Concept

Marginal distortion — where it appears

The stretch a wide flat picture gives its edges, which is correct from the station point and visible from anywhere else.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

051050100150field of view the picture was rendered at — degreeshow many times the depicted depth is stretchedthe screen subtends 49.3°100° → depth ×2.6027-inch monitor at 650 mmsubtends 49.3°

A wide field on a small screen

A picture rendered at a hundred degrees and shown on a screen that subtends forty-nine is being read from two and a half times its own station distance, so the depicted space is two and a half times too deep. The stretch at the edges everybody complains about is correct; the complaint is really that nobody is sitting where it would be invisible.

screen · screenfov
drawn: 1.309 : 1subtended at the station point: 1.000000000 : 1the faint circle is the mean radius, for comparisonthe drawn centre sits 3.42 px from the axis's own mark

The sixty-degree cone of vision

Every book says keep the subject inside a 60° cone. Measured, the marginal stretch the rule is nominally about is exactly zero from the station point — 1.000000000000 to 1, over 720 sampled points. The rule is a statement about the reader, and books do not obey it.

wrong · cone
d = 0d = 0.5d = 1d = 2d = 4d = 0straight48.06° angle×17.80 aread = 0.54.81% bend23.08° angle×4.66 aread = 17.21% bend16.84° angle×2.65 aread = 29.61% bend23.13° angle×1.57 aread = 411.54% bend32.71° angle×1.73 areaa general straight line · worst angle · area range, over the field130° acrossangular minimum at d = 1.00

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 on this site's own battery, the worst angular error over the field has a minimum, and the minimum is at 1.04.

curved · panini
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

Named alongside it

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

field of viewDemonstrationAnisotropyArea scalecone of visionConformalPicture surfaceStation pointStereographic projectionViewing distanceCircle preservingCylindrical projection

All concepts