A mathematical model for a comparative analysis of great circle vs. rhumb line navigation in the continental United States has been developed at the Avionics Engineering Center, Ohio University. A FORTRAN simulation of the model has been implemented on the IBM 370 computer. The simulation predicts pertinent navigation information for the two flight paths. The basis for the project, which is a part of an M.S. thesis, is to provide a data base for computing discrepancies between the two flight paths. This document briefly describes the model and discusses the implications of the results obtained

Kaul, R.

English

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TECHNICAL MEMORANDUM (NASA) 89
PATH DISCREPANCIES BETWEEN GREAT
CIRCLE AND RHUMB LINE
A simulation of a mathematical model
to compute path discrepancies between great
circle. and rhumb line flight paths is pre-
sented. The model illustrates that the path
errors depend on the latitude, the bearing
and the trip length of the flight.
by
Rajan Kaul
Avionics Engineering Center
Department of Electrical and Computer Engineering
Ohio University
Athens, Ohio 45701
December 1983
Prepared for
NASA LANGLEY Research Center
Hampton, Va 23665
(Contract NGR 36-009-017)
i
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(NASA-TM-85522) PATH DISCREPANCIES BETWEEN
	 N84--17168
GREAT CIRCLE A&D .RHUMB LINE (Chio Univ. )
38 p :HC AU2/MF A01	 CSCL 17G
Unclas
G3/04 18366
ti
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Table of Contents
A
Page
I. INTRODUCTION
	 1
II. BACKGROUND INFORMATION
	
2
III. PATH DISCREPANCIES
	
3
a
IV. THE MODEL
	
4
V. RESULTS AND CONCLUSIONS
	
5
i
VI. ACKNOWLEDGEMENTS
	
14	
4
VII. REFERENCES
	
15
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List of Figures
Figure Page	 y
1. Great Circle vs. Rhumb Line 8
2. Great Circle vs. Rhumb Line 9
3. Path Error vs. Trip Length of Average Bearing 76 0 . 10
4. Path Error vs. Trip Length of Average Bearing 52% 11
5. Path Error vs. Trip Length of Average Bearing 40°. 12
6. Path Error vs. Trip Length of Average Bearing 18 0 . 13
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I. INTRODUCTION
A mathematical model for a comparative analysis of great circle vs.
rhumb line navigation in the continental United States has been developed
at the Avionics Engineering Center, Ohio University. A FORMAN simulation
of the model has been implemented on the IBM 370 computer. The simulation
predicts pertinent navigation information for the two flight paths. The
basis for the projecry which is a part of an M.S. thesis, is to provide a
data base for computing discrepancies between the two flight paths. This
document briefly describes the model and discusses the implications of the
results obtained.
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II. BACKGROUND INFORMATION
The standard en-route navigation system used in the United States is
the VOR/DME. VHF Omnidirectional Range (VOR) is the basks for defining the
airways and is therefore an integral part of air traffic control proce-
dures. Two VOR stations are used to define a radial intersection, which is
the basis of the established Victor airways. These Victor airways follow a
flight path which maintains a path of constant heading (i.e. the course crosses
the meridians at the same angle). Thus, a rhumb line course appears as a
straight line on a Mercator projection.
Area/Random Navigation (RNAV) as defined in FAA Advisory Circular 90-
45A (1) is
...a method of navigation that permits aircraft operations on
any desired course within the coverage of station referenced
navigation signals or within the limits of self-contained
system capabilities.'
The principle advantage of RNAV is that it allows the navigator to fly
a great circle course between two points. The great circle course is the
shortest path between two points on the earth, and is formed by the inter-
section of the plane defined by the center of the earth and the points of
origin and destination, projected on the surface of the earth. The flight
path is projected as a curved line on a Mercator projection and hence
changes heading constantly.
Loran-C, OMEGA, GPS, Doppler, and INS are considered RNAV systems.
However, VOR in its basic form is not an RNAV system. A major concern is
that the discrepancies between the two flight paths may cause navigation
conflicts. This report attempts to quantize the discrepancies which depend
on various factors.
-2-
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III. PATH DISCREPANCIES
The magnitude of path discrepancies between the great circle and rhumb
line courses be,.ween two points increases as
1) the latitude increases,
2) the difference in their latitudes decreases and
3) the difference in their longitudes increases.
When plotted on a Mercator projection map, the two flight paths start
to diverge from the origin of the flight. The discrepancy is a maximum at
the mid-point of the flight, and then the flight paths start to converge
and meet at the destination point. The great circle path is always curved
away from the equator and therefore intersects the meridians at higher
latitudes than the rhumb line path. Also, the meridians converge as they
depart from the equator and are closer together at higher latitudes. Thus,
the deviation between the two paths increase . at higher latitudes because
the rhumb line path must curve rapidly to be able to fly a course of con-
stant meridian crossing angle.
The bearing of the flight path, which is a simple function of the lati-
tude and longitude, also has a considerable effect on the discrepancies
between the two flight paths. By definition, all lines of longitude are
also great circles. On a true north-south flight path the great circle and
rhumb line courses are exactly the same, However, as the bearing becomes
more easterly or westerly, errors between great circle and rhumb line paths
start to increase. The maximum path errors due to bearing alone are on a
true east-west course. It is important to note that on an east-west course
at the equator the great circle and rhumb line flight paths are the same
because the equator is also a great circle—path.
Finally, another factor that has an effect on path discrepancy is the
trip length, which is also a function of the latitude and longitude of the
two points. As mentioned earlier, the great circle and rhumb line courses
start to diverge from the point of origin of the flight and reach the point
of maximum divergence half way into the flight. Therefore, as the trip
length increases the paths simply diverge further till they reach the half-
way point.
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IV. THE MODEL
The mathematical model which was developed using the haversines formula
was obtained from Bowditch (2). The equations developed for tho model were
designed for computer applications. The model is based upon 4 spherical
approximation of the earth and adjusted using geodesic error equations for
the North American datum based on the Clarke ellipsiod. The equations
describing the model are given below.
^1 = Latitude of origin in degrees
X 1 = Longitude of origin in degrees
^2 = Latitude of destination in degrees
X 2 = Longitude of destination in degrees
_ ( ^1 + ^2 ) --- mid-latitude point
2
Dlo = a 2 - X, --- difference in longitude
Dlox = Interval of longitude measured from point of departure in degrees
A. RHUMB LINE rJUATIONS
1
r^rk
	 1FMK
Rhumb line bearing (relative from origin to destination)
-1	 ^2	 h1
ct = Tan	
^ Dlo'Cos^
Latitude of points on a rhumb line path
^(RL)	 1 + Dlox•CosVTana
-4-
tRhumb line distance
RLDIST - 60 1142 - 01)2 + (Dlo•CosT)2]
B. GREAT CIRCLE EQUATIONS	 '±
Initial course of the great circle path
	 !
C = Tan-1	 Sin(Dlo)	 1
os¢ 1 •Tan02 - Sinw Cos Dlo J
Vertex	 r
The vertex is the point of highest latitude on a great circle path
a
Lv = Cos -' [Cos(^1)•Sin(C)j
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Difference in longitude between vertex and point of origin
1
D1ov
	
Sin 1 
C 
Cos(C)^
Sin Lv t
Latitudes of points on great circle path
^(GC) - Tan-1 [Tan(Lv) • Cos(Dlox - Dlov)) i
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l
Great circle distance
GCDIST = 60 • Cos-1
 [Sin(^1)•Sin(^2) + Cos(01)•Co8(02)•Cos(Dlo)]
1
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C. GEODESIC ERROR EQUATIONS
The geodesic error equations between the model. and Clarke ellipsoid
ERROR(east) = [9.12'951 • Cos(T) - 2.92495 • Cos(3T)j • (a 2
 - 711)`180
ERROR(north) ® 0.37414 . 02 - W • (	 ) - 8.88543 • [Sin(2^ 2 ) - Sin(2^1)1
i
Similar equations were also used by Hogle, Markin and Toth in their report
'Evaluation of Various Navigation Concepts' (3).
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V. RESULTS AND CONCLJSIONS
The equations detailed in the previous section were modeled in FORTRAN
on the Ohio University IBM 370 computer. Figures 1 and 2 are simulations
of great circle and rhumb line paths on a mercator projection. The results
indicate that the discrepancy between the two .flight paths can be signifi-
cant. Figure 1 is an east-west flight path from Baltimore-Washington Interna-
tional to Los Angeles International airport. At the point of maximum deviation
the path discrepancy it. 126 nautical miles kor a trip length of 2017 nmi. On
the other hand, a nortl^-south flight path from St. Paul to Houston the discre-
pancy at the point of iiaximum deviation is only 5.25 nmi for a trip length of
903 nmi. (Figure 2.)
Since shorter trip lengths are of major importance to the pilot an
effort was made to quantify the path errors for flight paths of up to 500
nmi. Figures 3 through 6 illustrate how the factors mentioned earlier
effect the path discrepancies. The flights were simulated at a constant
bearing with the mid-point of the flight at the specified latitude. Com-
parisions of the plots indicate the compound effect bearing and latitude of
the mid-point of a flight have on the path errors. A flight at a mid-point
latitude of 35 degrees north and a relative bearing of 76 degrees has maxi-
mum path errors of 1.1 nmi for trip lengths of up to 300 %mi (Fig. 3).
However, a flight at the same mid-point latitude, but a relative bearing of
39 degrees has maximum path errors of more than 6 nmi for trip lengths up
to 450 nmi (Fig. 5).
It is of vital importance to note here that the errors mentioned in
this report are solely due to the discrepancies in the 3reat circle and
rhumb line paths. The errors do not account for reciever computational
errors or pilot errors. Another source of error may be discrepancies among
the earth models in various RNAV recievers. Offsets due to using different
navigation systems in the same airspace also may cause considerable dis-
parity (4). It is recommended that any decision made regarding this subject
must take into account all discrepancies mentioned above, and not just the
goemetrical errors.
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VI. ACKNOWLEDGEMENTS
This work presented in this technical memorandum has been supported by the
National Aeronautics and Space Administration at Langley Research Center under
grant number NGR 36-009-017. It was performed at Ohio University's Avionics
Engineering Center.
The author would like to acknowledge the help of Dr. Robert Lilley Asso-
ciate Director, and Mr. James Nickum project engineer at Avionics Engineering
Center, whose suggestions prowAd invaluable in all stages of this research.
J=
- 14-
VII. REVERENCES
[1]	 "Advisory Circular," Department of Transportation/Federal.
Aviation Adninstration, AC 90-45A, Feburary, 1975.
[2)	 Bowditch, N., "American Practical. Navigator," Defense Mapping
Agency Hydrographic Center, 1977.
(3)	 Hogle, L., Markin, K., and Toth, S., "Evaluation of Various
Navigation System Concepts," prepared for Department of
Transportation, Federal Aviation Adminstration, Report No.
FAA-EM-82-15.
[4]	 Ibid.	 +
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Path discrepancies between great circle and rhumb line

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