A simulation of a mathematical model to compute path discrepancies between great circle and rhumb line flight paths is presented. The model illustrates that the path errors depend on the latitude, the bearing, and the trip length of the flight

Kaul, Rajan

English

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I PATH DISCREPANCIES BETWEEN GREAT CIRCLE 
AND RHUMB LINE* 
Rajan Kaul 
Ohio Universi ty  
Athens, Ohio 
A s imula t ion  of a mathematical model t o  compute p a t h  d i s c r e p a n c i e s  
between g r e a t  c i rc le  and rhumb l ine  f l i g h t  p a t h s  i s  presented.  The model 
i l l u s t r a t e s  t h a t  t h e  pa th  e r r o r s  depend on t h e  l a t i t u d e ,  t h e  bear ing ,  and 
t h e  t r i p  l e n g t h  of t h e  f l i g h t .  
I. INTRODUCTION 
A mathematical model f o r  a comparative a n a l y s i s  of g r e a t  c i rc le  v e r s u s  
rhumb l i n e  naviga t ion  i n  t h e  c o n t i n e n t a l  United S t a t e s  has been developed 
a t  t he  Avionics Engineering Center,  Ohio Universi ty .  A FORTRAN simulat ion 
of t he  model has been implemented on the  IBM 370 computer. The s imula t ion  
p r e d i c t s  p e r t i n e n t  navigat ion information f o r  t h e  two f l i g h t  paths.  The 
b a s i s  f o r  t n e  p r o j e c t ,  svfiich is a p a r t  of a~ M,S. t h e s i s ,  is t o  provide a 
d a t a  base f o r  computing d iscrepancies  between t h e  two f l i g h t  paths.  This 
document b r i e f l y  descr ibes  t h e  model and d i s c u s s e s  t h e  impl ica t ions  of t h e  
r e s u l t s  obtained. 
11. BACKGROUND INFORMATION 
The s tandard  en-route navigat ion system used i n  t h e  United S t a t e s  i s  
t h e  VOR/DME. VHF Omnidirectional Range (VOR) is t h e  b a s i s  f o r  def in ing  t h e  
airways and is t h e r e f o r e  an i n t e g r a l  part of a i r  t r a f f i c  c o n t r o l  proce- 
dures .  Two VOR s t a t i o n s  are w e d  tc? defifie a radial i n t e r s e c t i o n ,  which is 
t h e  b a s i s  of t he  e s t a b l i s h e d  Victor  airways. These Victor  airways fol low a 
f l i g h t  pach which maintains a path of constant heading ( i . e .  t h e  course c rosses  
t h e  meridians a t  t h e  same angle).  Thus, a rhumb l i n e  course appears as  a 
s t r a i g h t  l i n e  on a Mercator pro jec t ion .  
Area/Random Navigation (RNAV) as defined i n  FAA Advisory C i r c u l a r  90- 
4 5 A  (1) is  
I . .  .a method of navigat ion tha t  permits  a i r c r a f t  opera t ions  on 
any des i red  course wi th in  the  coverage of s t a t i o n  referenced 
naviga t ion  s i g n a l s  or  w i t h i n  t h e  l i m i t s  of self-contained 
system c a p a b i l i t i e s . '  
The p r i n c i p a l  advantage of RNAV i s  t h a t  i t  allows the  navigator  t o  f l y  
a g r e a t  c i rc le  course between two points.  The g r e a t  circle course is t h e  
3; Published as TM 89 a t  Ohio University.  
39 
https://ntrs.nasa.gov/search.jsp?R=19870009094 2020-03-20T11:24:14+00:00Z
?-- i . '  
; .. . 
s h o r t e s t  path between two poin ts  on t h e  e a r t h ,  and is formed by t h e  i n t e r -  
s e c t i o n  of t h e  plane defined by t h e  c e n t e r  of t he  e a r t h  and t h e  p o i n t s  of 
o r i g i n  and d e s t i n a t i o n ,  pro jec ted  on t h e  s u r f a c e  of t h e  ear th .  The f l i g h t  
pa th  is projected as a curved l i n e  on a Mercator p r o j e c t i o n  and hence 
changes heading constant ly .  
Loran-C, OMEGA, GPS, Doppler, and INS are considered RNAV systems. 
However, VOR i n  i t s  b a s i c  form is  not an RNAV system. A major concern is  
t h a t  t h e  discrepancies  between t h e  two f l i g h t  paths may cause naviga t ion  
c o n f l i c t s .  This r epor t  a t t e m p t s  t o  quant ize  t h e  d iscrepancies  which depend 
on var ious  f ac to r s .  
111. PATH DISCREPANCIES 
The magnitude of path d iscrepancies  between t h e  g r e a t  c i rc le  and rhumb 
l i n e  courses between two po in t s  increases  as 
1) t h e  l a t i t u d e  increases  , 
2 )  t h e  d i f fe rence  i n  t h e i r  l a t i t u d e s  decreases  and 
3)  t h e  d i f fe rence  i n  t h e i r  longi tudes  increases .  
When plot ted on a Mercator p r o j e c t i o n  map, t he  two f l i g h t  paths  s t a r t  
t o  diverge from the  o r i g i n  of t h e  f l i g h t .  The discrepancy is a maximum a t  
t h e  mid-point of the  f l i g h t ,  and then t h e  f l i g h t  paths s t a r t  t o  converge 
and meet at t h e  d e s t i n a t i o n  point .  The g r e a t  c i rc le  path is always curved 
away from the equator and t h e r e f o r e  i n t e r s e c t s  t h e  meridians a t  higher  
l a t i t u d e s  than the rhumb l i n e  path. Also,  t h e  meridians converge as they 
depar t  from t h e  equat0.r and a r e  c l o s e r  toge ther  a t  higher  l a t i t u d e s .  Thus, 
t h e  deviat ion between t h e  two paths increase  a t  higher  l a t i t u d e s  because 
t h e  rhumb l ine  path must curve r a p i d l y  t o  be a b l e  t o  f l y  a course of con- 
s t a n t  meridian crossing angle.  
The bearing of t h e  f l i g h t  path,  which is a simple func t ion  of t h e  l a t i -  
tude and longi tude,  a l s o  has a considerable  e f f e c t  on t h e  d iscrepancies  
between t h e  t w o  f l i g h t  paths. By d e f i n i t i o n ,  a l l  l i n e s  of longi tude are 
a l s o  g r e a t  c i r c l e s .  On a t r u e  north-south f l i g h t  path t h e  great c i rc le  and 
rhumb l i n e  courses a r e  exac t ly  t h e  same. However, as t h e  bearing becomes 
more e a s t e r l y  or wester ly ,  e r r o r s  between g r e a t  c i r c l e  and rhumb l i n e  paths  
s ta r t  t o  increase.  The maximum path e r r o r s  due t o  bear ing alone a r e  on a 
t r u e  east-west course. It is  important t o  note  t h a t  on an east-west course 
a t  t h e  equator t he  g rea t  c i rc le  and rhumb l i n e  f l i g h t  paths  are t h e  same 
because t h e  equator is a l s o  a g r e a t  c i rc le  -path. 
F i n a l l y ,  another  f a c t o r  t h a t  has an e f f e c t  on path discrepancy is t h e  
t r i p  l eng th ,  which is  a l s o  a func t ion  of t h e  l a t i t u d e  and longi tude  of t h e  
two poin ts .  A s  mentioned e a r l i e r ,  t he  g r e a t  c i rc le  and rhumb l i n e  courses 
s tar t  t o  diverge from the  point  of o r i g i n  of t he  f l i g h t  and reach t h e  po in t  
of maximum divergence half  way i n t o  t h e  f l i g h t .  Therefore ,  as t h e  t r i p  
l e n g t h  increases  t h e  paths simply diverge f u r t h e r  till they reach the  ha l f -  
way point .  
I V .  THE MODEL 
The mathematical model which w a s  developed using t h e  haversines  formula 
w a s  obtained from Bowditch ( 2 ) .  The equations developed f o r  t h e  model were 
designed f o r  computer appl ica t ions .  The model i s  based upon a s p h e r i c a l  
approximation of t he  e a r t h  and ad jus ted  using geodesic  e r r o r  equat ions f o r  
t h e  North American datum based on the  Clarke e l l i p s o i d .  The equat ions 
d e s c r i b i n g  t h e  model are given below. 
$1 = Lat i tude  of o r i g i n  i n  degrees 
X1 = Longitude of o r i g i n  i n  degrees 
$2 = Lat i tude  of d e s t i n a t i o n  i n  degrees 
X 2  = Longitude of d e s t i n a t i o n  i n  degrees 
- ($1 + $2)  -- mid-lat i tude po in t  
2 
Dlo = X 2  - X 1  -- di f fe rence  i n  longi tude 
Dlox = I n t e r v a l  of longi tude measured from po in t  of depar ture  i n  degrees 
A. RHUMB LINE EQUATIONS 
Rhumb l i n e  bearing ( r e l a t i v e  from o r ig in  t o  d e s t i n a t i o n )  
L a t i t u d e  of po in t s  on a rhumb l i n e  path 
= $1 + Dlox'Cos$*Tana 
+(RL) 
41 
Rhumb l i n e  dis tance 
B. GREAT CIRCLE EQUATIONS 
I n i t i a l  course of the g rea t  c i r c l e  path 
1 Sin(D1o) = Tan-1 ( Cos$l*Tan+2 - Sin$l'Cos(Dlo) 
Vertex 
The v e r t e x  i s  the  point  of h ighes t  l a t i t u d e  on a g r e a t  circle pa th  
Dif fe rence  i n  longi tude between ve r t ex  and point  of o r i g i n  
-1 Cos(C) 
Dlov = Sin ~ T G i m l  
La t i tudes  of points on g rea t  c i r c l e  path 
-1 = Tan [Tan(Lv)*Cos(Dlox - Dlov)] 
Great c i rc le  dis tance 
GCDIST = 6 0 * C 0 s - ~  [Sin($ ) -S in (+2)  + COS($~)*~OS($~) -Cos(Dlo)] 
1 
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C. GEODESIC ERROR EQUATIONS 
The geodesic  e r r o r  equat ions between t h e  model and Clarke e l l i p s o i d  
71 
ERROR(east) = [9.12951'Cos(T) - 2.92495'Cos(3T)] *(A2 - A )*- 1 180 
ERROR(north1 = 0.37414*(+2 - $l)*(&) - 8.88543*[Sin(2+2) - Sin(2$1)] 
S i m i l a r  equat ions were a l s o  used by Hogle, Markin and Toth i n  t h e i r  r e p o r t  
'Evaluat ion of Various Navigation Concepts' ( 3 ) .  
V.  RESULTS AND CONCLUSIONS 
The equat ions d e t a i l e d  i n  t h e  previous s e c t i o n  were modeled i n  FORTRAN 
on t h e  Ohio Universi ty  IBM 370 computer. Figures  1 and 2 are s imulat ions 
of great c i rc le  and rhumb l i n e  paths on a iiieicatoi prc jcc t ien .  n - e  results 
i n d i c a t e  t h a t  t h e  discrepancy between the two f l i g h t  paths  can be s i g n i f i -  
cant .  Figure 1 i s  an east-west f l i g h t  path from Baltimore-Washington In te rna-  
t i o n a l  t o  Los Angeles I n t e r n a t i o n a l  a i rpo r t .  A t  t h e  point  of maximum d e v i a t i o n  
t h e  pa th  discrepancy is 126 n a u t i c a l  miles f o r  a t r i p  l e n g t h  of 2017 mi. On 
t h e  o t h e r  hand, a north-south f l i g h t  path from St.  Paul t o  Houston t h e  d iscre-  
pancy a t  t h e  point  of maximum d e v i a t i o n  is only 5.25 nmi f o r  a t r i p  l e n g t h  of 
903 nmi. (Figure 2.) 
Since s h o r t e r  t r i p  lengths  are of major importance t o  t h e  p i l o t  an 
e f f o r t  was made t o  quant i fy  t h e  path e r ro r s  f o r  f l i g h t  paths of up t o  500 
nmi. Figures  3 through 6 i l l u s t r a t e  how the faetcrs m,er?tioned e a r l i e r  
e f f e c t  t h e  path d iscrepancies .  The f l i g h t s  were simulated at a constant  
bear ing with t h e  mid-point of t h e  f l i g h t  a t  t h e  s p e c i f i e d  l a t i t u d e .  Com- 
par i sons  of t h e  p l o t s  i n d i c a t e  t h e  compound e f f e c t  bear ing and l a t i t u d e  of 
t h e  mid-point of a f l i g h t  have on t h e  path e r r o r s .  A f l i g h t  a t  a mid-point 
l a t i t u d e  of 35 degrees nor th  and a r e l a t i v e  bear ing of 76 degrees has maxi- 
mum path e r r o r s  of 1.1 mi f o r  t r i p  lengths of up t o  300 nmi (Fig.  3).  
However, a f l i g h t  a t  t h e  same mid-point l a t i t u d e ,  but a r e l a t i v e  bear ing of 
39 degrees  has maximum path e r r o r s  of more than 6 nmi f o r  t r i p  lengths  up 
t o  450 nmi (Fig. 5 ) .  
It is of v i t a l  importance t o  note  here t h a t  t h e  e r r o r s  mentioned i n  
t h i s  r epor t  are s o l e l y  due t o  the  discrepancies  i n  t h e  g r e a t  circle and 
rhumb l i n e  paths. The e r r o r s  do not  account f o r  r e c i e v e r  computational 
e r r o r s  or  p i l o t  e r r o r s .  Another source of e r r o r  may be d iscrepancies  among 
t h e  e a r t h  models i n  var ious  RNAV recievers.  O f f s e t s  due t o  using d i f f e r e n t  
naviga t ion  systems i n  t h e  same a i r space  a l s o  may cause considerable  dis-  
p a r i t y  (4).  It is recommended t h a t  any d e c i s i o n  made regarding t h i s  s u b j e c t  
must t ake  i n t o  account a l l  discrepancies  mentioned above, and not j u s t  t h e  
goemetr ical  e r r o r s .  
43 
V I .  ACKNOWLEDGMENTS 
This work presented i n  t h i s  t e c h n i c a l  memorandum has been supported by t h e  
Nat iona l  Aeronautics and Space Administration-Langley Research Center and t h e  
Federa l  Aviation Associat ion under g r a n t  number NGR 36-009-017. 
formed a t  Ohio Univers i ty ' s  Avionics Engineering Center. 
It w a s  per- 
The author would l i k e  t o  acknowledge t h e  h e l p  of D r .  Robert L i l l e y  Asso- 
c ia t e  Director ,  and M r .  James Nickum p r o j e c t  engineer  a t  Avionics Engineering 
Center ,  whose suggest ions proved inv5luable  i n  a l l  s t a g e s  of t h i s  research.  
V I I .  REFERENCES 
[ 11 "Advisory C i r c u l a r ,  " Department of Transpor ta t ion /Federa l  
Aviation Adminstration, AC 90-45A, Feburary, 1975. 
[ 21 Bowditch, N., "American P r a c t i c a l  Navigator," Defense Mapping 
Agency Hydrographic Center,  1977. 
[3]  Hogle, L. ,  Markin, K. ,  and Toth, S., "Evaluation of Various 
Navigation System Concepts," prepared fo r  Department of 
Transportat ion,  Federal  Aviation Adminstration, Report No. 
FAA-EM-82-15. 
[ 4 ]  Ib id .  
44 
Figure 1. Great c i r c l e  versus rhumb l i n e .  Baltimore - 
Washington t o  Los AxigePes. 
Figure 2. Great c i r c l e  ve r sus  rhumb l i n e .  
St. Paul  t o  Houston. 
45 
Figure 3 .  Path e r r o r s  ve r sus  t r i p  l e n g t h  a t  average bear ing  76'. 
Figure  4 .  Path e r r o r s  ve r sus  t r i p  l e n g t h  a t  average bea r ing  52 0 . 
g - 
D 
0 -0 I.__ 
x m  
t 
e 
E?.. 
E?.. 
a* 
W 
r. #- 
0 
N 
- 
Rl.-YS.D R l  BnNC.-Y3.31 
Rl.-YP.D R1 BnWC.-Yl.(n 
Figure 5. Path e r r o r s  versus  t r i p  l eng th  a t  average bear ing  40'. 
't 
Figure 6.  Path e r r o r s  versus  t r i p  l eng th  a t  average bear ing  18'. 
47 


Path discrepancies between great circle and rhumb line

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