The arcs the five-point construction actually draws

The arcs a curvilinear drawing uses fitted a general conic to the image of a straight world line under six named picture surfaces and found exactly one surface — stereographic — returns something close enough to a circle to matter, to nine decimal places. That settled what the technique of drawing straight edges as arcs amounts to as a mapping. It did not test the specific taught recipe for producing those arcs by hand: five vanishing points, a straightedge and a compass, and no mention anywhere in the instruction of what scale the radius is supposed to carry.

Which rule a fisheye obeys, from straightness alone built a fit that takes marks, is told which sets came from straight edges, and returns which of the four candidate rules — equidistant, equal-area, stereographic, orthographic, set out in full elsewhere in this collection — straightens them. That fit does not need a photograph. It needs marks and a claim about straightness, and the five-point construction supplies both without ever having passed through a lens.

Put the two together and the taught construction turns out to be exact — not approximately, to the arithmetic floor — under one specific radial scale, and off by several pixels under the one draughtsmen are actually told to use. The construction was always drawing a real projection. The instruction that accompanies it never named which one.

That gap between what a technique does and what its own instruction says it does is worth taking seriously rather than treating as pedantry, because the arcs a curvilinear drawing uses already found the same shape of gap in the surrounding practice: curvilinear perspective has been taught, as a recipe for handling a wide field, for sixty years without the surface it amounts to ever being named. This essay narrows that finding from the technique in general to the single most common named construction inside it, and asks the sharper question a general conic fit cannot: not merely “is there a surface these arcs resemble,” but “does the specific instruction a student is given — mark five points, graduate the radius evenly — actually produce it.”

What the construction specifies, and what it leaves to a draughtsman #

The recipe itself is short. Mark the horizon and a vertical through the centre; put a vanishing point for “straight ahead” at the centre of the disc, and vanishing points for up, down, left and right on the rim, ninety degrees from the centre in every direction. Any other straight direction in the scene gets a vanishing point somewhere on the disc, and the line it belongs to is drawn as the circular arc through that point and the two points already fixed for its own plane. Compass and straightedge suffice for every step.

What the recipe never specifies is the one thing a projection needs to be a projection: a rule assigning a radius on the disc to an angle off the forward direction. A draughtsman using an ordinary protractor or a scale ruled in equal divisions reads the disc as though radius were proportional to angle — which is the equidistant rule, chosen not because the construction asked for it but because it is the only scale an unmarked ruler can supply. That silent substitution is the entire subject of this essay: the construction fixes the arcs precisely, by a compass-and-straightedge procedure with no ambiguity in it, and leaves the scale to whatever happens to be lying on the drawing table.

This is a slightly different failure than the one measured for the technique in general. A general conic fit asks whether the arcs, as a set of curves with no radius attached at all, resemble a circle under some surface — a question with no scale in it to get wrong. Attaching numbers to the disc is a second, separate step, needed the moment anyone wants to read a distance, an angle, or a vanishing point off the finished drawing rather than merely admire its outline, and it is exactly the step the taught recipe is silent about.

Two readings of one drawing #

The arcs drawn above do not change when the assumed scale changes; only the angle a given radius is taken to represent does. That makes it possible to ask, of the identical set of drawn arcs, how far each of two candidate readings departs from the line’s true image.

Both curves start and end at the same two points for the same reason: on the axis and at the rim, every candidate scale agrees, because a diameter and the rim itself are landmarks no radial rule can disagree about. Between those two points the evenly graduated reading pays a real and growing price — 3.75 pixels at its worst, on a disc where the earlier essay’s photographic fit could resolve a rule from marks two orders of magnitude smaller — while the stereographic reading pays nothing at any tilt. One drawn object, two ways of attaching numbers to it, and only one of those two ways is describing what the compass actually put on the page.

The asymmetry between the two curves is worth dwelling on, because it is easy to misread the figure as showing two comparably good approximations. It does not. One reading is exact and the other is not; there is no tilt anywhere on the sweep where the evenly graduated curve does better than the stereographic one, and the gap between them is monotone through most of the range rather than merely occasional. That is the signature of a wrong assumption rather than a noisy measurement — a genuinely approximate method would scatter around some middle ground, and this one instead tracks a smooth, predictable curve away from a zero that a different reading reaches everywhere.

The independent check: fitting the four rules blind #

The conic fit two rungs below this one already said stereographic was the surface that makes sense of arcs-as-a-technique in general. The departure curve above says the specific five-point recipe’s arcs match it too, at the right scale. Neither of those is the identifiability fit this collection built for exactly this kind of question, so it is worth running that fit as a third, independent check.

A fit reaching the floor at exactly R/2 is not a coincidence dressed up as a measurement; it is the fit discovering, with no scene and no camera given to it, the same scale the departure curve found by direct comparison against a known line. Three separate routes — a general conic fit two rungs below, a direct comparison against a known true image, and a blind four-way identification — now agree on the same answer, which is the standard this site holds a claim to before calling it settled rather than merely observed once.

The reason is one line of geometry rather than a numerical accident. Two of the construction’s five vanishing points — up and down, or left and right — sit at opposite ends of a diameter of the disc, which is what a pair of perpendicular directions gives under any azimuthal scale. Under the stereographic rule specifically, two such antipodal points on the disc unproject to two antipodal points on the sphere of directions, and a plane through a point and its antipode always contains the sphere’s centre — so the circle that plane cuts is a great circle, and a great circle is exactly the set of directions a straight world line occupies. The construction’s arcs are great-circle images because two of the five points it is built from are antipodal, and stereographic is the one rule under which that antipodal relationship survives onto the page as a diameter.

That “antipodal points stay antipodal” property is not a fact special to this one construction; it belongs to stereographic projection generally, which is the same property stereographic keeps every angle already isolates from the other side, as the rule that preserves shape locally everywhere on the sphere rather than only along one favoured family of lines. What is specific to the five-point construction is only that a draughtsman’s own choice of layout — vanishing points for up, down, left and right, spaced a quarter-turn apart around the centre — happens to hand the construction two genuinely antipodal pairs for free, without anyone drawing the construction needing to know that antipodal points are the object being exploited.

The control: take three points away and there is nothing left to misread #

If the trouble above is specifically about curvature — about a disc trying to carry a sphere’s worth of directions — then removing the curvature from the question ought to remove the trouble along with it. The construction offers exactly that reduction on its own terms: run it with only the centre point and one pair of opposite vanishing points, and there is no longer a third direction for an arc to bend toward.

This is the control the whole essay needs, and it is worth reading as one. The five-point construction’s departure figures report a real, non-zero number precisely because there is a rule to get right and a rule the customary reading gets wrong. Take away the third vanishing point and the machinery that reported pixels of departure reports the floor instead — not because the construction improved, but because there is nothing left in a two-point drawing for a wrong scale to mismeasure. A flat plane has one honest reading of itself, and ordinary linear perspective is the special case where the disagreement this essay measures has nowhere to live.

No surface keeps everything is the reason a flat plane gets that one honest reading for free: it is the unique surface among the ones this collection measures that draws every straight world line as a straight line, so a flat two-point drawing needs no radial rule at all to be read correctly — a straightedge laid between two marks already is the correct instrument. The moment a third vanishing point is added and an arc has to be struck between it and its neighbours, that free ride ends, because now there is a curve rather than a line, and a curve has to be read at some scale for its shape to mean anything angular at all.

Why the customary reading is wrong by exactly this much #

The departure and stereo figures both report how much the evenly graduated reading misses by. What sets the size of that miss is the same fact which rule a fisheye obeys already quantified for photographs: the four candidate rules for turning an angle into a radius agree closely near the centre of a field and separate substantially toward the edge.

Reading a stereographic drawing on an equidistant scale is exactly the mismatch that 54 per cent divergence describes, applied to a construction rather than to a lens. Near the centre of the disc the two scales barely differ, which is why the axis lines in every figure above come out exact regardless of which reading is used; toward the rim they diverge substantially, which is why the departure curve’s worst point sits at an intermediate tilt rather than at either end. The five-point construction is usually recommended precisely for fields wide enough that this divergence is not small, which makes the mismatch a property of exactly the use case the technique is chosen for rather than an edge case around it.

The account of why the two rules diverge as a cubic in the angle off axis applies here without modification, because it is a fact about the pair of rules and not about whether a lens or a compass produced the picture. Near the disc’s centre both rules agree to first order, for the same reason every rule in that family does; the cubic correction that separates them is small there and grows toward the rim, so a mismatch invisible on the axis line becomes, at forty degrees of tilt, several pixels on a disc of a few hundred — the exact size this essay’s departure figure reports.

How much is at stake at the field this technique is sold for #

The five-point construction earns its place in a draughtsman’s toolkit at fields ordinary linear perspective cannot reach at all — well past a hemisphere, where a flat plane’s picture would need to run to infinity. It is worth seeing what the scale confusion costs at a field in that range rather than only at the 90° used above.

Equidistant and equal-area — the pairing behind a mirror ball’s own picture — are not the pair this essay’s own construction confuses, which is equidistant against stereographic, but the figure is the right instrument for the general point, because it is the same machinery run at a field past a hemisphere rather than at 90°. Whatever gap two candidate rules carry at a given field, that gap only grows as the field opens further, which is exactly the direction the five-point construction is pushed toward in practice. A technique recommended for its very wide reach is a technique whose scale-reading error is recommended right along with it, growing rather than shrinking with the field that makes the technique worth using at all.

The lines every azimuthal surface agrees about #

One more question is worth closing off before the essay’s limits: why do the two axis lines come out exact under every scale in every figure above, when nothing else does? The lines a surface leaves alone answers it for picture surfaces generally, and the same answer explains the construction’s two exact directions without needing a separate argument.

A line whose plane holds the optical axis is exactly the family every azimuthal rule — equidistant, equal-area, stereographic, orthographic alike — images as a straight line through the centre, because every one of those rules is built from the angle off the axis alone and such a line’s image inherits the world line’s own straightness by symmetry. The five-point construction’s up-down and left-right diameters are drawn as straight lines precisely because they belong to that family, and every candidate scale draws them correctly for the same reason every candidate rule agrees at the centre of the laws figure above: the disagreement between rules lives entirely off that one special family, which is exactly where the departure curve’s zeros sit.

What this does not settle #

The measurement above is about where a set of arcs, already drawn, actually sit relative to a straight line’s true image. It says nothing about several things a full account of the technique would need.

It says nothing about whether a wide-field picture built this way looks more natural to a viewer than the same field forced onto a flat plane, which is the usual argument made in its favour and is a claim about perception this collection has no machinery for — the same limit the eye is a picture surface too states for a retina rather than for a drawing. It says nothing about a real hand-drawn construction’s compass and measurement error, which would sit on top of the exact-arcs case measured here rather than replacing it: a draughtsman’s actual arc, struck slightly off centre or read against a slightly warped sheet, adds its own residual to whichever of the two curves above it is closer to, and nothing in this essay’s arithmetic bounds that separate source of error.

And it says nothing about constructions built from a different number of named points — seven-point or nine-point variants exist in the drafting literature — which would need their own fit rather than inheriting this one, because the antipodal-pair argument above is specific to vanishing points that sit at opposite ends of a diameter, and a construction with an odd arrangement of points need not have any such pair at all. A construction without an antipodal pair anywhere in its layout has no guarantee of landing on stereographic, or on any other single named rule, and the honest answer for one would need the same three-way check run again rather than assumed from this one’s result.

One method, a third subject #

The fit this essay leans on has now been run on a lens, on a schematic eye, and on a draughtsman’s construction that never passed light through anything. In every case the method is the same: take marks, accept nothing about their origin except which sets came from something straight, and ask which of four named rules straightens them. The eye is a picture surface too found the eye’s own geometry sitting near the equal-area rule, by coincidence of one animal’s proportions rather than by design. This essay finds the five-point construction sitting exactly on the stereographic rule, by a fact of its own geometry — two antipodal vanishing points — that holds regardless of what a draughtsman intends. Between the three, the fit has now told the difference between an approximate biological accident and an exact geometric one, which is a distinction no single figure of either kind could have shown on its own.

That distinction matters beyond this one construction, because taught geometry is full of instructions that specify a procedure and stay silent about what the procedure amounts to. A recipe that produces a correct answer for reasons nobody wrote into the recipe is not a lucky recipe; it is a piece of exact geometry wearing the clothes of a rule of thumb, and the fit this collection built for lenses is what let that geometry be read off a page that was never a photograph at all.