Sweeping the ${\bf cd}$-Index and the Toric $h$-Vector

We derive formulas for the ${\bf cd}$-index and the toric $h$-vector of a convex polytope $P$ from a sweeping by a hyperplane. These arise from interpreting the corresponding $S$-shelling of the dual of $P$. We describe a partition of the faces of the complete truncation of $P$ to reflect explicitly the nonnegativity of its ${\bf cd}$-index and what its components are counting. One corollary is a quick way to compute the toric $h$-vector directly from the ${\bf cd}$-index that turns out to be an immediate consequence of formulas of Bayer and Ehrenborg. We also propose an "extended toric" $h$-vector that fully captures the information in the flag $h$-vector.