Spherical Excess

spherical excess

[′sfir·ə·kəl ek′ses]

(mathematics)

The sum of the angles of a spherical triangle, minus 180°.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Spherical Excess

 

the difference between the sum of the angles of a spherical triangle and 180°—that is, the amount by which the sum is greater than the sum of the angles of a plane triangle.

The sum of the angles of a triangle formed by three geodesies on a surface of positive curvature—that is, on a convex surface—is always greater than two right angles and is equal to

π + ʃʃ K dS

where K is the total curvature of the surface and dS is the element of area of the surface. The spherical excess of a spherical triangle formed by great circles on a sphere of radius R is

where S is the area of the triangle. In the case of small spherical triangles on the surface of the earth, the quantity ∊, in seconds of arc, is given by the formula

Here, a and b are the lengths of two sides of the triangle, and C is the angle between the two sides.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.