Visual Plane
every picture states the point it is correct from
Concept

Equal-area projection — where it appears

The fisheye whose picture radius is twice the sine of half the angle off axis, giving equal solid angles equal areas and no fidelity of shape. Its area scale is the other exact zero in the comparison, which is why a count of pixels on it is a count of solid angle.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

02468020406080angle off the optical axis (degrees)area scale, relative to the centreplanecylinderstereographicequidistantequal-areaequirect.measured by differencing on the spherethe plane leaves the frame

Every fisheye is a different rule

The word "fisheye" names a shape of lens and not a projection. There are several, they disagree with each other by tens of per cent at the frame edge, and each is the right answer to a different question — one is a protractor, one is a counting instrument, one preserves shape. Which one a lens implements is a fact about that lens, and it is rarely printed on the barrel.

curved · Fisheye angle, worst over the sphere4.4e-8°anisotropy, worst1.000000023area scale, largest over smallest×255what a reader calls distortedthe third row, not the firstthe disc is 160° of the spheredrawn to 160° off axisthe first two rows are conformality

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.

curved · Littleplanet planeevery line—cylinderone parameter · parallelno meeting pointstereographicone parameter · concurrentthey meet to 1e-12equirect.one parameter · parallelno meeting point3 of 4 keep a curvethe signature has three values, not eight

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.

curved · Straightfamily 051015200204060how far off the fovea the line passes (degrees)departure from a straight chord, in degrees of visual angleschematic eyeconcentric retinaa line spanning ±65° of azimuth17.30° at 40° eccentricity

The eye is a picture surface too

A retinal sphere behind an off-centre nodal point takes every measurement this collection puts to a lens or a screen, and answers all of them. It is the equal-area fisheye to within 110 micrometres of retina rather than the equidistant one everybody draws it as, and a flat picture at its own correct distance leaves the identical arc on it, to 1.4e-14 degrees.

curved · Retina

Named alongside it

The objects these essays reach for when they reach for this one.

AnisotropyArea scaleConformalEquidistant projectionDemonstrationEquirectangularfield of viewFisheyenecessary, not sufficientPicture surfaceStereographic projectionBeltrami

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