Great Circle


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 delta and longitude lambda of (delta_1,lambda_1) and (delta_2,lambda_2) on a sphere of radius a, convert spherical coordinates to Cartesian coordinates using

 r_i=a[coslambda_icosdelta_i; sinlambda_icosdelta_i; sindelta_i].

(Note that the latitude delta is related to the colatitude phi of spherical coordinates by delta=90 degrees-phi, so the conversion to Cartesian coordinates replaces sinphi and cosphi by cosdelta and sindelta, respectively.) Now find the angle alpha between r_1 and r_2 using the dot product,


The great circle distance is then


For the Earth, the equatorial radius is a approx 6378 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


Then the combinations of the partial derivatives P, Q, and R are given by


The geodesic differential equation then becomes


However, because this is a special case of Q=0 with P and R explicit functions of v only, the geodesic solution takes on the special form


(Gradshteyn and Ryzhik 2000, p. 174, eqn. 2.599.6), where c_1 and c_2 are constants of integration. Now rewrite in the simpler form


rearrange, and take the sine of both sides,


Next, expand the left-hand side using a trigonometric addition formula and write cotv=cosv/sinv to obtain


Now multiply through by asinv and rearrange to obtain


This is the equation of the geodesic.

Identifying the first part of each term as the Cartesian coordinates x, y, z, respectively, (19) can immediately be recast as


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.

See also

Geodesic, Great Sphere, Loxodrome, Mikusiński's Problem, Oblate Spheroid Geodesic, Point-Point Distance--3-Dimensional, Pseudocircle, Small Circle, Sphere, Spheric Section

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Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, 1948.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 183 and 217, 1999.Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, pp. 24-25, 1965.Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, pp. 26-28 and 62-63, 1974.

Referenced on Wolfram|Alpha

Great Circle

Cite this as:

Weisstein, Eric W. "Great Circle." From MathWorld--A Wolfram Web Resource.

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