Refinable tri-variate C ¹ splines for box-complexes including irregular points and irregular edges

C ¹ splines over box-complexes generalize C ¹ degree 3 (cubic) tensor-product splines. A box-complex is a collection of 3-dimensional boxes forming an unstructured hexahedral mesh that can include irregular points and irregular edges where the layout deviates from the tensor-product grid layout. For example, an edge shared and enclosed by five boxes is irregular. Where the mesh is locally regular, the restriction of the space to each box is a polynomial piece of the C ¹ tri-cubic tensor-product spline, by default initialized as a C ² tri-cubic. Boxes containing irregularities have their polynomials binarily split into 2³ pieces to isolate the irregularity. The pieces join with matching derivatives. The derivatives are zero at irregularities, but these singularities are removable by a local change of variables. The space consists of 2³ linearly independent functions per box and is refinable.