Spherical triangle

Two corners for a globe by Martin Waldseemüller (around 1470–1522)

A lune , also spherical Zweieck , Kugelzweiseit , gusset or only Zweieck ( Digon called), is in the spherical geometry (spherical geometry) a set of points on a sphere that half of two great circles is limited.

Any two great circles on a sphere always intersect at exactly two opposite points. They divide each other into two semicircles and together the spherical surface into four spherical tangs. In the case of globes , spherical models of the globe , for example, the area enclosed by two meridians forms a spherical two- cornered shape, with the north and south poles of the globe being the corners.

The two corners of any spherical triangle are exactly opposite on the spherical surface. The side lengths are each half the circumference of a great circle. The two interior angles are the same size. The volume enclosed by a spherical triangle is a spherical wedge . A third great circle, which does not go through the corners, divides the spherical triangle into two spherical triangles .

formula

The following applies to the area of the spherical triangle ( is the surface of the entire sphere): ${\ displaystyle 4 \, \ pi \, r ^ {2}}$

{\ displaystyle A = {\ frac {\ gamma} {360 ^ {\ circ}}} \ cdot 4 \, \ pi \, r ^ {2} = {\ frac {\ gamma} {90 ^ {\ circ} }} \ cdot \ pi \, r ^ {2}}

Stand here

${\ displaystyle \ gamma}$ for the size of an interior angle (in degrees )
r for the radius of the sphere.

If given in radians , the formula can also be written as: ${\ displaystyle \, \ gamma}$

{\ displaystyle A = {\ frac {\ gamma} {2 \, \ pi}} \ cdot 4 \, \ pi \, r ^ {2} = \ gamma \ cdot 2 \, r ^ {2}}

Example: On the idealized globe, a spherical triangle which is bounded by two neighboring meridians (i.e. = 1 °) has the area ${\ displaystyle \, \ gamma}$

{\ displaystyle A = {\ frac {1 ^ {\ circ}} {90 ^ {\ circ}}} \ cdot \ pi \, (6370 \ \ mathrm {km}) ^ {2} = 1 {,} 4 \ \ mathrm {million \ km} ^ {2}}

literature

Bronstein-Semendjajew: Paperback of mathematics . Verlag Harri Deutsch, 2000, p. 167, ISBN 3-8171-2005-2

Web links

Commons : Kugelzweieck - Collection of images, videos and audio files

Individual evidence

↑ See definition of the spherical two-point in Guido Walz (Ed.): Lexikon der Mathematik . tape 4 . Springer-Verlag GmbH Germany, 2017, ISBN 978-3-662-53499-1 .

[1] See definition of the spherical two-point in Guido Walz (Ed.): Lexikon der Mathematik . tape 4 . Springer-Verlag GmbH Germany, 2017, ISBN 978-3-662-53499-1 .