Visual Plane
every picture states the point it is correct from
Surfaces that are not flat

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 other surface this site has.
19 min read 14 figures Taught, and never measuredThe surface is a choiceProjected, not constructed

Worth reading first: When the picture surface is not flat · Stereographic keeps every angle, and only stereographic.

There is a way of drawing very wide views that gets taught in art schools and never appears in any account of projection. It is usually called curvilinear or five-point perspective, and its central instruction is a single sentence: draw every straight edge of the world as a circular arc, bowing away from the centre of the picture, with the arc’s ends meeting at the two vanishing points on either side.

It is a good recipe. Pictures drawn that way look right in a way that a very wide flat picture does not, and a draughtsman can execute it with a compass and no arithmetic.

It has also never been told what surface it is. The books that teach it say the arcs are circles because straight lines “curve” in wide vision, which is an explanation of nothing, and the question of whether some picture surface produces exactly those arcs is not asked. This essay asks it, and the answer is a single surface.

Five great circles, imaged on the stereographicEach is fitted as a general conic and comes back a circle: |A−C|+|B| is 1e-9 of the fit's own scale. Stereographic is the only surface here that does this.fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples
Fig. 1 Five straight world lines imaged on the stereographic surface, each fitted as a general conic. A conic is a circle when A=CA = C and B=0B = 0; the fit reports ∣A−C∣+∣B∣|A - C| + |B| at 1e-9 of its own scale. Not approximately a circle — a circle, to the precision the fit can resolve.

What the question means

A straight line in the world is a set of directions from the eye, and that set is a great circle on the sphere of view. So asking what a picture surface does to a straight line is asking what it does to a great circle, and the question has an exact form for every surface at once.

The measurement uses machinery this site built two phases ago. Sample the great circle’s image, fit the general conic Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 to the samples by the same eigen-decomposition the circle-centre figures use, and ask how far the fit is from A=CA = C, B=0B = 0 — which is the condition for a conic to be a circle, up to the overall scale a fit is free in.

One measurement, and it gives a different correct answer on each surface. On a flat plane the fit reports a degenerate conic, which is what a straight line is. On stereographic it reports a circle. On everything else it reports a conic that is neither.

Five great circles, imaged on the flat planeEach images as a straight line, which is what the flat plane does to every great circle and is the reason a photograph's straight edges are straight.fitted conic: a straight line to 4e-16 pxflat planethe samples, fitted
Fig. 2 The same five lines on the flat plane. The fit reports a straight line, which is why photographs have straight edges and why the recipe is not needed for a narrow view.

The answer

Stereographic returns 9.7e-10. Every other surface in this site’s family returns 1.5 on the same lines, at the same tilt, from the same fit.

That is not a difference of degree between two candidates. It is a zero and a number of order one, and the gap between them is nine orders of magnitude.

So the recipe names stereographic projection, exactly and uniquely among the surfaces here. A draughtsman drawing straight edges as circular arcs is drawing a stereographic picture, whether or not the word has ever been used in the studio.

Five great circles, imaged on the equidistant fisheyeThe same conic fit reports 1.5e+0 — not a circle, and not a line either.fitted conic: 1.533 from being a circleequidistant fisheyethe samples, fitted
Fig. 3 The equidistant fisheye — the surveyor’s projection, in which radius is proportional to the angle off axis. Its images of straight lines are not circles and no compass draws them, and the same conic fit that calls stereographic’s a circle reports 1.5e+0 here.
Five great circles, imaged on the cylinderThe same conic fit reports 1.5e+0 — not a circle, and not a line either.fitted conic: 1.482 from being a circlecylinderthe samples, fitted
Fig. 4 And the cylinder, which is what a panoramic camera produces. Also not circles, by the same measurement — so the recipe is not a general fact about wide views, it is a fact about one surface.

What the fit is and is not doing

One objection has to be closed before the number means anything, because a conic fit is exactly the sort of measurement that can be made to say what its author wants.

The fit is not asked whether the image is approximately a circle. It is given six free coefficients — enough to describe any conic at all — and returns the one that best fits the sampled image with no preference for circles built in. The quantity reported, ∣A−C∣+∣B∣|A - C| + |B| divided by the fit’s own scale, is zero for a circle and nonzero for every other conic, and the fit has no way to make it small except by the image being one.

Nor is the fit given a small arc. The samples span a wide arc of the great circle, which matters: any smooth curve looks like a circle over a short enough piece, so a fit over a small window would return near-circularity for every surface and measure the window rather than the map.

And the same fit, unchanged, reports a straight line on the flat plane and a conic that is neither on the four remaining surfaces. A measurement that returned “circle” for everything would be worthless; this one returns three different answers on the same input, and all three are correct.

Why stereographic and nothing else

The reason is a property this site has already measured from a different direction, and it is worth connecting the two because they are usually taught as separate facts.

Stereographic projection is circle-preserving: every circle on the sphere maps to a circle in the plane, and lines are the limiting case. That is a stronger statement than conformality and it is what makes the recipe possible — great circles are circles on the sphere, so their images are circles in the picture.

Conformality follows from circle preservation rather than the other way round, and this site measured the conformality three phases ago: every right angle survives, everywhere, to arithmetic noise, and the anisotropy is exactly one. Nothing else in the family does either half.

So the studio recipe and the mathematical property are one statement. Draw straight edges as circular arcs and the picture is conformal — which is a considerably more interesting claim than the recipe makes for itself, because it says the drawing preserves every angle in the world and not merely that it looks acceptable.

Straight, or conformal — the corner that is emptySix picture surfaces plotted by how much they bend a straight line (across) against how far they are from preserving shape (up). The plane sits on the left edge and the top; stereographic on the bottom and the right. Nothing sits in the bottom-left corner, and by Beltrami's theorem nothing can.straight AND conformal — empty10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁰10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁰10¹10²bend of a straight line, as a fraction of its own chorddeparture from conformal (degrees, or % of anisotropy)planecylinderstereographicequidistantequal-areaequirect.sampled over a 120° fanlower left would be a surface with no cost
Fig. 5 The trade the family makes, plotted. The plane sits alone at the straight edge and stereographic alone at the conformal one, and the corner where a surface would be both is empty — by Beltrami’s theorem, and checked against the surfaces that exist.

What the recipe leaves out

Naming the surface immediately supplies three facts the recipe does not, and each of them is checkable against a drawing made with a compass.

Which circle. The recipe says the arc’s ends meet at the vanishing points and does not say what radius the arc has in between, so a draughtsman chooses by eye. The surface determines it: a great circle at a stated orientation has one image, and the fitted circle’s centre and radius come out of the map. A drawing whose arcs bow more or less than that is not a stereographic picture of anything.

Where the vanishing points are. In a stereographic picture the two ends of a world line’s image are the images of its two points at infinity, and they are at a stated angular distance apart — 180° of the sphere, mapped through the surface. The recipe treats their placement as a compositional choice.

How wide the picture goes. Stereographic images every direction except one, so a curvilinear drawing can in principle cover everything but a single point behind the draughtsman. That is an unusual property, and a recipe that stops at “wide” cannot say it.

One room at 120° across, cast onto six picture surfacesEach panel holds the same angular width of the same scene. The flat plane keeps every straight line straight and pays for it at the edges; the five curved surfaces bend the ground lines by 3.5% to 6.0% of their own length.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
Fig. 6 The same room on six surfaces, each scaled so the same angular width fills the same width of page. The recipe’s picture is the third panel, and the comparison is what makes it a choice rather than a correction.
How wide the picture gets as the field of view opensOn a flat plane the picture's half-width is tan(θ/2): it multiplies by 6.6 between 120° and 170° and is unbounded at 180°. On a cylinder it multiplies by 1.42 over the same range and keeps going past 180° without incident.0246850100150field of view across the picture (degrees)half-width of the picture, in focal lengthsplanecylinderstereographicequidistantcut off at eight focal lengthsthe plane crosses it at 166°
Fig. 7 And the reason a recipe of this kind exists at all. The flat picture’s half-width runs away as the field widens, so a very wide view has to be drawn on something else, and the recipe is a way of doing that without arithmetic.

The cost, which the recipe also does not state

Nothing is free, and the family’s trade says exactly what a curvilinear drawing pays.

Stereographic’s angular error is zero and its anisotropy is one; what runs away is area. Over a fan reaching 160° off axis, the area scale runs over a factor of 255, and taken to 176° over a factor of 4762. A shape drawn near the edge of a wide curvilinear picture is drawn at the right shape and at wildly the wrong size, and there is nothing in the drawing to say by how much.

That is the opposite failure from a flat wide picture, which keeps area better and destroys shape. The recipe’s picture and a wide photograph are two different answers to the same impossibility, and neither is the corrected version of the other.

Every angle exact, and unrecognisableA 360 photograph re-projected stereographically from below. Every crossing in the original crosses at the same angle here — the worst departure over 160° of the sphere is 4.4e-8° — while the area scale runs over a factor of 255. Conformal is not a synonym for undistorted.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
Fig. 8 The extreme case of the same surface. Every angle is exact to 4.4e-8° and the area scale runs over a factor of 255 — which is what a reader calls distorted, and it is the third row rather than the first.
Area scale across the picture, for six surfacesThe flat plane's area scale is sec³θ and leaves the frame before 70°. The equal-area fisheye is the flat line at 1, which is what its name asserts and what this measures.024680204060angle off the optical axis (degrees)area scale, relative to the centreplanecylinderstereographicequidistantequal-areaequirect.measured by differencing on the spherethe plane leaves the frame
Fig. 9 The area cost across the field, for the whole family. Stereographic’s curve is the one that climbs, and a curvilinear drawing’s edges sit on it.

What a draughtsman would have to do to be exact

Naming the surface turns the recipe into an algorithm, and the algorithm is short enough to state.

To place a world line’s arc: take its two points at infinity, map both through the surface — which for stereographic means a direction d\mathbf{d} goes to 2(dx,dy)/(1+dz)2(d_x, d_y)/(1 + d_z) in focal-length units — and mark them. Those are the arc’s ends, and they are the recipe’s two vanishing points, now computed rather than placed. Then take any third point of the line, map it, and draw the unique circle through the three. The radius follows; it is not a choice.

Two things about that procedure are worth noticing.

It needs no fourth point and no adjustment. Three points determine a circle, so a line’s whole image is fixed by three of its points — which is the same economy a straightedge has on a flat picture, where two points fix a line. A curvilinear drawing is therefore constructible in the same sense a linear-perspective drawing is, with a compass in place of a ruler and three points in place of two.

The two vanishing points are not free. In the recipe they are placed where the composition wants them; here they are the images of two antipodal directions, and their separation in the picture is determined by the line’s orientation relative to the axis. A drawing that placed them by eye has arcs that are circles and are not the images of any world line.

That second point is where a hand-drawn curvilinear picture actually departs from the surface it is imitating, and the departure is invisible in the drawing because every arc is still a circle. It is the same shape of error the site’s wrong field measures for the taught two-point cube: the construction produces something of the right kind and the wrong parameters, and looking at it does not help.

The straight lines a curvilinear picture still has

There is a detail the recipe gets right without saying so, and it is worth checking because it is the one place a compass drawing agrees with a straightedge.

Not every world line images as an arc. The lines whose images run through the picture’s centre come out straight — a circle through the projection point is a line — and on this surface those are exactly the lines whose direction passes through the optical axis. They form a one-parameter family, and every member of it is drawn with a ruler.

That is the same family every azimuthal surface has, and the essay that counts them finds that the count separates the plane from everything else and cannot separate the three fisheyes from each other. Here the practical form is simple: in a curvilinear drawing, the radial lines are straight and everything else is an arc, and a draughtsman who ruled a radial line has done the right thing.

The lines each surface leaves aloneA curved picture surface does not bend everything. Each panel draws the family of world lines the surface images as straight lines: two-dimensional for the plane, and a one-parameter family for every other surface here — running through the picture's centre on an azimuthal surface, and parallel on a cylindrical one.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
Fig. 10 The families drawn. Under stereographic they run through the picture’s centre and meet to 1e-12; on the cylinder they are parallel and never meet. A curvilinear drawing’s ruled lines are the third panel’s.

Where the recipe came from, and what it does not license

The construction has a history and this site takes no position on it. What can be said geometrically is narrow and worth saying precisely.

Somebody working by hand, wanting a very wide view, and unwilling to accept the flat picture’s edges, reached for the one curve a compass draws. That the curve happens to be exactly right for one of the six surfaces here is either a discovery made by eye or a coincidence, and the geometry has no standing to decide which. What it can say is that the recipe is not an approximation to something better: it is exact for a surface, and the surface is the one with the strongest preservation property in the family.

What the measurement does not license is the claim the recipe is usually attached to — that wide human vision is curvilinear, or that the arcs are what a viewer perceives. Those are questions about eyes, and this site has said since its foundation that it computes the geometry of pictures. The claim here is about a drawing, and it is that the drawing is a stereographic projection.

The same 84° view, projected onto a plane and onto a cylinderOn the plane every straight line stays straight (2e-13 px of bend) and the edges stretch; on the cylinder the stretch is even and straight lines bow by up to 863 px.flat picture plane — straight lines stay straightcylindrical picture surface — even stretch, bowed linesone scene, two picture surfacesneither is the distorted one
Fig. 11 The comparison the recipe exists to escape, drawn honestly. Neither picture is the distorted one; the flat one pays at its edges and the curved one pays in its lines, and choosing between them is choosing which cost to accept.
One family, one parameter, and where the convention sits in itThe same room on five members of the Panini family, each scaled so 130° of the world spans the same width. Every member keeps a vertical line vertical; only d = 0, the flat plane, keeps a general one straight. The worst angular error over the field is smallest at d = 1.00, at 16.84°.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
Fig. 12 And the compromise family, which is what the same problem looks like when it is solved with a parameter instead of a compass. Every member keeps verticals straight, none keeps a general line straight, and the value everybody uses is the one that minimises the worst angular error.
Six flat pictures, and what happens where two of them meetEach face is a flat picture at 90°, so a straight line inside one is drawn exactly straight — 1e-15 of its chord. Across a seam the two straight pieces meet at 1.80°. The shading is the area scale, which runs from 1 at a face's centre to 5.196 at its corner, with an anisotropy of 1.7321 there.leftfrontrightbackupdownacross the left/front seam: 1.80°, with each side straight to 7e-16corner area ×5.196anisotropy √3 = 1.7321 there
Fig. 13 The third answer: six flat pictures rather than one curved one. A straight line is exactly straight inside a face and kinks across a seam, which is a different failure from bending and the only one a compass cannot help with.

The recipe’s own boundary

There is a width past which the recipe stops being executable, and it is not the width at which it stops being exact.

Stereographic has an image for every direction but one — the point directly behind the draughtsman. As the field widens toward that point the picture’s radius grows without bound, in the same way a flat picture’s does toward 180°, and the half-width of the drawing runs away. So the recipe is exact everywhere and drawable over rather less than everything, and where it stops being drawable depends on how large a sheet is available rather than on any property of the surface.

That is a different kind of limit from the flat picture’s. A flat picture cannot represent 180° at all: the directions at exactly a right angle to the axis have no image, and the surface has run out. Stereographic has not run out anywhere except at a single point, and what has run out is the paper.

The practical consequence is the one a draughtsman meets: a curvilinear drawing covering, say, 250° is perfectly well defined and is several times the size of one covering 120°, at the same scale near the centre. Nothing in the recipe warns of it, because the recipe has no scale in it at all.

The cube map has a scale below which everything is straightThe fraction of world lines a cube map draws exactly straight, against how long a piece of the line is drawn. It falls from 95.8% at a 5° arc to 2.8% at 50°. Every other surface here gives the same answer at every length.0%25%50%75%100%0°10°20°30°40°50°world lines drawn exactly straightlength of the piece drawn, in degrees of arcsampled over 24×24 directionsno other surface here has a scale
Fig. 14 The comparison that makes the point. A cube map’s straightness depends on how long a line is drawn, which is a scale in the surface; a curvilinear drawing’s arcs are exact at every length, and what depends on the width is only how much paper it takes.

The short version

The taught curvilinear construction says to draw straight world lines as circular arcs. Fitting a general conic to the image of a straight line returns a circle to 9.7e-10 under stereographic projection, and returns 1.5 — a conic that is neither a circle nor a line — under every other surface in this site’s family.

So the recipe names stereographic exactly, and inherits its properties: every angle in the world survives, every anisotropy is one, and the area scale runs over a factor of 255 across a 160° field.

A compass drawing of a wide view is a conformal projection of it. That is a considerably stronger statement than the recipe makes for itself, and it comes with a cost the recipe never mentions.

Where this goes next

fitted conic: a circle to 1e-9stereographicdashed: the fit, not the samples 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. 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 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. 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.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

AnisotropyArea scaleBeltramiCircle preservingConformalConic fitDemonstrationDrawing systemEquidistant projectionfield of viewGnomonicPicture surfaceStereographic projectionStraight familyTaught and unmeasured