Cellwise assembly of intrinsic PL face fillings #
The fillings supplied by IntrinsicFaceFilling use one standard triangle for every maximal
intrinsic face. This file transports them back to the canonical barycentric realization and
glues the finite family. Coherence is proved from the common global one-skeleton replacement;
it is not stored as an extra compatibility assumption.
A standard-triangle filling transported to its intrinsic closed face.
Equations
- F.intrinsicMap x = F.map ↑((K.facePlaneHomeomorph t) x)
Instances For
The relative interior of a filled intrinsic face maps to the bounded complementary region.
A point of a face which is also carried by a set of at most two vertices lies on one of that face's three intrinsic edges.
The standard coordinate of a point shared by two distinct faces lies on the standard triangle frontier.
A point shared by distinct maximal intrinsic faces belongs to the global one-skeleton.
Lifting the standard coordinate of an intrinsic boundary point recovers that point.
The transported fillings of two faces agree at every point of their overlap.
A chosen maximal face containing a point of the canonical realization.
Equations
Instances For
The finite family of intrinsic face fillings, glued into one map.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The glued map restricts to the chosen certified filling on every maximal face.
The finite cellwise filling is continuous on the whole intrinsic realization.
On the global intrinsic one-skeleton, the cellwise filling is exactly the previously constructed simultaneous graph replacement.
The original embedding written in the standard coordinates of one intrinsic face. Values outside the standard closed triangle are irrelevant.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A one-skeleton point lying in a face has a standard coordinate on that face's frontier.
Conversely, the inverse face chart takes the standard frontier into the global one-skeleton.
The graph-side condition needed for injective cellwise assembly: the simultaneous global one-skeleton replacement avoids the bounded interior selected for every face.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The finite vertex-side condition used to establish graph-side compatibility.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A finite embedded intrinsic two-complex has one positive separation radius for every maximal face and every used vertex not belonging to that face.
Quantitative closeness and vertex-to-face separation put every nonincident vertex on the unbounded side of every replacement polygon.
Every filled face boundary is part of the single global replacement graph.
A replacement-graph point can lie on the polygon of a face only when its intrinsic source lies in that face. This is the exact-carrier consequence of boundary coherence and injectivity of the simultaneous graph replacement.
A point of a face which lies on the one-skeleton maps to that face's polygonal boundary.
Vertex avoidance places every vertex not incident to a face in the unbounded component of that face's replacement polygon.
Moise's finite vertex condition propagates along every intrinsic edge. Thus the whole simultaneous replacement graph avoids the bounded polygonal region selected for each face.
Quantitative control of the cellwise filling. If the original map oscillates by less than
η on every face and its replacement graph is ρ-close, the whole filled face is contained in
the closed ball of radius η + ρ about the original value.
Under graph-side compatibility, bounded interiors selected for distinct intrinsic faces are disjoint.
Once the global replacement graph avoids every selected face interior, the coherent cellwise filling is injective.
Vertex-side compatibility is sufficient to make the coherent cellwise filling a genuine topological embedding of the compact intrinsic realization into the plane.
The image of the coherent cellwise filling is exactly the finite union of its polygonal closed face regions.
The exact image of the coherent cellwise filling carries one conforming pure finite plane complex. The complex is built from a common arrangement of all face-polygon edge lines, rather than by taking a nonconforming union of the independently certified face meshes.
A compatible cellwise filling gives a finite plane triangulation of the intrinsic source, with the plane support coordinate equal to the filling map. This is the concrete output needed when a Radó chart replaces an abstract old patch by a polygonal patch in chart coordinates.