Whitney Forms for Spherical Triangles I: The Euler, Cagnoli, and Tuynman Area Formulas, Barycentric Coordinates, and Construction with the Exterior Calculus

Abstract

We establish the equivalence of the Tuynman midpoint area formula for a spherical triangle to the classical area formulas of Euler and of Cagnoli. The derivation also yields a variant of the Cagnoli formula in terms of the medial triangle. We introduce the three barycentric coordinates of a point within the spherical triangle as area ratios which sum to unity. The barycentric coordinates are the Whitney 0-forms, scalar functions over the domain of the triangle associated with each vertex. We then construct, by exterior differentiation of the barycentric coordinates, succinct expressions for the Whitney 1-forms associated with each geodesic side, or great circle arc. The Euler formula, in conjunction with that of Tuynman, facilitates the differentiation of a triangular area with respect to the position of a vertex. As both the Euler and Tuynman formulas may be expressed naturally in terms of the position vectors of the vertices on an embedded sphere, the Whitney constructions may be done in terms of vector-valued forms and without recourse to a particular projection or coordinate chart. Finally, we exhibit the Whitney 2-form of the triangle, which must be the product of a scalar function and the area 2-form of the sphere. We find an expression for this scalar function in terms of determinants of $3 \times 3$ matrices built from the vertex position vectors. It is a rational function in the Cartesian coordinates of a point. By construction it must be invariant under cyclic permutations of the three vertices, though it is not manifestly so. Also by construction the Whitney 2-form must integrate to one over the triangle. Thus the rational function is exactly integrable over its triangle when restricted to the sphere.