Measuring Solid Angles Beyond Dimension Three

The dot product formula allows one to measure an angle determined by two vectors, and a formula known to Euler and Lagrange outputs the measure of a solid angle in R3 given its three spanning vectors. However, there appears to be no closed form expression for the measure of an n-dimensional solid angle for n > 3. We derive a multivariable (infinite) Taylor series expansion to measure a simplicial solid angle in terms of the inner products of its spanning vectors. We then analyze the domain of convergence of this hypergeometric series and show that it converges within the natural boundary for solid angles. 1. The Normalized Solid Angle Measure There are a number of equivalent ways to extend the concept of planar angle measure to higher dimensions. When the solid angle is a simplicial cone in R3 spanned by unit vectors a,b, and c, the measure of solid angle is defined to be the area of the spherical triangle abc on the unit sphere (see Fig. 1). According to Eriksson [1], both Euler and Lagrange knew how to calculate this solid angle measure V via the triple product: tan V 2 = |a · (b × c)| 1 + a · b + b · c + a · c . More generally, if K is a surface in Rn for which every ray from point C intersects K exactly once, then the measure of the solid angle formed by the solid cone of rays that emanate from C and intersect K is the volume of intersection of the solid cone and a unit sphere centered at C (see Fig. 2) [4]. We prefer to normalize the solid angle measure so that the cone of rays emanating from the origin (all of space) has solid angle measure 1. Write voln for the usual volume form in Rn , Bn for the unit ball in Rn , and Sn for the