An angle criterion for conical mesh vertices

A vertex in a mesh surface with planar faces may have the property that offsetting all the face planes incident with the vertex by a constant distance leads to planes which intersect again in a common point. This is equivalent to the property that the planes, consistently oriented via the connectivity of the mesh, are tangent to an oriented cone of revolution. We show that for vertices of valence 4, this conical property is characterized in terms of the interior angles of the faces adjacent to the vertex: The two sums of opposite angles are equal. For a convex vertex this angle criterion follows directly from known results in spherical geometry concerning convex spherical quadrilaterals. For other types of vertices, however, the occurrence of non-convex spherical quadrilaterals makes it necessary to exhaustively enumerate and study a number of cases. The present short note resolves this combinatorial difficulty and proves that all conical vertices are characterized by this same angle criterion. This result is especially relevant in the context of modeling with conical meshes.