Stereographic projection — where it appears
Named by 7 essays across one field — each of them below, with the objects they name alongside it.
Stereographic keeps every angle, and only stereographic
One surface in the family preserves shape exactly — every right angle stays a right angle and both its arms are magnified equally, to the last bit the arithmetic has. It also sends every circle in the world to a circle in the picture, which the site's existing conic fit can be pointed at and asked to confirm without being told what it is looking at.
The arcs a curvilinear drawing uses
The taught way to draw a very wide view by hand is to run every straight edge of the world as a circular arc. That recipe has been repeated for sixty years without a surface attached to it, and it turns out to name one exactly — fitting a general conic to the image of a straight line returns a circle to nine decimal places under stereographic projection and returns nothing like a circle under any of the other standard picture surfaces.
One parameter between two surfaces
Wide architectural views are usually made on a projection with a number attached to it — a family running from the flat plane at one end toward the cylinder at the other, with everybody using the value one. That value has never been given a geometric defence. Measured across the family with the same battery of tests as every other surface, the worst angular error over the field has a minimum, and the minimum is at 1.04.
Conformal is not undistorted
The most distorted-looking picture in ordinary circulation is the little planet — a 360 photograph re-projected from below, with the ground curled into a ball. Its worst angular error over 160 degrees of the sphere is 4.4e-8 degrees, which is arithmetic noise. Every crossing in the original crosses at exactly the same angle in the result, and what has gone is area, over a factor of 255.
The lines a surface leaves alone
Only the plane draws every straight line straight, which is easy to measure and which gets read as though a curved surface bent everything. It does not. Every curved picture surface here keeps a one-parameter family of world lines exactly straight, and the family is a curve on the sphere of lines rather than a region of it — so a grid of samples finds none of it, which is what the first version of this measurement reported.
Which rule a fisheye obeys, from straightness alone
Four candidate rules for a fisheye lens part by 54.0 per cent at 45° off axis, and a plumb-line fit shown no scene, no camera and no calibration target can still name which one took a photograph — reliably from about 45° of half-field. Below that the four are indistinguishable in the marks, and naming one collapses to guessing.
The arcs the five-point construction actually draws
The taught five-point construction draws circular arcs between five vanishing points and instructs a draughtsman to graduate the radius evenly. Read that way, the arcs miss a straight line's true image by up to 3.75 pixels on a 300-pixel disc. Read at the stereographic scale instead, the same arcs are exact to 4.3e-13 pixels — the construction was always drawing one projection, and the taught scale was never it.
Named alongside it
The objects these essays reach for when they reach for this one.
ConformalAnisotropyfield of viewCircle preservingDemonstrationEquidistant projectionPicture surfaceArea scaleConic fitEquirectangularStraight familyBeltrami