How Many Great Circles Could You Draw That Are Parallel To The Great Circle In The Figure

Intersection of the sphere and a aeroplane which passes through the heart point of the sphere

"Swell Circumvolve" redirects here. For the novel by Maggie Shipstead, see Great Circumvolve (novel).

A corking circle divides the sphere in two equal hemispheres

A peachy circumvolve, as well known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on whatsoever given sphere. Whatever diameter of any smashing circle coincides with a diameter of the sphere, and therefore all great circles have the aforementioned heart and circumference as each other. This special instance of a circle of a sphere is in opposition to a minor circle, that is, the intersection of the sphere and a airplane that does not laissez passer through the center. Every circumvolve in Euclidean 3-space is a great circle of exactly ane sphere.

For near pairs of distinct points on the surface of a sphere, there is a unique corking circle through the two points. The exception is a pair of antipodal points, for which in that location are infinitely many bully circles. The minor arc of a great circle betwixt ii points is the shortest surface-path between them. In this sense, the minor arc is analogous to "straight lines" in Euclidean geometry. The length of the minor arc of a smashing circumvolve is taken as the distance between two points on a surface of a sphere in Riemannian geometry where such slap-up circles are called Riemannian circles. These corking circles are the geodesics of the sphere.

The disk bounded past a great circumvolve is called a great disk: information technology is the intersection of a 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 ^{northward + i}.

Derivation of shortest paths [edit]

To evidence that the small arc of a peachy circle is the shortest path connecting two points on the surface of a sphere, 1 tin 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 non intersect either pole, except possibly at the endpoints, tin can be parametrized past

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

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

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

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

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

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

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

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

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

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

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

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

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

which is a aeroplane through the origin, i.eastward., the centre of the sphere.

Applications [edit]

Some examples of great circles on the angelic sphere include the celestial horizon, the angelic equator, and the ecliptic. Great circles are also used as rather authentic approximations of geodesics on the Earth's surface for air or sea navigation (although it is non a perfect sphere), as well as on spheroidal angelic bodies.

The equator of the idealized earth is a bully circle and whatever tiptop and its opposite peak class a slap-up circle. Another great circle is the 1 that divides the land and water hemispheres. A great circle divides the earth into ii hemispheres and if a smashing circle passes through a signal it must pass through its antipodal point.

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

Run into likewise [edit]

• Great-circle altitude
• Swell-circle navigation
• Great ellipse
• Rhumb line

External links [edit]

• Great Circle – from MathWorld Bully Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999
• Great Circles on Mercator's Chart past John Snyder with additional contributions by Jeff Bryant, Pratik Desai, and Carl Woll, Wolfram Demonstrations Project.
• Navigational Algorithms Paper: The Sailings.
• Nautical chart Work - Navigational Algorithms Nautical chart Piece of work complimentary software: Rhumb line, Great Circle, Composite sailing, Meridional parts. Lines of position Piloting - currents and coastal fix.

Source: https://en.wikipedia.org/wiki/Great_circle

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