Drag the gold point around the manifold and watch it appear in and disappear from each chart.
What to try. Drag the gold point p inside the blue region, then into the green region, then into the overlap, then outside both.
- When p is in Oα (blue), the chart ψα(p) appears in the left ℝ² panel.
- When p is in Oβ (green), the chart ψβ(p) appears in the right ℝ² panel.
- In the overlap Oα∩Oβ (red), both charts describe the same point with different coordinates — that is what the transition map ψβ∘ψα−1 relates.
- Outside both open sets, p still lives on the manifold but has no chart representation here: the maps ψα, ψβ are undefined there. A proper atlas would supply additional charts covering the remaining regions.