“spherical triangle”的概念、定义、习题解题方法及翻译-数学辞典

单词

spherical triangle

释义

spherical triangle

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

the cosine rule states that

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

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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