What Is a Mesh

A polygon mesh is a finite set of vertices, edges, and faces that approximates the surface of an object. Triangles dominate today because GPUs are optimized for them and any planar polygon can be triangulated. Quad meshes are common in modeling because they subdivide cleanly and represent flow lines naturally.

1. Explicit Representation

Each face stores its vertices' coordinates directly: F1 = ((x1,y1,z1), (x2,y2,z2), (x3,y3,z3))

• Pro: Trivial to render face by face.
• Con: Each shared vertex is repeated -> 3x to 6x duplication; updating a vertex requires touching every face that uses it.

2. Pointer to Vertex List (Indexed Mesh)

Two arrays:

• Vertex array: V[i] = (x_i, y_i, z_i).
• Face array: F[j] = (i_a, i_b, i_c), indices into V.
• Pro: No duplication of coordinates; updating a vertex updates everywhere.
• Con: Edge queries (find faces sharing an edge) are O(F).
• This is the standard format for OpenGL element arrays, glTF primitives, .obj files.

3. Explicit Edge Representation (Winged-Edge / Half-Edge)

Three arrays: Vertices, Edges, Faces, with cross-pointers.

• Each edge knows its two end vertices, its left and right faces, and pointers to neighboring edges.
• Pro: O(1) topology queries (next edge around a face, faces sharing an edge, vertex valence).
• Con: Larger memory footprint, more complex updates.
• Used by 3D modeling tools (Blender's BMesh, Maya) and subdivision algorithms.

Comparison Table

Mesh Quality Metrics

• Manifold: every edge belongs to exactly two faces (closed) or one face (boundary).
• Watertight: no holes; useful for 3D printing and constructive solid geometry.
• Aspect ratio of triangles: thin slivers cause numerical and shading problems.
• Vertex normal averaging: per-face normals are flat-shaded; smoothing requires averaging neighboring face normals into per-vertex normals.

Storage Order Matters

Reordering vertices by vertex cache locality (e.g., Forsyth's algorithm) can double GPU throughput because nearby triangles share vertex shader work via the post-transform cache.