OBSERVATION CHAMBER / ACTIVE
Discrete Curvature Mesh Observatory
Read vertex angle defects, boundary geodesic curvature, Euler characteristic, and the Gauss–Bonnet residual on deterministic triangle meshes.
ROUTE 68 / GEOMETRY / CURVATURE
01 / READ THE MESH
Local angles, global topology
Drag a vertex to edit · drag empty space to rotate · arrows rotate
Observation contract Kv = 2π − Σθ for interior vertices; kg = π − Σθ on the boundary; χ = V − E + F; εGB = ΣK + Σkg − 2πχ.
Canonical renderer: depth-sorted Canvas 2D. The numeric contract remains visible if Canvas is unavailable. No external library or API is used.
03 / READ THE INVARIANTS
Mesh readout
- Preset
- PLANAR DISK
- Mesh
- 13 V / 24 E / 12 F
- Selected vertex
- —
- Select a vertex · K
- —
- Select a boundary vertex · kg
- —
- Euler χ
- 1
- Total curvature
- 6.28319 rad
- Global GB residual
- 0.00e+0 rad
- Finite
- PASS
The selected vertex is the local probe. The residual is recomputed after every valid drag, so local edits can be compared with the unchanged global identity.
Advanced diagnostics and acceptance checks
Self-test checks disk χ=1 and 2π, sphere χ=2 and 4π, torus χ=0 and 0, finite values, residual ≤ 1e−8 rad, and rejection of a zero-area triangle.
Observation report
Formula note
For a triangulated surface, use the sum of incident face angles at each vertex. Interior angle defect is Kv = 2π − Σθ. For a boundary vertex, the signed geodesic contribution is kg = π − Σθ. The discrete Gauss–Bonnet check is ΣK + Σkg = 2πχ.
OBSERVATION GUIDE
Local probe, global identity
Read one selected vertex and then read the whole mesh. The planar disk exposes a boundary term, the sphere concentrates positive interior defect, and the torus balances positive and negative contributions at χ = 0.
- Use INTERIOR K for angle defects inside a closed mesh, or BOUNDARY kg for the disk edge.
- Switch between PLANAR DISK, SPHERE, and TORUS. χ and the Gauss–Bonnet target change with topology.
- Drag a vertex. The geometry changes, but the global residual stays within the finite-mesh tolerance when the faces remain valid.
This is a finite Euclidean triangle-mesh model. It is not a smooth-surface estimator, a physical measurement, or a full remeshing system.
Runs inside the browser with no upload and no registration.
OBSERVATION POLICY
This observatory runs a deterministic triangle-mesh model inside the browser. Mesh coordinates, calculations, reports, and images stay local. It does not measure a physical surface or upload observations.
Route
This chamber belongs to Shape & Space.
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