Stereographic Projection

Project from the north pole N through a sphere point S onto the equatorial plane, giving Q. Select the direction, a point or a source curve; Parameters contains the active coordinates and curve controls. Changes update immediately. Drag to rotate; Ctrl/Meta + wheel zooms, a plain wheel scrolls the page. Reset frames the sphere and visible plane.

For a unit sphere the forward map is u = x/(1 − z), v = y/(1 − z); the inverse is (x,y,z) = (2u,2v,u²+v²−1)/(1+u²+v²). Near the north pole stable equivalent formulas avoid cancellation, and inverse inputs are scaled to avoid overflow. N has no finite image; nearby finite points remain finite. S/Q coincide on the equator. On the southern hemisphere the ray continues through Q to S.

The equator maps to a unit circle. A latitude circle can use any latitude from −89.999999° to 89.999999°; its image radius is tan(45° + latitude/2). At 30° the radius is √3. A meridian through N maps to a line. Green is the sphere curve, violet its plane image and pink the projection ray. The plane window is u,v ∈ [−3,3]: outside points are hidden, rays and image chords are clipped to the window, and numerical coordinates remain unchanged. Source curves use 128 segments, while image type/radius are analytical.

Play explicitly moves a curve point at the chosen speed. Pause retains its position in the current view; changing inputs or leaving pauses it. Raw user-entered values restore, including invalid drafts. Computed playback position, camera, results and playback state are not saved: reload restores the last position you entered yourself. Without WebGL the numerical output remains available.

PNG exports the current view and pauses playback. Metadata includes the actual point/curve parameters, exact projection output and current camera. An input change cancels a pending outdated export. All calculations stay in the browser; this is a geometric unit-sphere demonstration, not a geographic map application.