The sphere bargains with the plane
Stereographic Sea
Drag the gold buoy on either surface. Its route, neighbourhood, and exact inverse move together; near the coral pole, the plane runs out of room.
Circles cross intact.
The equator and a southern latitude arrive as exact circles.
At the gold buoy, local scale is being calculated.
Pointer or touch moves either gold copy. With Canvas focused, arrows move the plane copy; Shift takes larger steps, Home returns south, and Enter advances the authored sequence.
The structural reveal
Shape survives. Scale does not.
Every spherical circle becomes a circle—or a line.
The sphere's circle is an intersection with a plane. After projection its equation remains quadratic with equal squared terms: a plane circle. If the original circle contains the north projection pole, that quadratic term cancels and the generalized circle passes through infinity as a line.
Angles keep their meeting.
The map stretches every tangent direction at one point by the same local factor. The right-angle chapter makes that bargain visible: the two curves bend differently across each surface but retain their 90° crossing.
The price is paid near the pole.
With north-pole projection onto the equatorial plane, local length scale is 1/(1−z), and local area scale is its square. The equator is 1×; z = 0.9 is 10× in length and 100× in area; the pole itself has no finite image.