DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING ENGINEERING
GREAT CIRCLE DISTANCE
The Great-Circle distance or Orthodromic distance is the shortest distance between any two points on the surface of the earth measured along a path on its surface, assuming earth to be a perfect sphere. (Figure 1)
Figure 1: Great Circle Distance between two points on the earth
GREAT CIRCLE If a sphere is cut by a plane at any arbitrary distance and any arbitrary angle, the section is always a perfect circle. The diameter of this circle will be less than or equal to the diameter of the earth.
Figure 2: Section of a sphere forming a Great Circle
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The largest diameter of the section circle is achieved when the intercepting plane passes through the center of the earth, in which case the diameter of the circle is maximum and equal to the diameter of the earth (Figure 2). Such circles are called Great Circles. It is also to be noted that the Great Circles will always have their centers exactly at the center of the earth.
SHORTEST PATH AND GREAT-CIRCLE The shortest path between any two points on the earth surface is the path along the Great Circle that passes through both the points. For any two points on the earth there could be only one unique GreatCircle passing through both of them. Note: The exception to this are when both the points are exactly opposite sides of the sphere and when both the points are at exact same location. Both these cases have infinite number of Great-Circles passing through them.
CALCULATION OF THE GREAT CIRCLE DISTANCE Let us considering the earth at the center of the co-ordinate system with the North-Pole on the positive z-axis and 0˚ longitude (Greenwich Meridian) in the direction of x-axis x-axis. (Figure 3) Now consider two points A and B on the earth surface. Let the co-latitude co-latitude and longitude of point point A be δ A and λ A respectively. And the co-latitude and longitude of point B be δ B and λ B respectively. Note: The co-latitude is the angle measured from the z-axis (North Pole). It is related to the actual latitude σ by
--- (1)
Let us now draw a line from the origin to point A. This line forms the position vector r A for the point A in the co-ordinate system. Similarly construct the position vector r B for the point B. Since we know that the points A and B are located on the surface of the earth, the magnitudes of the vectors r A and r B
will be equal to the radius of the earth.
| r A | = | r B | = R
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--- (2)
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DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING ENGINEERING
Where R ≈ 6371 km ≈ 3959 miles (the Mean Radius of Earth)
Figure 3: Spherical co-ordinate system for the Earth
For mathematical convenience let us now resolve the position vectors r A and r B into rectangular co-ordinate system (Cartesian coordinate system). To do this we need to know the x, y and z component of the points A and B. ’
If A If A the projection of the point A on the x-y plane, plane, then from Figure 4; OA’ = R sin ( δ A )
--- (3)
z A = R cos ( δ A )
--- (4)
x A = OA’ cos ( λ A ) = R sin ( δ A ) cos ( λ A )
--- (5)
y A = OA’ sin ( λ A ) = R sin ( δ A ) sin ( λ A )
--- (6)
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DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING ENGINEERING
Figure 4: Resolving Point A into Rectangular Coordinates
Hence; r A
= R sin ( δ A ) cos ( λ A ) x + R sin ( δ A ) sin ( λ A ) y + R cos ( δ A ) z
--- (7)
Similarly for the point B; r B
= R sin ( δ B ) cos ( λ B ) x + R sin ( δ B ) sin ( λ B ) y + R cos ( δ B ) z
--- (8)
Since the shortest path from A to B on the surface of the earth is along a Great Circle, the radius of this arc (path) will be R, the radius of the earth. Hence the length of this arc could be found by knowing the angle α subtended by this arc (Figure 5).
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Figure 5: The Great-Circle through the points A and B
From equation (7) and (8), the dot product of the vectors r A and r B is
r A
2 2 . r B = R sin ( δ A ) cos ( λ A ) sin ( δ B ) cos ( λ B ) + R sin ( δ A ) sin ( λ A ) sin ( δ B ) sin ( λ B )
+ R
r A
2
cos ( δ A ) cos ( δ B )
2 . r B = R sin ( δ A ) sin ( δ B ) { cos ( λ A ) cos ( λ B ) + sin ( λ A ) sin ( λ B ) }
+ R 2 cos ( δ A ) cos ( δ B )
r A
--- (9)
2 2 . r B = R sin ( δ A ) sin ( δ B ) cos ( λ A - λ B ) + R cos ( δ A ) cos ( δ B )
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--- (9a)
--- (9b)
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DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING ENGINEERING
r A
2 . r B = R { sin ( δ A ) sin ( δ B ) cos ( λ A - λ B ) + cos ( δ A ) cos ( δ B ) }
--- (10)
However from the principle of vector dot product; r A
. r B = | r A | | r B | cos ( α )
--- (11)
Using equation (2) α
2 = cos-1 ( ( r A . r B ) / R R )
--- (12)
Substituting equation (12) in equation (11) α
= cos-1 { sin ( δ A ) sin ( δ B ) cos ( λ A - λ B ) + cos ( δ A ) cos ( δ B ) }
If
α
--- (13)
is in radians then the length of the arc from A to B is given by
D = R α
--- (14) Where
R ≈ 6371 km ≈ 3959 miles (the Mean Radius of Earth)
Therefore the Great-Circle Distance between two points on the earth surface is given by; -1 D = R cos { sin ( δ A ) sin ( δ B ) cos ( λ A - λ B ) + cos ( δ A ) cos ( δ B ) }
--- (15)
USING ACTUAL LATITUDE INSTEAD OF CO-LATITUDE
It is to be noted that the above equations uses the co-latitudes ( δ A and δ B ) rather than the actual latitudes of the locations. Equation (15) is valid for all possible locations on the earth. However, if we replace the co-latitude with actual latitude values σA and σ B of the points A and B respectively, the following cases are need to be considered separately.
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Both the location A and B are on the northern hemisphere (i.e. above Equator). Hence Case (i): from the equation (1) δ = 90˚- σ Hence; -1 D = R cos { sin (90˚- σ A ) sin (90˚- σ B ) cos ( λ A - λ B ) + cos (90˚- σ A ) cos (90˚- σ B ) } -1 D = R cos { cos ( σ A ) cos ( σ B ) cos ( λ A - λ B ) + sin ( σ A ) sin ( σ B ) }
Case (ii): equation (1)
--- (16)
Both A and B are on the southern hemisphere (i.e. below Equator). Hence from the δ = 90˚+ σ
Hence; D = R
-1
{ sin (90˚+ σ A ) sin (90˚+ σ B ) cos ( λ A - λ B ) + cos (90˚+ σ A ) cos (90˚+ σ B ) }
D = R
-1
{ cos ( σ A ) cos ( σ B ) cos ( λ A - λ B ) + (- sin ( σ A ) ) (- sin ( σ B )) }
D = R cos
-1
Case (iii):
{ cos ( σ A ) cos ( σ B ) cos ( λ A - λ B ) + sin ( σ A ) sin ( σ B ) }
--- (17)
When A is above the Equator and B is below the Equator
Using equation (1); -1 D = R cos { sin (90˚- σ A ) sin (90˚+ σ B ) cos ( λ A - λ B ) + cos (90˚- σ A ) cos (90˚+ σ B ) }
D = R cos
-1
Case (iv):
{ cos ( σ A ) cos ( σ B ) cos ( λ A - λ B ) - sin ( σ A ) sin ( σ B ) }
--- (18)
When A is below the Equator and B is above the Equator
Using equation (1); D = R cos -1 { sin (90˚+ σ A ) sin (90˚- σ B ) cos ( λ A - λ B ) + cos (90˚+ σ A ) cos (90˚- σ B ) } D = R cos
-1
{ cos ( σ A ) cos ( σ B ) cos ( λ A - λ B ) - sin ( σ A ) sin ( σ B ) }
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--- (19)
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DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING ENGINEERING
Considering all the above cases the e quations (16), (17), (18) and (19) can be generalized as follows;
-1 D = R cos { cos ( σ A ) cos ( σ B ) cos ( λ A - λ B ) ± sin ( σ A ) sin ( σ B ) }
---(20)
[ Use + when both A and B are on the same hemisphere and use – when A and B are on different hemispheres. ]
Note: If either A or B is on the Equator then it doesn’t matter whether you use equation (16), (17), ( 18) or (19) since the term sin ( σ A ) sin ( σ B ) will become zero.
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