Curvilinear face-to-face triangulation

Let $M$ be a compact smooth two-manifold, possibly disconnected, nonorientable, or with boundary. A curvilinear face-to-face triangulation of $M$ is finite data $(V,E,F,\varphi )$ with the following properties.

1. $V$ is a finite subset of $M$. Each $e\in E$ is the image of a regular $C^2$ embedding ${\gamma }_e:[0,1]\to M$. Its endpoints are two distinct vertices in $V$; its relative interior contains no vertex. Distinct edges meet only at common endpoints.
2. For each $f\in F$, ${\varphi }_f:{\bar{\mathrm{\Delta }}}^2\to M$ is a homeomorphism onto a closed subset $F_f$. It sends the three distinct vertices of the standard closed triangle to three distinct points of $V$, and sends each full side homeomorphically onto an edge of $E$. The open face ${\varphi }_f(({\mathrm{\Delta }}^2{)}^{\circ })$ lies in $\mathrm{Int}M$ and is open there; $F_f$ is its closure in $M$. The three boundary edges are regular $C^2$ embedded arcs. At an interior edge point, its incident face has the local one-sided half-disk chart of Regular oriented surface regions with corners. At a boundary edge point, the edge lies in $\mathrm{\partial }M$ and the open face approaches it from the inward side in a boundary half-space chart. At a face vertex, orient the two incident boundary arcs by the reference triangle and let $v_{-}$ and $v_{+}$ be their nonzero one-sided velocities into and out of the vertex. Require $v_{+}\mathrm{\ne }-c{v}_{-}$ for every $c>0$, matching the regular-region convention. Separately, require the face to occupy the closure of one local sector bounded by those arcs; a zero signed turn is allowed. At a boundary vertex read that sector relative to the half-space chart. The orientation used on a face is induced by its reference triangle only; no orientation of $M$ is required.
3. The faces cover $M$. Two distinct face images are disjoint, meet in one common vertex, or meet in one common full edge. An edge is either contained in $\mathrm{\partial }M$ or has relative interior in $\mathrm{Int}M$; each interior edge is incident to exactly two faces and each boundary edge to exactly one. The union of the boundary edges and their vertices is exactly $\mathrm{\partial }M$.
4. For a vertex $v$, its finite link graph has one vertex for each incident edge germ and one edge for each incident face corner. The link is a circle if $v\in \mathrm{Int}M$ and a closed interval if $v\in \mathrm{\partial }M$; for a boundary vertex the interval endpoints are precisely the boundary edge germs.

The objects in $V$, $E$, and $F$ are counted once each. In particular, a boundary edge has one incident face but contributes one edge to $\mathrm{\mid }E\mathrm{\mid }$. A prescribed boundary edge need only be a regular $C^2$ arc; it need not be a geodesic.

Jost, Compact Riemann Surfaces: An Introduction to Contemporary Mathematics, §2.3.A, Definition 2.3.A.1, printed pp. 31–32 (PDF pp. 43–44), lines 1776–1793, defines a finite topological subdivision by triangular subsets, homeomorphisms from planar triangles, and the disjoint/common-vertex/full-edge intersection condition. Jost's definition is topological and is stated for closed surfaces. The local piecewise-$C^2$ edges, ordinary corners, vertex links, and boundary subcomplex required here are additional explicit clauses, using the earlier library regular-region definition for the face charts.