Persistent Homology

Edges appear when their Euclidean length is at most ε; a triangle appears when all three edges are present. ε is an edge-distance threshold, not a disk radius. Filled faces can overlap in the planar drawing without adding connections to the abstract complex.

Calculate once, then move ε. β₀ counts current connected components; β₁ counts current independent loops over F₂. The barcode records their intervals [birth, death). Zero-duration pairs are hidden. An arrow means the feature survives at the chosen maximum; it does not prove survival forever. Only H₀/H₁ are computed, with a limit of 24,000 triangles.

The square has one loop from ε=1 until ε=√2, when its diagonals and faces arrive. PNG exports the current complex; CSV exports positive-duration intervals with the filtration limit. A blank death is censored at that limit.

After the first calculation, valid settings update automatically. Click or drag in the barcode to set ε. Settings accepts 2–80 own points (one x y pair per line); distances use their original coordinate units. Separate decimal-comma coordinates with spaces or semicolons. The PNG includes the current points, ε and filtration limit as metadata. Reset restores the default inputs. Only inputs are retained; the complex is recalculated after navigation.