Construction of Simple Quadrangulations of the Sphere at Skeletal Joints

We present a method for constructing a simple quadrangulation of the sphere from an unordered set of $n$ direction vectors, with application to parametric volumetric models of skeletal joints. We first select a quadrangulation encoding the joint topology by exhaustively enumerating candidate dual graphs, prioritizing convexity and regularity of the resulting quads. We then realize this topology on the sphere without crossings by solving a conflict-constrained assignment problem with lazy cyclic-order separation, and recover the corresponding primal graph. Experiments on direction sets from spherical Fibonacci sampling of size $3$ to $20$ show that the method robustly produces valid, high-quality quadrangulations across a range of input configurations. The resulting primal graph is directly converted into exact rational Bézier shell and column volumes, yielding a compact and precise volumetric representation of the joint. The approach extends naturally to multi-joint assemblies, enabling the construction of truss-based lattice structures within 3D domains.