An inverse perspective is a leaning plane

This site has one operation for questions of this kind, and it has had it since the foundation phase. A drawn quadrilateral is the image of exactly one planar shape, up to scale, given a focal length and a principal point. Back-project its four corners onto the plane they came from and read off the sides and the angles.

Applied to the taught two-point cube it returns a parallelepiped. Applied to a divergent construction it returns something the standard accounts do not predict.

The recovery says: a rectangle, at every splay #

Run depictedRectangle on the constructions from the previous essay and the answers are almost boringly uniform.

Convergent, splay 0.62: four corners at 90.000000°, near edge equal to far edge to twelve decimal places.

Divergent, splay 1.32: four corners at 90.000000°, near edge equal to far edge to twelve decimal places.

The splay is not a taper. A divergent construction does not depict a trapezium that widens with depth, which was the expected answer and is the one the “vanishing point behind the viewer” story implies. It depicts a rectangle, exactly as a convergent construction does.

What differs between them is what the plane is doing.

The tilt is what the splay sets #

The reconstruction fixes the plane’s normal from the two in-plane directions the drawing determines, and the angle between that normal and the optical axis is the plane’s tilt.

Sweeping the splay: at 0.62 the recovered plane leans 53.5° away from the viewer; at 1.32 it leans 36.3° toward. The map is smooth on either side of the splay of 1, where the construction is a parallel projection with no tilt to report and the reconstruction correctly refuses.

So the free parameter a one-point construction offers is not the shape of the thing depicted — that is pinned to a rectangle by the two parallel edges — but the inclination of the surface it lies on. Convergence is a surface leaning away; divergence is a surface leaning toward.

Why one pair of parallel edges forces the shape #

The reason the shape comes back a rectangle every time is worth working out, because it is not obvious and it is the whole mechanism.

A one-point construction has its near and far edges parallel on the page. Parallel on the page means their vanishing point is at infinity, and a point at infinity in image direction $(d_x, d_y)$ corresponds to the camera-space direction $(d_x, d_y, 0)$ — a direction perpendicular to the optical axis.

So one of the plane’s two in-plane directions is forced to be perpendicular to the axis. The other comes from the sides’ vanishing point and is whatever the splay makes it. The plane’s normal is their cross product; and because the first direction has no depth component, the reconstructed near and far edges are both parallel to it and both get the same scale factor from the back-projection — which makes them the same length.

Four right angles follow for the same reason: the two in-plane directions come out perpendicular whenever the sides’ vanishing point lies on the axis, which a symmetric construction guarantees.

This is a satisfying piece of geometry and it also explains why the finding is not visible without doing the computation. Everything about it lives in the reconstruction, and the reconstruction is a step nobody takes when the question is “why do these lines spread”.

The repair this forced in the site’s own machinery #

There is a piece of housekeeping here that is worth recording because of what it says about coverage.

depictedRectangle has been on this site since the foundation phase and had never been asked about a one-point construction. It required both pairs of opposite edges to meet, and asserted rather than returning when either pair was parallel — so it refused the commonest kind of perspective drawing there is, whose transversals are horizontal and parallel on the page by construction.

Every quadrilateral it had ever been handed happened to be two-point, so the refusal never fired and the limitation was invisible. It took thirty seconds of this field to hit it.

The repair is not a special case: a pair of edges parallel in the image is a vanishing point, at infinity, and the direction it stands for is read off the edge itself with the same formula the finite case uses. That is now in lib/scene.js with the reason written into the docstring, and the case is in this phase’s gate.

The general form of the lesson is one the fleet keeps relearning: a function’s untested branch looks exactly like a function without that branch, and the way it gets found is a new question rather than a new test.

Two objections, both worth answering #

“The reconstruction assumes the four corners are coplanar.” It does, and that assumption is load-bearing. A drawn quadrilateral could be the image of four points in general position, in which case there is no plane and no shape to recover. What licenses the assumption here is that the object being depicted is a table top, a book or a step — a flat thing — and the recovery is answering if this is a flat thing, which flat thing is it.

That is the same conditional every one of this site’s single-view recoveries carries. The height from a photograph assumes the reference and the subject are both vertical; the rectification assumes the four marks are coplanar. A single view determines nothing without an assumption, and the useful thing is to state which one.

“A splay could also be a trapezoidal object seen straight on.” It could, and that is a genuine second solution rather than an objection. A trapezium that really is wider at one end, lying flat, projects with a splay too. So a drawn splay has at least a two-parameter family of explanations: a rectangle on a plane at some tilt, or a trapezium on a plane at some other tilt, and so on.

What breaks the tie is not geometry — it is that the other edges of the object usually settle it. A table drawn with a splayed top and parallel legs is not a trapezoidal table, because the legs would splay too. That kind of reasoning is what the multi-view field replaces with a second picture, and a single icon does not have one.

So the honest statement of the finding is conditional and remains sharp: among the flat rectangles, a divergent construction depicts one on a plane leaning toward the viewer, and the lean is computable. Nothing is claimed about which of the alternatives the painter had in mind.

The control is the argument #

Everything above is a computation on a drawing, and a computation on a drawing is only as good as the model behind it. The control is what makes it an argument about the world.

tiltedSquarePicture builds a square — a real one, in space, four equal sides and four right angles — on a plane tilted 45° toward the camera, and projects it with an ordinary pinhole from lib/camera.js. No construction, no drawing, no choice of splay.

The picture that comes out has its near edge at 155 px and its far edge at 197 px. A splay of 1.267. The effect that inverse perspective is supposed to require an impossible projection for is what an ordinary camera does to an ordinary square on a leaning plane.

And running the recovery on that photograph returns a square — ratio 1.000000000, corners $1.4 \times 10^{-14}$° from right angles — which closes the loop: the machinery that says a divergent drawing depicts a rectangle is the machinery that correctly identifies a real rectangle from a real photograph of one.

The sweep, and what the shape of it says #

Running the tilt from leaning-away through fronto-parallel to leaning-toward, and reading the splay off each photograph, produces a curve worth looking at.

At a tilt of 20° away, the splay is about 0.89 — mild convergence. At 45° away, 0.79. At 70° away, 0.73 and flattening: a plane leaning steeply away approaches an edge-on view and the splay approaches a limit rather than going to zero.

Through zero tilt the splay passes exactly 1, because a fronto-parallel square photographs as a square.

At 20° toward, 1.12. At 45° toward, 1.27. At 70° toward, 1.37.

Two things fall out. The curve is smooth through 1, so there is no qualitative transition at the parallel case in the world — the transition is in the picture’s vanishing point, which runs off to infinity and returns, while the object it describes turns continuously. And the splay is bounded, roughly between 0.7 and 1.4 for tilts within 70° of fronto-parallel, which is a useful sanity check: a drawn splay of 3 would not be a plane at any tilt and would need a different explanation.

Icon splays sit comfortably inside the bounded range, which is not evidence of anything by itself and does rule out the objection that the effect is too large to be a tilt.

What a lectern is #

The tilt at the splay icons typically use is around 30 to 40 degrees, and it is worth naming the objects that lean at that angle, because the list is short and specific.

A lectern. An open book on a stand. A writing desk with a sloped top. A footstool tipped toward the viewer. A dish or a paten presented rather than set down. A throne’s seat seen from below.

Those are, with some consistency, the objects in icons that show the divergence — and they are objects that in life are tilted toward a viewer, because they are made to be presented, read from, or offered. An open gospel book on a stand leans toward the reader by design.

This is not offered as an explanation of the convention, and it would be a poor one: the divergence in icons is applied more broadly than that list, it is often applied to objects that are plainly horizontal, and the tradition’s own account of what it is doing is not a geometric one. What the observation does establish is narrower and worth having: a divergent drawing is not a picture of nothing. It is a correct picture of a leaning plane, the lean is computable from the splay, and for a real class of objects the computed lean is the lean those objects have.

The ambiguity that remains, stated #

A single view determines shape up to scale and no further, and this reconstruction inherits that exactly.

The recovered rectangle’s proportions are determined; its size is not. The plane’s tilt is determined; its distance is not. And the whole reconstruction is conditional on the focal length and principal point assumed — a drawing carries no intrinsics, so the tilt reported is the tilt under the assumed camera, and a different assumed focal length gives a different tilt for the same drawing.

That last dependence is worth quantifying rather than waving at, and the next section does it: a longer assumed focal length makes a given splay evidence of a steeper tilt, and the amount is a single arctangent. A drawing with no camera is a drawing with a one-parameter family of interpretations, and every one of them is a rectangle on a plane at some tilt.

So the finding survives the ambiguity, which is the useful thing about it. Which leaning plane depends on what camera is assumed. That it is a rectangle on a leaning plane does not.

How much the assumed camera moves the answer #

The tilt is not read off the drawing; it is read off the drawing and an assumed focal length, and the arithmetic joining the two is short enough to write down.

The near and far edges are parallel on the page, so one in-plane direction is $(1, 0, 0)$ — perpendicular to the optical axis, as the mechanism above requires. The sides meet at a point $v$ pixels from the principal point along the page’s vertical, so the other in-plane direction is $(0, v, f)$. Their cross product is the plane’s normal $(0, -f, v)$, and the angle between that and the optical axis $(0,0,1)$ is

$$\text{tilt} = \arctan\!\left(\frac{f}{|v|}\right).$$

Only the ratio $f/|v|$ enters. The drawing supplies $|v|$; the assumption supplies $f$; nothing else in the picture matters at all.

That expression can be run backwards against the two tilts this essay already reports, which is worth doing because neither the focal length nor the principal point is quoted anywhere. The splay of 0.62 puts the meeting point at $y = -170$ and the recovered lean at 53.5°; the splay of 1.32 puts it at $y = 924$ and the lean at 36.3°. Two equations, two unknowns — a solve rather than a check, and it should be labelled as one — and they give a principal point at $y = 214.5$, which is the construction’s own centre, and a focal length of 521 px. The camera the reconstruction has been assuming all along is an ordinary wide one, about 47° across the construction’s height.

With it in hand the dependence is a table rather than an intuition. Holding the drawing fixed at the splay of 1.32, so that $|v| = 709.5$ px:

Halving the assumed lens gives a plane leaning twenty degrees; quadrupling it gives one leaning seventy. The recovered tilt is not a measurement of the drawing at all — it is a measurement of the drawing under an assumption that contributes more to the answer than the drawing does, and a drawn icon supplies no evidence whatever about which value to use.

Two things survive that, and they are the two the finding actually rests on.

The sign does not move. The lean is toward the viewer exactly when $v$ is on the far side of the principal point from the far edge, which is exactly when the splay exceeds 1. No focal length changes that, because $f$ enters only through its magnitude. So the previous essay’s classification is intrinsics-free, and this essay’s is not.

And the shape does not move. The rectangle came from the near and far edges being parallel on the page and from the construction’s symmetry, neither of which mentions $f$. Every assumed camera returns four right angles and equal near and far edges; they differ only in how steeply the rectangle is leaning. That is the exact form of what one picture can and cannot supply: proportions determined, everything with a length or an angle to the camera in it open.

The expression also says what the gap at a splay of 1 is and is not. As the splay approaches 1 from either side the meeting point runs off to infinity, $|v|$ grows without bound, and the tilt goes to zero — from both sides, continuously. So the map from splay to tilt has no discontinuity there and its value is a fronto-parallel plane. What the gap marks is a second reading available at that one splay and nowhere else: a drawing with parallel sides is equally the perspective image of an untilted rectangle and the parallel projection of a rectangle at any tilt at all. The figure leaves it blank because two answers is not the same as one, not because the curve breaks.

What this leaves for the next essay #

The field’s first two essays have taken one convention apart and found an ordinary object underneath it. That is worth doing once and is not a method, because it worked here for a reason that will not generalise: a divergent construction is a single quadrilateral, and a single quadrilateral is exactly the input this site’s oldest recovery function takes.

The conventions in the previous field are not like that. A handscroll is not a quadrilateral; a removed roof is not a quadrilateral; an aspective figure is several. Each needed machinery built for it, and each came back with a different kind of answer — a miss in metres, a spread in percentage points, a ratio of totals.

What the remaining essays do is put all of those answers on one table and ask whether the table says anything the individual measurements did not. The answer is one exclusion, one absence of an ordering, and a warning about where the columns came from.

The convergent end, for symmetry #

The sign is the only thing that separates the two cases, and the essay is worth ending on the other one. A construction whose sides meet above the far edge — a splay below 1 — hands the recovery a quadrilateral of exactly the same kind and gets back an answer of exactly the same kind: four right angles, near edge equal to far, and a plane leaning away rather than toward. Nothing in the machinery distinguishes them, which is why the classification the previous essay makes from the drawing alone survives every assumption this one has had to add on top of it.