mathematics//computational geometry//polygon mesh

A polygon mesh is a representation of a surface as a set of **vertices** (points in space), **edges** joining pairs of them and **faces** (flat polygons, usually triangles) bounded by those edges, and it is the common format in which 3D shapes travel between tools: from a scanner to a CAD program, from a modeller to a game engine or a simulator, from a slicer to a 3D printer (the STL file is nothing but a list of triangles). A curved surface becomes many small flat pieces, and its fidelity is set by how many there are and where they sit.

A polygon mesh is a representation of a surface as a set of vertices (points in space), edges joining pairs of them and faces (flat polygons, usually triangles) bounded by those edges, and it is the common format in which 3D shapes travel between tools: from a scanner to a CAD program, from a modeller to a game engine or a simulator, from a slicer to a 3D printer (the STL file is nothing but a list of triangles). A curved surface becomes many small flat pieces, and its fidelity is set by how many there are and where they sit.

Two kinds of information live in a mesh. The geometry is where the vertices are; the topology is which vertices form which faces and which faces share an edge. Most of a mesh's quality is in the topology: a surface is manifold when every edge is shared by exactly two faces (no dangling flaps, no three faces meeting on one edge), and watertight when it has no holes, which is what a printer and a volume computation need. Normals, the direction each face points outward, tell a renderer which side is outside and a physics engine which way a collision pushes.

More faces is not better, better-placed faces is.

A flat wall needs two triangles and a fillet needs dozens; a mesh that spends its budget evenly is heavy where it does not matter and coarse where the curvature lives. Artists and meshers concentrate faces on curves and edges, which is also what makes a mesh deform and texture cleanly.

A face count is a budget: a real-time simulator for a drone fleet may allow a few thousand triangles per object, a film asset millions. Decimation reduces the count while keeping the shape; subdivision refines it.

Meshes from scans or from isosurface extraction are dense, irregular triangle soups; remeshing turns them into the regular faces a designer can edit.

A mesh describes a surface only. Mass, material, tolerances and the solid inside come from elsewhere (a CAD solid model, a finite-element mesh of volume elements), which is why a mesh describes how a part looks and leaves its engineering definition to those other models.

Generative models that write meshes token by token (3D generation) inherit all these criteria: a generated mesh is checked for manifoldness, holes and inverted normals before it is used.