The solid angle subtended by a surface is defined as the surface area of a unit sphere covered by the surface's projection onto the sphere. This can be written as(1)where is a unit vector from the origin, is the differential area of a surface patch, and is the distance from the origin to the patch. Written in spherical coordinates with the colatitude (polar angle) and for the longitude (azimuth), this becomes(2)Solid angle is measured in steradians, and the solid angle corresponding to all of space being subtended is steradians.To see how the solid angle of simple geometric shapes can be computed explicitly, consider the solid angle subtended by one face of a cube of side length centered at the origin. Since the cube is symmetrical and has six sides, one side obviously subtends steradians. To compute this explicitly, rewrite (1) in Cartesian coordinates using(3)(4)and(5)(6)Considering the top face of the cube, which is located at and has sides..
Let , , and be the sides of a spherical triangle, then the spherical defect is defined as
Let a spherical triangle have sides of length , , and , and semiperimeter . Then the spherical excess is given by
Let a spherical triangle be drawn on the surface of a sphere of radius , centered at a point , with vertices , , and . The vectors from the center of the sphere to the vertices are therefore given by , , and . Now, the angular lengths of the sides of the triangle (in radians) are then , , and , and the actual arc lengths of the side are , , and . Explicitly,(1)(2)(3)Now make use of , , and to denote both the vertices themselves and the angles of the spherical triangle at these vertices, so that the dihedral angle between planes and is written , the dihedral angle between planes and is written , and the dihedral angle between planes and is written . (These angles are sometimes instead denoted , , ; e.g., Gellert et al. 1989)Consider the dihedral angle between planes and , which can be calculated using the dot product of the normals to the planes. Assuming , the normals are given by cross products of the vectors to the vertices, so(4)(5)However, using a well-known vector identity..
The Schwarz triangles are spherical triangles which, by repeated reflection in their indices, lead to a set of congruent spherical triangles covering the sphere a finite number of times.Schwarz triangles are specified by triples of numbers . There are four "families" of Schwarz triangles, and the largest triangles from each of these families are(1)The others can be derived from(2)where(3)and(4)(5)
Let a spherical triangle have angles , , and . Then the spherical excess is given by
A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. The spherical triangle is the spherical analog of the planar triangle, and is sometimes called an Euler triangle (Harris and Stocker 1998). Let a spherical triangle have angles , , and (measured in radians at the vertices along the surface of the sphere) and let the sphere on which the spherical triangle sits have radius . Then the surface area of the spherical triangle iswhere is called the spherical excess, with in the degenerate case of a planar triangle.The sum of the angles of a spherical triangle is between and radians ( and ; Zwillinger 1995, p. 469). The amount by which it exceeds is called the spherical excess and is denoted or , the latter of which can cause confusion since it also can refer to the surface area of a spherical triangle. The difference between radians () and the sum of the side arc lengths , ,..
The function giving the volume of the spherical quadrectangulartetrahedron:(1)where(2)and(3)
A closed geometric figure on the surface of a sphere which is formed by the arcs of great circles. The spherical polygon is a generalization of the spherical triangle. If is the sum of the radian angles of a spherical polygon on a sphere of radius , then the area is
Let a spherical triangle have sides , , and with , , and the corresponding opposite angles. Then(1)(2)(3)(4)(Smart 1960, p. 23).
Let a spherical triangle have sides , , and with , , and the corresponding opposite angles. Then(1)(2)(3)(4)These formulas are also known as Delambre's analogies (Smart 1960, p. 22).
The study of figures on the surface of a sphere (such as the spherical triangle and spherical polygon), as opposed to the type of geometry studied in plane geometry or solid geometry. In spherical geometry, straight lines are great circles, so any two lines meet in two points. There are also no parallel lines. The angle between two lines in spherical geometry is the angle between the planes of the corresponding great circles, and a spherical triangle is defined by its three angles. There is no concept of similar triangles in spherical geometry.
Given a Schwarz triangle , replacing each polygon vertex with its antipodes gives the three colunar spherical triangles(1)where(2)(3)(4)
The difference between the sum of the angles , , and of a spherical triangle and radians (),The notation is sometimes used for spherical excess instead of , which can cause confusion since it is also frequently used to denote the surface area of a spherical triangle (Zwillinger 1995, p. 469). The notation is also used (Gellert et al. 1989, p. 263).The value of the excess is the solid angle (in steradians) subtended by the spherical triangle, as proved by Thomas Hariot in 1603 (Hopf 1940).The equation for the spherical excess in terms of the side lengths , , and is known as l'Huilier's theorem,where is the semiperimeter.