The arcs a curvilinear drawing uses

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.

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 $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ to the samples by the same eigen-decomposition the circle-centre figures use, and ask how far the fit is from $A = C$, $B = 0$ — 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 — and the fisheyes are a family, not one rule — it reports a conic that is neither.

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.

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, $|A - C| + |B|$ 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.

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.

Which circle, exactly #

The first of the three omissions can be closed completely, and the answer is short enough for a draughtsman to use.

Measure the picture in units where the 90° cone images as the unit circle — which for stereographic is the natural unit, since a direction $\theta$ off axis lands at radius $\tan(\theta/2)$ and $\theta = 90°$ gives exactly 1. A straight world line is a great circle, and a great circle is fixed by the angle $\beta$ its plane’s normal makes with the view axis. Taking the two ends of the image along the diameter through the picture’s centre, at radii $\tan(45° \pm \beta/2)$, gives the image circle’s centre and radius directly:

$$c \;=\; \tan\beta, \qquad R \;=\; \sec\beta.$$

So the arc’s bow is not a matter of taste at all: one angle fixes both numbers. And the two are not independent, because

$$R^{2} - c^{2} \;=\; \sec^{2}\beta - \tan^{2}\beta \;=\; 1.$$

Every straight world line images as a circle that cuts the 90° circle at right angles. That single identity is the whole of “which circle”, and it is checkable with a compass: draw the boundary circle of the 180° field, and every edge in the picture must be an arc orthogonal to it.

A reader who recognises that condition has met it elsewhere. Circles orthogonal to a fixed circle are the geodesics of the Poincaré disc, so a curvilinear drawing’s arcs are hyperbolic straight lines, and the studio recipe is drawing in a non-Euclidean geometry without saying so. The two limiting cases are the ones the recipe already knows: $\beta = 0$ gives $c = 0$, $R = 1$ — the boundary circle itself, which is the image of a line at 90° all the way round — and $\beta \to 90°$ sends both to infinity, leaving a straight line through the picture’s centre, which is why radial edges are drawn straight and is the straight family this surface keeps.

The second omission goes the same way. A line’s two points at infinity are antipodal directions, and antipodal directions image at radii $\tan(\theta/2)$ and $\cot(\theta/2)$ on opposite sides of the centre — a pair whose radii multiply to one. The two vanishing points of any edge are inverse points in the 90° circle, so placing one places the other, and a drawing that treats their positions as a compositional choice has at most one of them right. That is the same statement where parallel lines meet makes on a flat plane, transported to a surface where the two images are at finite distance rather than one of them being off the page.

None of this is available from the recipe, and all of it follows from naming the surface — which is the argument for naming it. The recipe is a good drawing procedure and a bad description; once the surface is identified, the procedure acquires a compass construction, a placement rule for its vanishing points, and the conformality measured separately as the reason it looks right. What it still does not acquire is a size, and that is the price the family charges rather than a gap in the recipe.

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.

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 $\mathbf{d}$ goes to $2(d_x, d_y)/(1 + d_z)$ 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.

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 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 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.