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 on the sphere its circumference is \(2\pi\sin r\), not \(2\pi r\). The chart stretches that circle sideways by \(r / \sin r\), and the stretch comes from the curvature.
Shrink the radius and the ratio \(\sin r / r\) climbs toward \(1\) like \(1 - r^2/6\), so the stretch falls with the square of the radius: halve the patch and the distortion drops to a quarter. At small enough scale the patch is, to the eye, 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, and the photograph softens. A 2 km/pixel version exists, but at 21600×10800 it is too large for most graphics cards to hold as one image and would need to be cut into tiles — future work.
The graticule and the patch grid are computed rather than sampled, so they stay sharp at every radius.