“spherical triangle”的概念、定义、翻译、参考文献-科学参考

spherical triangle

Mathematics

A triangle on a sphere, with three vertices and three sides that are arcs of great circles. The angles of a spherical triangle do not add up to 180°. In fact, the sum of the angles can be anything between 180° and 540°. Consider, for example, a spherical triangle with one vertex at the North Pole and the other two vertices on the equator of the Earth. The area of a spherical triangle on a sphere of radius r equals r²(α‎+β‎+γ‎–π‎), where the triangle’s angles are α‎,β‎,γ‎, measured in radians; this is Girard’s Theorem.

Given a spherical triangle on a sphere of unit radius, with angles α‎,β‎,γ‎ and sides of length a,b,c, the (spherical) sine rule states that

$\frac{sin α}{sin a} = \frac{sin β}{sin b} = \frac{sin γ}{sin c},$

the cosine rule states that

$cos a = cos b cos c + sin b sin c cos α,$

with similar versions for β‎ and γ‎, and the dual cosine rule states that

$cos α = - cos β cos γ + sin β sin γ cos a .$

Note that when a,b,c are small enough that the approximations sina ≈ a and cosa ≈ 1 − a²/2 apply, the sine rule and cosine rule approximate to the Euclidean versions.

单词	spherical triangle
释义	spherical triangle Mathematics A triangle on a sphere, with three vertices and three sides that are arcs of great circles. The angles of a spherical triangle do not add up to 180°. In fact, the sum of the angles can be anything between 180° and 540°. Consider, for example, a spherical triangle with one vertex at the North Pole and the other two vertices on the equator of the Earth. The area of a spherical triangle on a sphere of radius r equals r²(α‎+β‎+γ‎–π‎), where the triangle’s angles are α‎,β‎,γ‎, measured in radians; this is Girard’s Theorem. Given a spherical triangle on a sphere of unit radius, with angles α‎,β‎,γ‎ and sides of length a,b,c, the (spherical) sine rule states that $\frac{sin α}{sin a} = \frac{sin β}{sin b} = \frac{sin γ}{sin c},$ the cosine rule states that $cos a = cos b cos c + sin b sin c cos α,$ with similar versions for β‎ and γ‎, and the dual cosine rule states that $cos α = - cos β cos γ + sin β sin γ cos a .$ Note that when a,b,c are small enough that the approximations sina ≈ a and cosa ≈ 1 − a²/2 apply, the sine rule and cosine rule approximate to the Euclidean versions.
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