Measuring solid angles beyond dimension three

Abstract

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. © Springer 2006.