No solid casts an aspective figure

Assembled from several views measured what an aspective figure keeps: each part of a standing body drawn from the direction that identifies it best — head in profile, eye and shoulders turned frontal — so that a single compromise viewing direction would keep at most 58% of what the composite keeps of every part at once. The pond with its trees laid flat measured a related convention exactly, by treating it as a fold: a plan hinged open, every length exact, every dihedral angle spent. Both essays ask what the convention keeps. This one asks the question this collection puts to every drawing on the site — what solid, from what direction, projects to this — and answers it for a figure built, by construction, to have no single answer.

No orthographic direction projects a rigid body onto the marks an aspective figure draws, and the residual that proves it is not a small number chosen to look convincing — it is the minimum over every direction a full sweep can try. The better half of the finding is what remains once that negative result is granted: the figure is not a failed projection of nothing. It is an exact projection of a body cut and rotated at its own joints, and the rotations recovered from the marks alone are exactly the angles the convention names.

The picture #

The construction is five parts of one standing body — head, eye, shoulders, hips, legs — each a small rigid landmark set turned about its own joint by the angle the convention gives it: nought degrees for the head and the legs, a right angle for the eye and the shoulders.

The construction is orthographic rather than perspective throughout, which is itself a decision worth naming: an aspective figure is drawn with no vanishing point and no convergence anywhere in it, part by part, which is exactly the family of drawing this field has already measured as needing no camera at all. What makes the figure worth asking a question of is that each part carries its own turn, so a single rigid rotation applied to the whole assembly — the operation any orthographic photograph of a single real body would apply — cannot reproduce more than one part’s turn correctly at a time. The 2.6% is what is left over once the best possible compromise rotation has been found and applied to every part at once.

It is worth being exact about what is and is not rigid here, because the figure is not an arbitrary distortion of a body — every individual part is drawn correctly. Turning the head about its own joint by a stated angle is a rotation, and a rotation is an isometry: it changes nothing about the lengths and angles within that one part, so the head, taken alone, is drawn exactly as a real head would photograph from the direction the convention turns it to. The failure is not inside any one part; it is that the five isometries applied are five different rotations rather than one rotation shared across all five, and a single rigid body photographed from a single station point can only ever supply one shared rotation, whatever it is. An aspective figure is, part by part, a set of perfectly faithful local records assembled under a global rule no single camera obeys.

The question this collection always asks #

Every convention in this field earns the same test eventually: given only the marks a drawing makes, is there a rigid body and a viewing direction that would have produced them.

Answering it requires fitting an orthographic projection — a scale, a horizontal origin shared across every part, and a single rotation — to the aspective figure’s own marks, and reading off how far the fitted projection’s own points land from the marks it is being fitted to. That residual, 2.6% of the drawn figure’s own height, is the headline number this whole family of drawings is built to explain, and every figure below either measures its scale or accounts for it constructively.

The fit has three genuine degrees of freedom and no more: one shared scale, one shared horizontal origin, and one rotation, applied identically to every part. Seven numbers no picture can name catalogues, for a full perspective recovery, exactly which quantities a picture alone can never pin down regardless of how good the fit is — a gauge freedom rather than a measurement failure — and the same discipline applies here in a smaller way: nothing about this fit could ever recover where in the world the body actually stood, only the rotation and the internal proportions its marks are consistent with, because an orthographic projection is silent about absolute position along its own viewing axis by construction. What it is not silent about, and what the 2.6% residual measures precisely, is whether one rotation explains all five parts, which is a genuinely different question from where the body stood and one this fit answers with no gauge ambiguity attached to it at all.

The control: what a genuine single view fits to #

A residual of 2.6% means nothing without knowing what a true single-view drawing of a rigid body would score on the same test, so the essential control is to run the identical fitting routine on a body that really was drawn from one direction.

Seven point six times ten to the minus thirteen pixels is not “very good” in the sense that 2.6% is bad and this is merely better. It is the arithmetic floor of the fitting routine itself — the residual left over from floating-point rounding once the fit has genuinely nothing left to explain — and the aspective figure’s 2.6% sits thirteen orders of magnitude above it. That gap is the whole of the claim stated as one comparison: a real single-view drawing and an aspective figure are not close together on a scale of “how well does a rigid view explain this,” one of them is at the floor an orthographic fit can reach and the other is nowhere near it.

The floor holds at every angle, not just one #

A control that only works at one particular angle would leave open the possibility that 34° is somehow a special or lucky direction for the fitting routine, so the honest next step is to run the identical control again from a different direction entirely.

Both floors sit within a factor of two of each other — 7.6e-13 against 8.7e-13 pixels — which is exactly the kind of variation floating-point rounding alone should produce and nothing more. The control is not “34° happens to fit well.” It is “any single angle applied uniformly fits to the arithmetic floor,” which is the actual content of the claim that an aspective figure’s 2.6% residual is doing real work rather than measuring an artefact of the fitting procedure itself.

The constructive half: recovering the convention’s own angles #

A residual that proves no single view works is, by itself, only half an answer — it says what the figure is not. The better half is showing what it is, and the same fitting machinery, asked a different question, supplies it directly.

This is the finding the negative result was clearing the ground for. An aspective figure is not a rigid body drawn badly from one direction; it is an exact projection of a body that has been rotated at its own joints, and the rotations a regression recovers from nothing but the drawn marks are exactly nought, ninety, ninety, nought and nought degrees — the convention’s own list, to the last digit the fit can report. The 2.6% single-view residual and the 0.0° angle-recovery residual are two readings of the same object, and only together do they say what an aspective figure is: not one view, and not an arbitrary assembly either, but a specific, recoverable set of rotations about a specific, shared set of joints. Three views do not fix the solid measures the more familiar direction of this question — how many photographs of an unknown body are needed before its shape is determined — and this is close to its mirror image: here the body’s shape at each part is already known, and what is being recovered is the rotation, one angle at a time, from marks that were never meant to agree on a single one.

An ambiguity is not an uncertainty names a distinction worth being explicit about here, because the single-view search above and the angle recovery could, in principle, be confused for the same kind of finding read two ways. They are not. The single-view residual reports that no rotation fits well — a genuine non-existence, the same shape of finding as a least-squares system with no solution at all, rather than a family of nearly-equally-good rotations any of which would do. The angle recovery reports the opposite situation entirely: a uniquely identified set of five angles, each pinned down to the last digit a floating-point fit can report, with no family of alternative solutions competing for the answer. A search that found many directions tied for second-best, or a recovery that left the angles only loosely constrained, would both be ambiguities in that essay’s specific sense, and neither describes what either figure here actually shows.

The search, swept properly #

“No direction does better than 2.6%” is a claim about a minimum over every possible orthographic viewing direction, and a claim about a minimum proved by trying a handful of directions is not proven at all — the whole caution this essay’s own catalogue insists on. The proper version sweeps.

Two numbers from two different searches appear across this essay and they are not the same measurement: the 2.6% quoted for the aspective figure itself comes from a refined search — twenty steps followed by local refinement — while the 3.0% floor in this sweep comes from a coarser, thirty-six-step grid run for the purpose of drawing a full curve across every yaw rather than for chasing the single best value as far as it will go. The two numbers should not be read as disagreeing, and they should not be quoted as though they were one figure: 2.6% is this essay’s best estimate of the true minimum, obtained by refining around the region a coarse search finds promising, and 3.0% is what the same class of search finds without that refinement, at every yaw rather than only near the best one. What the sweep proves that the refined fit alone cannot is the shape of the whole curve — that nothing anywhere across the full circle of yaw comes close to the control’s own floor, which sits at 3.4e-3% under the identical coarse search. A single best-effort number could in principle be a fluke of wherever the optimiser happened to start; a swept curve with no dip anywhere near the control’s floor cannot be.

A different composite, the same ceiling #

The aspective figure’s own five parts are one particular composite. The claim that a single view cannot keep everything several perpendicular aspects keep at once is more general than this one body, and it is worth seeing measured on a different composite entirely.

This is the same finding as the sweep above, arrived at from the opposite direction on a different object. There, a specific body was fitted and found to have no direction reaching its own composite’s total; here, a family of composites is built directly from perpendicular aspects and the ceiling a single view can reach is derived algebraically before any search is run, and the search is only there to confirm the algebra rather than to discover the number cold. Both routes agree that a single orthographic view is bounded well below what a composite made of several genuinely different directions can keep, and the bound is a fact about how many mutually perpendicular directions are being asked for, not a fact peculiar to a standing human figure with a head and two legs.

The honest limit #

Every number in this essay is a statement about orthographic projection — a parallel view with no station point and no focal length. Nothing here says what the equivalent search would find under a perspective camera instead, where each part’s own distance from the camera would additionally scale its projected size, adding a further parameter the orthographic fit does not have to contend with and potentially finding a still-worse residual for a genuinely aspective figure, or conceivably a better one for an aspective figure drawn at a modest field of view where perspective and parallel projection nearly coincide. That comparison is not made here.

It is also worth being precise about what kind of recovery this is and is not, next to recovering the camera, which fits a station point and a focal length to a perspective picture’s own vanishing points. That recovery works because a perspective picture’s converging lines are themselves evidence of a specific station point, and the fit is well conditioned exactly where the convergence is strong. An orthographic fit has no convergence to read at all — every one of its rays is parallel by definition — so the evidence the rotation-and-scale fit above relies on comes entirely from the relative positions of the marks within and across the five parts, and what two parallel views leave free is the essay to read for what stays undetermined even when two such views, rather than one, are available: a parallel recovery is a different kind of problem from a perspective one throughout, not merely perspective’s problem run without a horizon.

Nor does anything above say how many joints, or which rotations, an aspective convention is free to choose. The five-part body here is turned by exactly the angles assembled from several views already measured — nought and ninety degrees only — and the angle-recovery figure confirms those specific values are recoverable to the last digit. A figure turned by some other set of angles at its joints would still fail the single-view test, for the same structural reason, but the recovered angles would of course be different ones, and nothing here claims every aspective convention in every tradition uses only right angles and rest.

What this is an instance of #

The question this essay asks — what solid, from what direction, projects to this — is the one three views do not fix the solid asks about an unknown body from known photographs, run here in reverse against a known body and an unknown, non-existent single view. The negative half of the answer, that no direction reaches the composite’s own total, is assembled from several views’s own finding restated as a search rather than as a closed-form bound: that essay computed 58% analytically for a comparable case, and this one confirms the same shape of limit by trying every direction directly and finding none of them close.

The constructive half — that the figure is nonetheless an exact projection of something, once the something is understood to be a jointed body rather than a rigid one — is this essay’s own addition to that pair, and it is the reading the pond with its trees laid flat already gives for a different aspective convention: a fold-out plan is not a bad photograph of a garden, it is an exact isometry of one, cut and rotated about its own hinge lines, with the fold’s own angle spent rather than lost to error. An aspective figure of a standing body is the same statement one level less flat — cut at the joints instead of at a hinge, rotated by a stated angle instead of laid fully open — and the fact that a rigid single view cannot reach it is not evidence the convention is careless. It is evidence the convention is doing something a single view was never able to do, stated as precisely as this collection states everything else it measures.

What perspective gave up reads every convention in this field as a trade, and an aspective figure states its own trade in exactly the vocabulary that essay uses for perspective’s: a station point buys a single, consistent viewing direction at the cost of every part not facing that direction being foreshortened away, and an aspective figure spends the single viewing direction to buy back each part’s own best face, at the cost of the 2.6% no rigid recombination can ever recover. Neither a single photograph nor an aspective figure is the more complete record of a standing body; each keeps one of the two things a drawing of several distinguishable parts might want, and this essay’s own numbers — 2.6% against 7.6e-13 pixels, 3.0% against 3.4e-3%, five recovered angles against the convention’s own five — are the price of that choice, measured rather than merely named.