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

Panorama — where it appears

A picture made by an eye that turns without moving, unbounded in bearing and still a projection through one point. It is a projection onto a curved surface, so it can run all the way round at the cost of bending every line that is not a vertical or the horizon.

Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.

fitted conic: 1.467 from being a circlecylinderthe samples, fitted

The cylinder, and the price of going all the way round

A cylindrical picture can hold three hundred and sixty degrees, keeps every vertical vertical, and bows every horizontal. Its cost is a stretch of sec φ in elevation, which is also the equirectangular surface's cost exactly — two surfaces that are always described as different and are identical in the one respect anybody notices.

curved · Cylinder flat picture plane — straight lines stay straightcylindrical picture surface — even stretch, bowed linesone scene, two picture surfacesneither is the distorted one

When the picture surface is not flat

A flat picture plane keeps straight lines straight and stretches the edges without bound. A cylindrical one spreads the stretch evenly and bends every straight line that is not through the axis. Neither is the distorted one — they are answers to different questions, and the choice decides what a wide view can be.

viewing · Surface the far field — where the stitch was fitted2.2 m — 3.2 px out24 m — 0.3 px outthe sky registers to 1e-13 pxthe foreground does not — up to 3.2 px

The eye is a place, not a point

Rotate a camera about the wrong point and the sky still stitches perfectly while the foreground slides. The misregistration falls as one over the distance, exactly — which is what says the fault is the pivot and not the lens.

lens · Pupil plane — bend 0.0%cylinder — bend 6.0%stereographic — bend 3.5%equidistant — bend 4.5%equal-area — bend 5.1%equirect. — bend 5.7%120° across in every panelsame scene, same angle, six surfaces

A scroll is not a panorama

Both draw straight world lines as curves, and one of them is a projection. A rotating eye keeps its centre exactly however far it turns; a translating eye has none at all. Curvature and centrelessness are independent properties, and conflating them is the standard mistake about both objects.

scroll · Panorama leftfrontrightbackupdownacross the left/front seam: 1.80°, with each side straight to 7e-16corner area ×5.196anisotropy √3 = 1.7321 there

Six flat pictures of everything

There is one way to photograph the whole sphere and keep every straight line straight, and it is to stop using one surface. Six flat pictures at ninety degrees cover everything, each of them a perfect pinhole, and the price is paid entirely at the seams — where a straight line does not bend but kinks, by an angle that reaches 45 degrees and is exactly zero for the lines lying in the seam's own plane.

curved · Cubemap paraxial pupilthe stopa curved front elementcrossings 15.0 · 14.5 · 13.5 · 11.6 mm

The entrance pupil walks with the angle

The place a picture is a projection from is not a point in a wide-angle design. Chief rays traced through a strongly curved front element cross the axis 15.07 mm behind its front vertex when they are nearly on the axis, and 4.23 mm nearer the front at 80° of field. So no pivot makes a wide panorama seam clean: at one metre, pivoting at the paraxial pupil leaves 4.39 arcminutes of misregistration along a seam, and the best pivot still leaves 1.41.

lens · Pupil 0.313100510how far away the subject is, in metreshow far the pupil moves forward, in millimetres0.50 mthe whole walk with field anglea 50 mm lens, focused as one pieceequal at 0.50 m

Focusing moves the pivot past its best place

Focusing a fifty-millimetre lens to one metre carries its entrance pupil 2.63 millimetres forward of the camera body, and to half a metre 5.56 — which is more than the whole 5.53 that the pupil walks with field angle, so past a subject at 502 millimetres the focus decides where the pupil is. A panorama head aligned at infinity and used at a metre leaves 7.46 arcminutes along its seam; aligned at four metres it leaves 2.21, better than pivoting at the pupil at all.

lens · Pupil the pivottangent circle, 30.0 mmno single viewpoint — the rays miss by 24.98 mm6 frames · pivot 60 mm off

The pivot that is not the eye

A camera bolted to a tripod turns about the screw under its baseplate, and the light crosses somewhere else. Every ray of the panorama that results passes the pivot by e sin γ — the offset times the sine of how far off its own frame's axis the ray points — so the picture has a radius where a projection would have a point.

curved · Pivot 0204060204060frames in the panoramadistance from the pivot (mm)π/β = 9.5across the seamup the framepivot 60 mm · frame 38° tallfloor 19.53 mm

The parallax you cannot shoot away

A stitched panorama's parallax has two halves and they do not behave alike. The one across the seam falls as the sine of half a frame spacing, so more frames buy it off; the one up the frame is the sine of half the frame's own height, and no quantity of shooting touches it. They cross at π over β, which has no pivot error in it at all.

curved · Pivot the pupil, and the pivotsubject at 34°the filmcorrect from 15 cm, at 160 mm wideagrees with the cylinder to 1.1e-16

The camera that is a cylinder

A swing-lens camera turns its lens about its own entrance pupil and sweeps a slit across film bent into a circle concentric with it. Compute where the light lands, unroll the film, undo the pinhole's inversion, and the result is not similar to the cylindrical picture surface — it is the same map, to the arithmetic floor. What it pays instead is detail, and a shear on anything that moves.

curved · Pivot -500500.5001distance to the thing being stitched (log₁₀ metres)what the stitch leaves behind (px)2.86 m80 mm baselinezero at one depth each

A rig is right on one surface

Several cameras looking outward have several centres, and no warp registers all of a scene. The residual is a disparity, so the depth the stitch is computed for has an exact optimum — the harmonic midpoint of the depth range, not the middle of it — and the arithmetic middle costs a factor of 2 z_far over the sum, which tends to two.

curved · Stitchdepth column 62 mmcorrect from 19 cm, at 160 mm wide0.966% hidden, to 15.3° from the nadir

The hole a rig cannot fill

A two-lens spherical rig covers every direction between its two lenses and still cannot see 0.470 per cent of the sphere directly beneath it, reaching 12.8 degrees from straight down — its own tripod, standing exactly where neither lens can look. No arrangement of lenses removes it, because it is not a gap in coverage; it is the rig occluding itself.

curved · Nadir

Named alongside it

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

centre of projectionParallaxEntrance pupilPicture surfaceCylindrical projectionDemonstrationStitchinginstrument limitEquirectangularfield of viewSpherical panoramaAnisotropy

All concepts