however, s i n c e t h e c y l i n d e r i s i n c o n t a c t w i t h t h e g l o b e a l o n g t h e a r c o f a g r e a t c i r c l e or c u t s t h e g l o b e c o m p a r a t i v e l y c l o s e t o t h e a r c o f a g r e a t c i r c l e , t h e o r t h o d r o m e on t h e s e maps w i l l i n p r a c t i c e b e r e p r e s e n t e d b y a s t r a i g h t l i n e . T h e d i s t o r t i o n s o f l e n g t h s o n f l i g h t maps o f o b l i q u e t a n g e n t i a l p r o j e c t i o n s do n o t exceed 0.5%; f o r i n t e r s e c t i n g p r o j e c t i o n s t h e y do n o t .exceed 0.8%-1.2%.

/30

Fig. 1.17. Isogonal Oblique (Tangential) Cylindrical Projection.

Isogonal Oblique Fig. 1.18. (Intersecting) Cylindrical Projection.

I s o g o n a l t r a n s v e r s e and c y l i n d r i c a l G a u s s i a n p r o j e c t i o nThe a x i s o f t h e c y l i n d e r i n G a u s s i a n p r o j e c t i o n s i s p e r p e n d i c u l a r t o t h e a x i s of r o t a t i o n o f t h e E a r t h ( g l o b e ) . The c o n s t r u c t i o n o f maps w i t h t h i s p r o j e c t i o n i s s i m i l a r t o t h e c o n s t r u c t i o n o f maps with oblique cylindrical projections. F o r e x a m p l e , a f l i g h t map o n a s c a l e o f 1:1,000,000 f o r L e n i n g r a d - K i e v h a s b e e n c o m p i l e d o n such a projection. H o w e v e r , on t h e w h o l e , i s o g o n a l t r a n s v e r s e c y l i n d r i c a l G a u s s i a n p r o j e c t i o n i s u s e d f o r c o m p i l i n g maps o n a l a r g e s c a l e , where t h e s p e c i a l p r i n c i p l e s o f c o n s t r u c t i o n are used.A s p h e r o i d ( E a r t h ' s e l l i p s o i d ) i s t a k e n as t h e f i g u r e from which t h e E a r t h ' s s u r f a c e i s p r o j e c t e d , w h i l e t h e t a n g e n t i a l c y l i n d e r on w h i c h t h e E a r t h ' s s u r f a c e i s p r o j e c t e d h a s a n e l l i p t i c a l b a s e a c c o r d i n g t o t h e form of t h e E a r t h ' s e l l i p s o i d .

The e n t i r e E a r t h ' s s u r f a c e i s d i v i d e d b y m e r i d i a n s i n t o z o n e s , e a c h o f w h i c h h a s a l a t i t u d e o f 6 O a n d i s p r o j e c t e d o n t o i t s own c y l i n d e r which i s t a n g e n t i a l t o t h e E a r t h ' s s u r f a c e a l o n g t h e mid d l e meridian of t h e g i v e n zone. Thus, i n o r d e r t o p r o j e c t t h e whole s u r f a c e o f t h e E a r t h , it is necessary t o t u r n t h e e l l i p t i c a l c y l i n d e r mentally around t h e25

a x i s of t h e E a r t k ' s e l l i p s o i d t h r o u g h 6O a t a t i m e , In Figure 1.19, a , t h e p r o j e c t i o n o f o n l y one zone f o r 6 O l o n g i t u d e i s shown, w h i l e i n Figure 1.19, b y the unrolling of a semicylinder after its rotat i o n around t h e E a r t h ' s a x i s i n o r d e r t o p r o j e c t several zones i s shown. W i t h s u c h a p r o j e c t i o n , a l l maps a r e c o n s t r u c t e d o n t h e scales: 1:500,000, 1:200,000, 1:100,000, 1:50,000, and 1:25,000. The l a t t e r a r e e s s e n t i a l l y c h a r t s .

/31

Fig.

1.19.

Isogonal Transverse-Cylindrical Gaussian Projectioh.

E a c h z o n e o n maps w i t h a s c a l e o f 1:200,000 a n d l a r g e r h a s i t s own s p e c i a l X a n d Y(Z) r e c t a n g u l a r c o o r d i n a t e s y s t e m , w h i c h i s c a l l e d t h e Gaussian kilometer system. M e r i d i a n s a n d p a r a l l e l s on maps o f t h i s p r o j e c t i o n a r e c u r v e d l i n e s a n d d o n o t c o i n c i d e w i t h t h e Gaussian system. The v e r t i c a l l i n e s o f t h e r e c t a n g u l a r Gaus s i a n system a r e p a r a l l e l t o t h e c e n t r a l m e r i d i a n o f t h e zone and do n o t c o i n c i d e with o t h e r meridians o f t h e zone. The a n g l e b e t w e e n t h e v e r t i c a l l i n e X o f t h e G a u s s i a n s y s t e m and t h e l i n e t o t h e o b j e c t ( p o i n t ) i s c a l l e d t h e d i r e c t i o n a l angle. I n order t o o b t a i n t h e t r u e o r magnetic d i r e c t i o n ( a n g l e ) , t h e a n g l e s o f t h e convergence o f t h e system w i t h t h e t r u e and magnetic In addition, the m e r i d i a n s a r e i n d i c a t e d o n t h e e d g e o f t h e map. v e r t i c a l s e c t i o n o f a map ( f r a m e ) a l w a y s r u n s i n t h e d i r e c t i o n o f the true meridian, By m e a n s o f t h e G a u s s i a n s y s t e m a n d f i g u r e s i n t h e f r a m e s o f t h e maps, it i s p o s s i b l e t o determine t h e d i s t a n c e from t h e e q u a t o r and from t h e c e n t r a l meridian o f t h e zone t o t h e o b j e c t ( p o i n t ) . D i s t o r t i o n s o f l e n g t h s o n t h e s e maps a r e i n s i g n i f i c a n t a n d d o n o t exceed 0.14% a l o n g t h e edges o f t h e zone i n t h e l a t i t u d e which i s e q u a l t o z e r o (140 m a t 1 0 0 k m ) . Maps o n a n i s o g o n a l t r a n s v e r s e - c y l i n d r i c a l G a u s s i a n p r o j e c t i o n are used b o t h i n a v i a t i o n f o r a d e t a i l e d o r i e n t a t i o n and l o c a t i o n

26

o f t a r g e t s , a n d i n many b r a n c h e s o f t h e n a t i o n a l e c o n o m y f o r l i n k i n g p r o j e c t s , equipment, and r a d i o engineering f a c i l i t i e s i n a l o c a t i o n , 1 3 2 f o r determining geodesic r e f e r e n c e p o i n t s , and f o r a c c u r a t e geodesic c a l c u l a t i o n s o f d i s t a n c e s and d i r e c t i o n s , e t c .Conic P r o j e c t i o n s

Conic p r o j e c t i o n s a r e c o n s t r u c t e d by p r o j e c t i n g t h e s u r f a c e o f t h e E a r t h ' s spheroid (globe) on a tangent o r i n t e r s e c t i n g cone, w i t h i t s subsequent u n r o l l i n g t o form a p l a n e s u r f a c e ( F i g . 1 . 2 0 , a ) .

( a ) Tangent ( i n t e r Construction of Conic P r o j e c t i o n s : F i g . 1 . 2 0 . s e c t i n g ) c o n e ; ( b ) U n r o l l i n g o f t h e C o n e t o Form a P l a n e .

A c c o r d i n g t o t h e p o s i t i o n s of t h e axes o f t h e g l o b e a n d c o n e , c o n i c p r o j e c t i o n s can be normal, t r a n s v e r s e , and oblique. However, i n o u r p u b l i c a t i o n s n o r m a l p r o j e c t i o n s a r e g e n e r a l l y u s e d when t h e a x j s o f t h e cone c o i n c i d e s w i t h t h e a x i s of t h e g l o b e . I n a normal c o n i c p r o j e c t i o n , m e r i d i a n s a r e r e p r e s e n t e d by s t r a i g h t l i n e s , w h i l e p a r a l l e l s a r e r e p r e s e n t e d by a r c s o f c o n c e n t r i c circles (Fig. 1.20, b). From F i g u r e 1 . 2 0 , a , i t i s e a s y t o s e e t h a t t h e r a d i u s o f a p a r a l l e l of tangency (p,) c a n b e e x p r e s s e d by t h e E a r t h ' s r a d i u s :

where R i s t h e r a d i u s o f t h e E a r t h ( g l o b e ) and $ 0 i s t h e l a t i t u d e of t h e p a r a l l e l o f tangency. The e q u a t