Assembled from several views

An Egyptian figure is drawn head in profile, eye frontal, shoulders frontal, hips and legs in profile, feet in profile. A pond in an Egyptian garden scene is drawn as a rectangle in plan with its trees laid out flat around it, pointing away in four directions.

The usual description of this is aspective: each part is shown from the aspect that best identifies it. That description is exactly right and it is qualitative, and the quantity it is about turns out to have a closed form.

The quantity, defined #

A part of a figure is a roughly flat piece with a normal: the plane of a face in profile, the plane of a pair of shoulders, the surface of a pond. The share of that part which reaches the picture from a viewing direction $d$ is

$$|d \cdot n|$$

which is 1 when the part is seen face-on and 0 when it is seen edge-on. This is the foreshortening factor, and it is the same quantity the parallel field measures as an axis scale — the projected length of a unit vector under an orthographic projection.

Call it identifiability, because that is what it is doing: an eye seen in profile is a wedge, a pair of shoulders seen edge-on is a line, and neither identifies anything.

Aspective takes each part from its own direction, so every part scores exactly 1, by construction. A single view scores $\sum_i |d \cdot n_i|$, maximised over $d$. The whole content of the convention is the gap between those two numbers.

The closed form, and why it is √k #

For $k$ parts with mutually perpendicular normals, the maximum of $\sum |d \cdot n_i|$ over unit $d$ is $\sqrt{k}$, achieved by the direction making equal angles with all of them.

The derivation is one line of Cauchy–Schwarz: with the normals as an orthonormal basis, the sum is $\sum |d_i|$ over the components of a unit vector, and $\sum |d_i| \le \sqrt{k}\,\|d\| = \sqrt{k}$, with equality when every component is $1/\sqrt{k}$.

So the best single view keeps $1/\sqrt{k}$ of each part, and the convention’s advantage over it is exactly $k : \sqrt{k}$.

For three mutually perpendicular aspects the best single view retains 57.7% of each — and $1/\sqrt{3} = 0.5774$ is the same number that gives isometric projection its axis scale of 0.8165, which is $\sqrt{2/3} = \sqrt{1 - 1/3}$. The two are the same computation seen from opposite ends: isometric asks how much of a unit axis survives the equal-angle direction, and this asks how much of a unit normal projects onto it. That an Egyptian convention and a nineteenth-century engineering drawing standard turn on one number is not a connection between the two traditions; it is a fact about three perpendicular directions.

The parts are not all perpendicular, and the formula handles it #

The five parts in the figure above are not five mutually perpendicular normals. Head and legs share a profile plane; eye and shoulders share the frontal plane; the pond is in plan. Three distinct axes, with weights 2, 2 and 1.

The maximum then becomes $\sqrt{\sum_a w_a^2}$ over the axis groups, which is $\sqrt{4 + 4 + 1} = 3$ exactly. The swept maximum comes back at 2.9999 — the difference is the sweep’s angular resolution — and the per-part scores are 0.668, 0.669, 0.669, 0.668 and 0.326, which is the equal-angle direction weighted toward the axes carrying two parts each.

So the advantage of the composite over the best single view is 5 : 3, or 1.667×, on this figure. A figure with more parts on more distinct axes would show a larger gap; a figure with all its parts on one plane shows none.

That last case is the control, and it is in the gate: a figure whose parts all share a normal is fully served by a single view, and the sweep returns 5.000000 of 5. Without it, the advantage above could be a property of the arithmetic rather than of the convention.

Why the maximum is at the equal-angle direction #

The closed form deserves a second look, because the direction that achieves it is a familiar one and the reason it is familiar is not a coincidence.

For $k$ perpendicular unit normals, the sum $\sum |d \cdot n_i|$ is maximised when $d$ makes equal angles with all of them, and its components are all $1/\sqrt{k}$. For $k = 3$ that direction is $(1, 1, 1)/\sqrt{3}$ — the body diagonal of a cube, which is exactly the isometric viewing direction.

So the best single view of a figure with three perpendicular aspects is an isometric one. That is a pleasing result and it has a practical form: an isometric drawing is the single orthographic view that treats three perpendicular faces as fairly as possible, which is precisely why engineering drawing uses it as its one-view option and reserves the three-view layout for when fairness is not enough.

The relationship between the two numbers is worth writing out. Isometric’s axis scale is $\sqrt{2/3} = 0.8165$: how much of a unit axis survives. The identifiability here is $1/\sqrt{3} = 0.5774$: how much of a unit normal projects onto the view direction. They are $\sqrt{1 - 1/3}$ and $\sqrt{1/3}$, two halves of one Pythagorean identity, and they answer two different questions about the same direction.

What is actually being traded #

It is worth being exact about what the composite gives up, because “it is not a projection” is true and not very informative.

It gives up coherence. There is no viewing direction, and therefore no viewer position, from which the assembled figure is what would be seen. The parts do not fit together as a solid seen from anywhere.

It gives up occlusion as a depth cue. In a single view, one part in front of another hides it, and the hiding says which is nearer. A composite has to decide overlaps by convention, and the convention is not a depth statement.

And it gives up the possibility of measuring across parts. Each part is at true scale in its own plane; the relationship between two parts’ scales is a matter of drawing rather than of projection. So the height of a head relative to the width of a pair of shoulders is a choice, and in Egyptian relief it is a very carefully regulated choice — the canon of proportions is a grid, and the grid is doing the job the projection is not.

That last point is the one worth carrying. A system that abandons the single projection does not thereby abandon exactness; it moves the exactness somewhere else. Here it moves into a proportional canon, which is a rule about relative sizes applied across parts that no projection relates. A drawing made to that canon is measurable, and what makes it measurable is the canon rather than the geometry.

The model’s limits, which are larger here than elsewhere #

This is the essay in the phase whose model is furthest from its subject, and the distance should be stated rather than glossed.

A part is not a flat plate with a normal. A head is a solid; the “profile plane” is an idealisation of the direction from which its outline is most characteristic. The identifiability measure $|d \cdot n|$ is exact for a flat part and is a proxy for anything else.

Identifiability is not projected area. What makes an eye recognisable from the front is not merely that more of it reaches the picture; it is that the frontal view carries features the profile does not. Projected area is a measurable stand-in for a perceptual property, and the stand-in is the part of this that is doing work the geometry cannot check.

And the convention was not an optimisation. Nobody swept a sphere. What the measurement establishes is that the arrangement the convention arrived at is the one an optimisation would have chosen, and that the margin over the alternative is a specific number rather than a matter of opinion.

That third point is the useful one and it is worth stating in the general form, because this field keeps producing it. A convention can be shown to be optimal for a stated objective without anybody having optimised anything — and the finding is then about the objective rather than about the makers. Here the objective is “show as much of each identifying aspect as possible”, and what the measurement establishes is that aspective is exactly optimal for it and a single view is 60% as good.

If that objective is the wrong description of what the pictures were for, the measurement is still correct and simply about something else. Which is the honest position for a geometric account of a convention to be in.

The same optimisation, elsewhere on this site #

The shape of the argument here — no single choice serves every requirement, and the best compromise is quantifiable — is one this site has now made three times on unrelated material, and the repetition is worth naming.

No picture surface is both straight and conformal: six surfaces, three properties, and the measurement is that nothing achieves all three, with stereographic exact on angles and the equal-area fisheye exact on area and neither exact on both.

A carpet and the people on it want optical axes ninety degrees apart, so no camera supplies both effects.

And here, $k$ perpendicular aspects allow one view at most $\sqrt{k}$ of them.

Three exclusions, three different subjects, one form: a requirement vector that cannot be satisfied simultaneously, and a closed form for the best compromise. The last field of this site is about what that form has in common, and it ends by turning it on perspective.

What the assembly costs in the site’s own terms #

The composite is not one map, so it does not appear as a row in this field’s comparison table — and it is worth working out what its row would say if the table could hold it.

A centre: no, and in a stronger sense than any other row. A parallel system’s rays are parallel, so its centre is at infinity, which is at least a definite answer. A composite’s rays belong to several different bundles pointing in several different directions, and asking for one point they pass through is not a question with an at-infinity answer either. The bundles do not even share a direction.

True measure: yes, within each part. Each aspect is an orthographic projection, so each part’s own ratios are exact. Across parts there is no projection to preserve anything, so the question does not arise.

Diminution: no. Every part is orthographic.

Bounded depth: no, for the same reason.

Straight lines: yes, within each part.

The pattern is the same in three of the five cells: yes, locally; the question does not arise, globally. That is what a composite is, stated in the table’s vocabulary, and it is why the table cannot hold it — the table’s unit is a map from a world to a page, and a composite has no such map.

The number, restated #

The whole of this essay is one comparison and it is worth having in one line.

A composite keeps every part’s identifying aspect at full extent, by construction: k of k. The best single view keeps √(Σ w²) of them, which for the five-part figure measured here is 3 of 5, and for k mutually perpendicular aspects is √k of k.

So the convention’s advantage grows as the square root of the number of distinct aspects a subject has. A figure with two aspects gains 41%; with three, 73%; with five perpendicular ones, 124%. Subjects with many distinct characteristic directions gain most, which is a testable statement about which subjects the convention is applied to hardest.

What a technical drawing does about the same problem #

One modern comparison, because it makes the point that the trade is not historical.

An engineering drawing of a part shows three orthographic views — front, top, side — laid out in a standard arrangement. That is aspective. Each view is a projection along a different direction; each is at true scale in its own plane; none of them is what anybody would see; and the relationship between them is a convention (first-angle or third-angle) rather than a projection.

The reason is the same reason. A single view of a machined part foreshortens most of its features, and a feature seen at 58% of its extent cannot be dimensioned reliably. Three views at 100% each can. The drawing gives up being a picture of anything in order to be a measurement of everything, and it has done so continuously since the eighteenth century, in a culture that had perspective available and understood.

That is the strongest available argument that aspective is a solution rather than a stage. The same problem, met by two traditions four thousand years apart, produced the same answer, and one of the two traditions had the alternative in hand and declined it.

Two comparisons worth carrying #