Great circle

In

Any

formed by the two points and the center of the sphere.

A great circle is the largest circle that can be drawn on any given sphere. Any

small circle

, and is the intersection of the sphere with a plane not passing through its center. Small circles are the spherical-geometry analog of circles in Euclidean space.

Every circle in Euclidean 3-space is a great circle of exactly one sphere.

The

ball

and a plane passing through its center. In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space R^{n + 1}.

Derivation of shortest paths

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

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 ${\displaystyle \phi }$ is allowed 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.

Applications

Some examples of great circles on the

.

The

.

The Funk transform integrates a function along all great circles of the sphere.

See also

• 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.