Area scale — where it appears
Named by 34 essays across 10 fields — each of them below, with the objects they name alongside it.
No picture surface keeps everything
A picture has to be cast onto something, and every candidate surface destroys something different. Six of them are measured here on the same three properties, and the corner of the plot where a surface pays nothing is empty — not because nobody has thought of one, but because a theorem says there is none.
What the removed roof buys
The Japanese convention of drawing an interior with its roof lifted off is usually explained as a way of seeing inside. What it actually buys is uniformity — every room reports the same share of its floor, to the last sample, where the eye that frames the same building reports three different numbers.
Measuring a room off the page
A perspective picture of a floor has to be rectified before anything on it can be measured, and the rectification is a fit that amplifies the marking error. An oblique picture of the same floor is already rectified — the page is the plan, at one scale, and a ruler on the paper is a ruler on the ground.
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.
The sky inside a cone
From under water the whole sky — every direction out to the horizon — arrives inside a cone of 48.61°. Outside it the surface is a mirror. That cone is a picture surface, and it has a distortion no surface in the curved field has — an area scale that runs to zero.
A mirror ball is an equal-area fisheye
Photograph a mirror ball from far enough away and its rule is ρ = R·sin(θ/2), which is the equal-area fisheye — not an approximation to it, the rule. Measured, the departure falls from 4.27% of the picture's radius at 3 radii to 0.01% at 2000, while the next-best named rule stays 21% out at every distance. And the ball reflects 100.0% of the directions there are, which no designed surface does.
What a 360-degree photograph actually is
The format every spherical camera writes preserves nothing — not straightness, not shape, not area — and it is the right choice anyway, for a reason that has nothing to do with looking at it. An equirectangular file is a lookup table of directions, and the picture only exists at the moment something re-projects a piece of it.
Assembled from several views
An Egyptian relief takes each part of a figure from the direction that identifies it — head in profile, eye and shoulders frontal, a pond in plan. What that buys is exactly measurable: any single viewing direction keeps at most √k of k perpendicular aspects, so the best compromise view retains 58% of each.
What perspective gave up
The field ends by turning its own battery on the system it has been comparing everything against. Four quantities a pinhole destroys that the other systems keep, each measured by the same computation as the systems it is set against, and each the price of the one thing perspective has and they do not.
The cylindrical mirror unbends it
Put a mirrored cylinder in the middle of the sheet and the light path from the eye to the paper bends once. The map that results is not a homography and not a projection in the plane sense at all — it varies its scale by a factor of seven across the design, which is why the smear is unreadable and why the mirror can put it back.
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.
A yes in the table is a price
The comparison table says isometric, cavalier, the elevation and the plan oblique all keep measure. Priced on four hundred boxes, with each picture handed its own best ruler, the four charge 0%, 0%, 33.3% and 0% for an edge — and a pinhole charges 39.5%, only 6.2 points more than the elevation it is filed against. Turn the boxes and three of the four yeses cost something; only the plan oblique's stays free.
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.
The screen is a picture surface too
A curved television is one of the six named picture surfaces sitting in a living room, and from its own axis it delivers azimuth in proportion to the picture, exactly. What it is shown is a rectilinear picture from a sofa, and the difference is not a matter of degree — a flat screen from any seat shows a homography of the intended picture, so it is a correct picture of a transformed scene, and a curved one shows a map that is not a homography from any seat at all.
A page is bounded by a divide, not a centre
A pinhole draws the whole of an infinite ground in a bounded patch of page — each doubling of distance half the one before — while a handscroll spends the same page on every doubling and an isometric drawing spends three quarters of its page on the last one. It is tempting to credit the centre. A crossed-slits camera, whose rays miss any common point by 0.46 m, is bounded too: what does it is dividing by depth in both directions of the page.
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.
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.
The corner sees an ellipse
A circular pupil viewed from off the axis is foreshortened by the cosine, so the blur patch a corner receives is an ellipse of axis ratio 0.920 at the edge of a full-frame picture with a 50 mm lens — and the light through it falls as the fourth power of the same cosine, 0.480 stops. Both are geometry, both happen to a perfect lens, and no design removes either.
The ceiling that is not a plane
Paint the same design for the same eye onto a floor and onto a barrel vault, then fit the best possible homography to each set of marks. On the floor it misses by femtometres, because the map is a collineation and four marks determine every other. On the vault it misses by half a metre, and no choice of four marks helps — which is where every projective construction made for a floor stops applying.
An area, out of one photograph
A patch of ground comes back at 5.205 m² from one photograph, to 1.8 × 10⁻¹⁴, through a homography built from four marks and their four known positions. What is worth knowing is how it degrades — the patch's extent across the picture is read with an error growing as the depth, its extent into the picture with an error growing as the depth squared, and at twenty-eight metres the two are sixteen times apart — which is the depth divided by the camera's height.
Counting is a measurement
A tiled floor gives its area with no reference length at all — count the tiles and multiply. The count is an integer, so it is exact wherever it can be made, which is a completely different error law from the rectifier's smooth decay. And the distance at which it fails is set by the tile's depth edge, which foreshortens as one over the depth squared, so 18.7 m for a 62 cm tile, where the across edge alone would have allowed 217.
The true shape of a cut
A plane through a box makes a hexagon of 2.8996 m². The front view draws it at 1.9966 m² — the true area times the cosine, 0.6886 — and gets its corners wrong as well, the worst by 15.60°, because a foreshortening scales one direction and not the other. The area is recoverable with one number and the angles are not, which is why a section gets a view of its own.
The floors that unroll
A ridged floor curves visibly and can be laid flat without stretching anything — 7.4e-9 of strain across the patch. A dished floor curves less and cannot be laid flat by any means whatever. The difference is one number, Gaussian curvature, and it is the number Gauss proved no bending can change: 0 for the ridge, 0.0144 per square metre for the dish, and no cleverness in the flattening touches it.
The third column is area
This field has measured what each picture surface does to straight lines and to shape. Both are questions for somebody looking at the picture. Somebody counting in it wants a third column, and the same projections have been returning it all along without anybody asking: the equal-area fisheye holds a square degree at one printed area to 8e-8 across 80° off axis, while a flat plane inflates it 191-fold.
Counting cloud by counting pixels
A sky camera looks up and something counts the white pixels. On an equal-area fisheye that answer is right to 0.01%, which is the grid's own error. On an equidistant one it is 9% low, on stereographic 27% low, and on an ordinary flat lens 77% low — against a cover that is known exactly, because the clouds here are caps whose solid angles add. Weighting each pixel by the surface's area scale repairs every one of them to better than a fifth of a per cent.
No view draws a curved plate true
The auxiliary view is descriptive geometry's answer to a foreshortened plane — turn until the plane is parallel to the paper and it draws at true shape. A bent plate has no such direction and a dished one has none twice over: the best view of the first is out by 1 − cos w and the best of the second by 1 − cos²w, worse by exactly 1 + cos w, because its normals need two parameters rather than one.
The stretch decides the band
A design band chosen by geometry — the rays that meet the object — includes rays that graze along it, and a grazing ray lands two design points twenty times further apart than the design says. Cap the stretch at four and a bare floor keeps 55 per cent of its design, a corner 80, and the top of a descending flight all of it.
A picture that can be printed
Two screens are fed the surface that is exactly right for each, so nothing about the viewer is left in the answer. What remains is whether the picture can be made flat before it goes up — and a cylinder unrolls while a sphere does not, so the dome's picture is stretched by eighteen per cent between its middle and its rim before anybody sits down.
Where a surface spends its pixels
A picture surface is a budget before it is anything else, and the six named ones distribute the same marks over the same directions quite differently. The flat plane lays 25.0 times as many on a square degree at the edge of a 70° field as on one at the centre; the equal-area fisheye is flat to 8.3e-6 per cent.
A pole is a line
An equirectangular surface sends one direction to a whole edge, so an 8° cap of sky at the pole takes 4.50 per cent of the marks against a 0.49 per cent share of the world. The worst singularity is not the pole at all — the equidistant fisheye's antipode costs ×17.9 — and only the cube map, which never holds a sphere in one chart, is bounded.
One face, one scale
A design carried from a point onto a flat face varies in scale by nearly six across that one face; the same design carried along a direction varies by 1.000000000. The stretch is reported here as the two singular values of the local map rather than as one directional difference, which is the honest form and which the previous round owed.
The kink at a seam
A cube map is six flat pictures, so every great circle is drawn exactly straight inside a face — to 3e-15 of its chord — and breaks at the join. The break is a kink and not a bend, it is exactly zero at a seam's midpoint whatever the slant, and it is bounded by 2·atan(½) = 53.130° at the corner.
Shot on one surface, shown on another
Two picture surfaces are two charts of the same pencil of rays, so a reprojection between them is a change of coordinates and loses no geometry at all — bit-exact at all 408 sampled directions. What it costs lies elsewhere — 70 per cent of the source has nowhere to go, and the target wants ×5.49 the marks at its edge.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
AnisotropyPicture surfaceDemonstrationfield of viewFisheyeConformalDrawing systemEquirectangularEquidistantForeshorteningJacobianSolid angle