the length of a curve from to is a functional of the curve given by According to the Euler–Lagrange equation, is minimized if and only if , where is a -independent constant, and From the first equation of these two, it can be obtained that . Integrating both sides and considering the boundary condition, the real solution of is zero. Thus, and can be any value between 0 and , indicating that the curve must lie on a meridian of the sphere. In Cartesian coordinates, this is which is a plane through the origin, i.e., the center of the sphere. Applications Some examples of great circles on the celestial sphere include the celestial horizon, the celestial equator, and the ecliptic. Great circles are also used as rather accurate approximations of geodesics on the Earth's surface for air or sea navigation (although it is not a perfect sphere), as well as on spheroidal celestial bodies. The equator of the idealized earth is a great circle and any meridian and its opposite meridian form a great circle. Another great circle is the one that divides the land and water hemispheres. A great circle divides the earth into two hemispheres and if a great circle passes through a point it must pass through its antipodal point. The Funk transform integrates a function along all great circles of the sphere. See also Small circle Circle of a sphere Great-circle distance Great-circle navigation Great ellipse Rhumb line References External links Great Circle – from MathWorld Great Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999 Great Circles on Mercator's Chart by John Snyder with additional contributions by Jeff Bryant, Pratik Desai, and Carl Woll, Wolfram Demonstrations Project. Navigational Algorithms Paper: The Sailings. Chart Work - Navigational Algorithms Chart Work free software: Rhumb line, Great Circle, Composite sailing, Meridional parts. Lines of position Piloting - currents and coastal fix. Elementary geometry Spherical trigonometry Riemannian geometry Circles Spherical curves