Great circle

To prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it.

Consider the class of all regular paths from a point ${\displaystyle p}$ to another point ${\displaystyle q}$. Introduce spherical coordinates so that ${\displaystyle p}$ coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by

${\displaystyle \theta =\theta (t),\quad \phi =\phi (t),\quad a\leq t\leq b}$

provided we allow ${\displaystyle \phi }$ to take on arbitrary real values. The infinitesimal arc length in these coordinates is

${\displaystyle ds=r{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}\,dt.}$

So the length of a curve ${\displaystyle \gamma }$ from ${\displaystyle p}$ to ${\displaystyle q}$ is a functional of the curve given by

${\displaystyle S[\gamma ]=r\int _{a}^{b}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}\,dt.}$

According to the Euler–Lagrange equation, ${\displaystyle S[\gamma ]}$ is minimized if and only if

${\displaystyle {\frac {\sin ^{2}\theta \phi '}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}}=C}$,

where ${\displaystyle C}$ is a ${\displaystyle t}$-independent constant, and

${\displaystyle {\frac {\sin \theta \cos \theta \phi '^{2}}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}}={\frac {d}{dt}}{\frac {\theta '}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}}.}$

From the first equation of these two, it can be obtained that

${\displaystyle \phi '={\frac {C\theta '}{\sin \theta {\sqrt {\sin ^{2}\theta -C^{2}}}}}}$.

Integrating both sides and considering the boundary condition, the real solution of ${\displaystyle C}$ is zero. Thus, ${\displaystyle \phi '=0}$ and ${\displaystyle \theta }$ can be any value between 0 and ${\displaystyle \theta _{0}}$, indicating that the curve must lie on a meridian of the sphere. In a Cartesian coordinate system, this is

${\displaystyle x\sin \phi _{0}-y\cos \phi _{0}=0}$

which is a plane through the origin, i.e., the center of the sphere.

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.

• Great ellipse
• Rhumb line

• 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 Archived 2018-10-16 at the Wayback Machine 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.