A planar domain whose boundary consists of finitely many straight line segments joined at vertices, giving a piecewise-linear boundary with corners and edges but no curved arcs.

Principle

The piecewise-linear structure reduces geometric questions to finite combinatorial and linear pieces: boundary regularity is encoded by vertex angles and edge adjacency, which govern reflection, singularity, and approximation properties.

Demonstration

A canonical demonstration is a simple polygon in the plane: solving Laplace's equation with Dirichlet data exhibits corner singularities whose strength depends on interior angles, and finite element meshes align to linear edges for exact boundary representation.

Misapplication

Assuming smooth boundary estimates (like classical Schauder interior regularity up to the boundary) without accounting for corners leads to overoptimistic regularity claims; singular behavior concentrates at vertices.

Consequence

Proper treatment yields explicit descriptions of singular solutions near vertices, simple meshing strategies for numerical methods, and combinatorial control of topological features (e.g., Euler characteristic via triangulation).

Reversal

The reversal is a domain with continuously curved boundary (C^1 or smoother) where corner-induced singularities disappear and different regularity and approximation techniques apply.

Boundary

Scope is planar polygonal regions (possibly with holes) and excludes domains with curved boundary segments or fractal perimeters; also excludes infinite-sided polytopes and higher-dimensional polyhedral generalizations unless specified.

Semantic Tension

Tension arises between polygonal domains viewed combinatorially (as planar graphs) and analytically (as PDE domains): combinatorial simplicity eases some arguments but may hide analytic singularities requiring local analysis.

Synthesis

A polygonal domain is a finite, piecewise-linear planar region whose combinatorial boundary data (edges and vertex angles) control analytical phenomena like corner singularities and guide practical discretization for computation.