Rips complex at filtration ε
Sublevel set of the filtration parameter (disk radius)
Filtration controls
Range 0–200 (depends on point cloud)
5–30 vertices
Vertices — sample points. Disks B(pi, ε): a 1-simplex (edge) joins pi, pj when ‖pi − pj‖ ≤ 2ε; a 2-simplex (triangle) appears when all three pairwise edges exist; a 3-simplex (tetrahedron on four vertices) appears when all six pairwise edges exist. Each resample uses the selected point cloud model.
Persistence diagram
Cumulative diagram up to the current filtration value ε
Diagram options & legend
- H0 — finite (b, d), b = 0
- H1 — d ≤ ε (dead)
- H1 — d > ε (alive)
- H2 — d ≤ ε (dead)
- H2 — d > ε (alive)
- 3-simplex fill in Rips panel
- Current ε
Each point (b, d) is a generator in persistent homology (b = birth, d = death). H0 (blue): connected components. H1 (green): 1-dimensional classes. H2 (purple): 2-dimensional classes from 3-simplices. Points with d − b ≫ 0 lie far above the diagonal {b = d}. Hover diagram points for explanations; green highlights a triangle, purple a 3-simplex in the Rips panel.
Help: reading the diagram
Reading the persistence diagram
- H0 — each off-diagonal blue point (0, d) records a connected component merging at filtration d.
- H1 — green points (b, d) with b < d are 1-dimensional classes; persistence is d − b.
- H2 — purple points record 2-dimensional classes filled in by 3-simplices (four mutually adjacent vertices).
- Diagonal — instant death (d − b = 0). See persistence diagram and persistent homology.
Filtration & ε
Both panels share the same ε. The Vietoris–Rips complex grows monotonically; diagram points with b ≤ ε accumulate. Orange lines mark the current ε.