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Given two $n$-vertex polygons, $P=(p_1, \ldots, p_n)$ lying in the $xy$-planeat $z=0$, and $P'=(p'_1, \ldots, p'_n)$ lying in the $xy$-plane at $z=1$, abanded surface is a triangulated surface homeomorphic to an annulus connecting$P$ and $P'$ such that the triangulation's edge set contains vertex disjointpaths $\pi_i$ connecting $p_i$ to $p'_i$ for all $i =1, \ldots, n$. The surfacethen consists of bands, where the $i$th band goes between $\pi_i$ and$\pi_{i+1}$. We give a polynomial-time algorithm to find a banded surfacewithout Steiner points if one exists. We explore connections between bandedsurfaces and linear morphs, where time in the morph corresponds to the $z$direction. In particular, we show that if $P$ and $P'$ are convex and thelinear morph from $P$ to $P'$ (which moves the $i$th vertex on a straight linefrom $p_i$ to $p'_i$) remains planar at all times, then there is a bandedsurface without Steiner points.

Reconstructing a Polyhedron between Polygons in Parallel Slices