A sphere is not flat — put otherwise, it is non-Euclidean. If we zoom in to a small enough patch, it appears flat; this is what it means for a surface to be locally Euclidean. Drag the point to move the patch, or shrink the radius to watch the curvature drain out of it.
The chart is the patch flattened by the exponential map: a point at geodesic distance \(\rho\) and bearing \(\alpha\) from \(p\) is drawn at Cartesian coordinates \((\rho\cos\alpha,\ \rho\sin\alpha)\). Distance along any straight line out of the centre is exactly right, so a circle of geodesic radius \(r\) becomes a circle of radius \(r\) — but its circumference is \(2\pi\sin r\), not \(2\pi r\). That shortfall is the curvature, and it is the only thing stopping the chart from being an honest map.
Shrink the radius and the ratio \(\sin r / r\) climbs toward \(1\) like \(1 - r^2/6\). That quadratic is the whole idea: the error vanishes faster than the patch does, so at small enough scale every neighbourhood is indistinguishable from a piece of \(\mathbb{R}^2\).
The Blue Marble base map is 5400×2700 — about 8 km/pixel at the equator. The chart magnifies whatever the patch covers into a fixed square, so as the radius falls the same screen area is fed by fewer and fewer source pixels: below roughly \(r = 5^\circ\) there are fewer texels than screen pixels and the photograph visibly blocks up. A 2 km/pixel version exists, but at 21600×10800 it exceeds what a browser will decode into a canvas.
The graticule and the patch grid are computed rather than sampled, so they stay sharp at every radius. What degrades is the photograph running out of detail, not the geometry — which is convenient, because the geometry is the part making the argument.