The Dimension of Trivariate C¹ Splines on Generic Bipyramid Cells

A bipyramid cell is a tetrahedral partition with one interior vertex v₀, n boundary vertices that are coplanar with v₀ and surround it, and two further boundary vertices on opposite sides of this base plane. Colvin, DiMatteo and Sorokina determined the dimension of the space S¹_d(Δ) of C¹ splines of degree at most d on such a cell Δ in their collinear and coplanar cases. In the generic case they gave lower and upper bounds and conjectured that the dimension equals the upper bound whenever the two bounds differ. We prove an exact formula for every generic bipyramid cell. Let m be the number of distinct lines spanned by the n interior edges in the base plane. Then dim S¹_d(Δ) = 1, 4, 11 for d = 0, 1, 2, and for d ≥ 3 it equals 2n·C(d,3) + 6d − 1 + (4 − m)₊ if m ≥ 3, and 8·C(d,3) + 8d − 3 − (4 − d)₊ if m = 2. For d ≥ 3 these are exactly the upper bounds as printed in the Oberwolfach abstract of Colvin, DiMatteo and Sorokina, and for d = 2 the upper bound is attained unless m = 3. So the conjecture holds in every case except m = 3, d = 2, where the dimension is 11 and the printed upper bound is 12; the smallest example has a triangular base. We could not access the journal version of their paper; if its bounds for d ≥ 3 are the printed ones and it states this one case differently, the conjecture holds as stated there, and in any case the exact formula answers the question. An elementary cofactor argument shows directly that dim S¹_2(Δ) = 11 for every generic cell. The spline space splits into homogeneous components; the dimensions of those of degree at least 5 also follow from a general formula of Alfeld, Neamtu and Schumaker, so the new part of the formula is the determination of the components of degrees 3 and 4. The proof reduces the problem to bivariate splines on the planar fan in the base plane: each half of the cell projects onto this fan, and gluing the two halves forces the trace on the base plane to be C² across every ray. The same reduction gives a uniform proof of their formulas in the collinear and coplanar cases. The formula agrees with Macaulay2 values of DiPasquale and Villamizar for n = m = 5 and with exact rational computations. This is an unrefereed note.