Dimension of trivariate C1 splines on bipyramid cells

• We introduce bipyramid cells in R3R3.
• We improve the lower bound on the dimension of C1C1 splines for bipyramids.
• We derive a new upper bound that is equal to the known lower bound in most cases.
• We use tools from algebraic geometry and Bernstein–Bézier analysis.

We study the dimension of trivariate C1C1 splines on bipyramid cells, that is, cells with n+2n+2 boundary vertices, n of which are coplanar with the interior vertex. We improve the earlier lower bound on the dimension given by J. Shan. Moreover, we derive a new upper bound that is equal to the known lower bound in most cases. In the remaining cases, our upper bound is close to the known lower bound, and we conjecture that the dimension coincides with the upper bound. We use tools from both algebraic geometry and Bernstein–Bézier analysis.