A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called small circles. A great circle becomes a straight line in a gnomonic projection (Steinhaus 1999, pp. 220-221).
The shortest path between two points on a sphere, also known as an orthodrome, is a segment of a great circle. To find the great circle
(geodesic) distance between two points located at latitude 
and longitude 
of 
and 
on a sphere of
radius 
, convert spherical coordinates
to Cartesian coordinates using
![r_i=a[coslambda_icosdelta_i; sinlambda_icosdelta_i; sindelta_i].](/images/equations/GreatCircle/NumberedEquation1.svg) |
(1)
|
(Note that the latitude 
is related to the colatitude 
of spherical
coordinates by 
,
so the conversion to Cartesian coordinates
replaces 
and 
by 
and 
, respectively.) Now find the angle 
between 
and 
using the dot product,
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(2)
|
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(3)
|
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(4)
|
The great circle distance is then
![d=acos^(-1)[cosdelta_1cosdelta_2cos(lambda_1-lambda_2)+sindelta_1sindelta_2].](/images/equations/GreatCircle/NumberedEquation2.svg) |
(5)
|
For the Earth, the equatorial radius is 
km, or 3963 (statute) miles. Unfortunately, the
flattening of the Earth cannot be taken into account
in this simple derivation, since the problem is considerably more complicated for
a spheroid or ellipsoid
(each of which has a radius which is a function of latitude). This leads to extremely complicated expressions
for oblate spheroid geodesics and geodesics
on other ellipsoids.
The equation of the great circle can be explicitly computed using the geodesic formalism. Convert to spherical coordinates by writing
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(6)
|
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(7)
|
Then the combinations of the partial derivatives 
, 
, and 
are given by
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(8)
|
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(9)
|
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(10)
|
The geodesic differential equation then becomes
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(11)
|
However, because this is a special case of 
with 
and 
explicit functions of 
only, the geodesic solution takes
on the special form
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(12)
|
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(13)
|
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(14)
|
 |  | ![-tan^(-1)[(cosv)/(sqrt((a/(c_1))^2sin^2v-1))]+c_2](/images/equations/GreatCircle/Inline59.svg) |
(15)
|
(Gradshteyn and Ryzhik 2000, p. 174, eqn. 2.599.6), where 
and 
are constants of integration.
Now rewrite in the simpler form
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(16)
|
rearrange, and take the sine of both sides,
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(17)
|
Next, expand the left-hand side using a trigonometric addition formula and write 
to obtain
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(18)
|
Now multiply through by 
and rearrange to obtain
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(19)
|
This is the equation of the geodesic.
Identifying the first part of each term as the Cartesian coordinates 
,
, 
, respectively, (19) can immediately be
recast as
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(20)
|
which shows that the geodesic giving the shortest path between two points on the surface of the equation lies on a plane that passes through the two points in question and also through center of the sphere.
