Regular parametrized surface patches on compact Jordan parameter regions

A compact Jordan parameter region is a compact Jordan measurable set $D\subseteq {\mathbb{R}}^2$ that is the closure of its nonempty connected interior $D^{\circ }$ (The Riemann integral of a bounded function over a bounded Jordan measurable set, Interior, closure, boundary, limit point, isolated point and dense subset of a metric space). Its boundary has content zero (A bounded set in ${\mathbb{R}}^m$ is Jordan measurable iff its boundary is null, equivalently of content zero).

A regular parametrized surface patch is the image of a $C^1$ map from a compact Jordan parameter region whose parameter cross product is nonzero on the region's interior and for which no interior parameter point shares its image with a distinct point of the whole region. Seam identifications and rank failures may occur only on the boundary. Here $C^1$ is the Euclidean componentwise class of $C^k$ Euclidean maps and diffeomorphisms. More precisely, the parametrization $\phi$ is defined on an open neighbourhood of $D$, ${\phi }_u\times {\phi }_v\mathrm{\ne }0$ on $D^{\circ }$, and no point of $D^{\circ }$ has the same image as a distinct point of $D$. The chosen pair $(D,\phi )$ is part of the patch data.