Notes / G.02
Connectivity before coordinates.
Robust geometry processing starts with topology—neighbors, edges, faces, and boundaries—then applies local operators to update the mesh's geometric positions.
01 / Mental model
A mesh is two coupled structures.
Connectivity answers who is adjacent to whom; geometry assigns a position to each vertex. Keeping those roles separate makes subdivision, smoothing, and editing rules easier to reason about and debug.
02 / Core mechanism
Local neighborhoods define the operator.
Many filters compute a new vertex from a weighted combination of its one-ring neighbors.
The choice of neighborhood and weights determines whether the operator smooths, sharpens, interpolates, or preserves features.
03 / Implementation
Make topology failures visible.
- Build explicit adjacency or a half-edge representation.
- Handle boundaries and extraordinary vertices with dedicated rules.
- Check consistent winding and non-manifold input before processing.
- Recompute normals only after the geometric update is complete.
04 / Evidence
Judge shape, not only triangles.
Compare silhouettes, feature preservation, surface smoothness, and volume before and after each operation. Shrinkage, pinching, or shading seams usually expose a weighting, boundary, or connectivity mistake.