Three views do not fix the solid

A machine part is specified by three drawings: a front view, a top view and a side view, each an orthographic projection along one of three perpendicular directions. The convention is old, it is taught first, and it is the reason a drawing office exists. The implicit claim is that the three together say what the part is.

They do not, and the gap is not a technicality about pathological shapes.

What a view can and cannot say #

The asymmetry is in one sentence, and everything else here follows from it.

A view can forbid a cell of space, and it cannot require one. If the front view is blank at some height and horizontal position, then nothing in the part lies anywhere along that line of sight, and a whole column of space is emptied. If the front view is filled there, all it says is that something lies somewhere along that column — and it says nothing whatever about where.

So the three views are three sets of prohibitions. Intersect the three, and what is left is the largest solid consistent with all of them: the set of cells whose shadows are filled in all three pictures. Every solid drawing those views is inside it, and the intersection is computable directly from the pictures with no reasoning about the shape at all.

That largest solid has a name outside drawing offices — it is the visual hull — and it is what a shape-from-silhouette reconstruction returns. What it is not is the part.

The witness with a closed form #

A search that removes cells until it cannot is a witness to a gap at that grid size, and it is as good as the search. Something better is available here, and it is worth having because it makes the ratio a fact rather than a result.

Take an $n \times n \times n$ grid and keep the cells where

$$k \equiv i + j \pmod n.$$

Along $z$, every column $(i, j)$ has exactly one cell in it, so the front view is a full square. Along $y$, given $(i, k)$ there is exactly one $j$ with $k \equiv i+j$, so the top view is full too. Along $x$, given $(j, k)$ there is exactly one $i$. All three views are solid squares — the cube’s own three views — from $n^2$ cells rather than $n^3$.

So for every $n$ there is a solid with the cube’s three views and $1/n$ of its material. No search, no grid size at which it happens to work, and no bound on how large the ratio gets.

A hole is visible and a dimple is not #

The block in the first figure has a hole through its base, and the hole is in all three views. That is not luck, and the difference between a hole and a dimple is the cleanest available statement of what a silhouette carries.

A through hole interrupts the line of sight. Look along its axis and the outline has a gap in it, because there is genuinely nothing anywhere along those columns — the prohibition a view issues is exactly the right kind of statement about a hole, so the hull has the hole in it and the reconstruction is right there.

A dimple interrupts nothing. Every column through it still meets material further along, so no view is blank anywhere over it, and the hull fills it. The reconstruction is wrong there and no view can be added that makes it right.

So the deficiency is not uniform over features. A part made of through cuts, steps and slots is very nearly determined by its three views; a part with a pocket, a blind hole or a cast recess is not, and the drawing office’s sections are aimed at exactly those.

The small solid is a witness, not a minimum #

The right-hand solid in the figures above is found by a stated procedure: start from the hull, take the cells in index order, and drop each one if the three views are unchanged without it. That is a greedy pass and it is worth being exact about what it proves.

It proves the gap is at least that large, on that input. It does not prove the solid is the smallest one, and calling it the minimum would be claiming an optimum the procedure makes no attempt to reach — the answer depends on the order the cells are visited in, which is why the order is written down rather than left to whatever a hash table produced.

The closed-form sheet is the other half, and the two together are the honest statement: a search says this input has a gap of this size, and the sheet says the ratio is $n$ for every $n$. Neither one alone would do.

The sheet is a minimum after all #

The essay above is careful to call the greedy solid a witness rather than a minimum, and it is right to be. For the cube, though, the closed-form sheet is the minimum, and a counting argument settles it in two lines.

Three full views mean every axis-parallel line of the grid has to contain at least one filled cell. There are $3n^{2}$ such lines — $n^{2}$ in each direction — and every cell lies on exactly three of them, one per direction. So a set of cells covering all the lines must satisfy

$$3\,|S| \;\ge\; 3n^{2}, \qquad |S| \;\ge\; n^{2},$$

and the diagonal sheet $k \equiv i + j$ achieves it. So $n^{2}$ is exactly the smallest solid with the cube’s three views, and the ratio $n$ is exactly the largest gap three full views permit — not a lower bound from a search, and not improvable.

The same counting gives a bound for any three views, not just full ones. Every cell covers one filled square in each view, so a solid consistent with views of $f_1$, $f_2$ and $f_3$ filled squares needs at least $\max(f_1, f_2, f_3)$ cells. The true minimum sits between that and the hull, and it reaches the bound exactly when the filled squares of the three views can be paired up into a system of distinct representatives — a Latin-square condition, which the cube’s views satisfy and a general part’s need not.

So the honest statement of the gap has three levels rather than two: at most the hull, at least the largest view’s area, and equal to the second of those precisely when a combinatorial matching exists. The block at $n = 7$ — hull 315, witness 95 — sits between them, and its true minimum is a matching problem rather than a geometry one.

Two consequences worth carrying out of the counting.

In the continuum the minimum volume is zero. Letting $n$ grow, the sheet’s share is $1/n$, so the limit of the construction is a surface: a set of measure zero whose three orthographic silhouettes are solid squares. Three views of a part are therefore compatible with a part of no material at all, which is the strongest form of the deficiency and the one that makes clear it is not about coarse grids.

And the gap is a covering number rather than a shape. What the three views constrain is a covering problem on $3n^{2}$ lines, and every geometric feature — the hole that survives, the dimple that does not — is that problem’s structure seen from outside. A through hole removes lines from the problem entirely, so nothing can hide there; a dimple removes none, so the covering is free to fill it. That is the same distinction a silhouette’s unreachable term measures for a smooth object, where the concavity is not poorly sampled but never sampled, and it arrives here as a line that was never in the constraint set.

What the drawing office actually does about it #

The convention is not naïve, and reading this result as “engineering drawings are wrong” would be reading it backwards.

A real multiview drawing carries far more than three silhouettes. Hidden edges are drawn dashed, which converts each view from a shadow into a statement about every face along the line of sight rather than about the outermost one. Section views cut the part open and draw what is inside the cut. Dimensions pin lengths the views only imply. Notes name the features — a through hole, a counterbore, a chamfer — in words, because words are cheap and unambiguous.

Every one of those exists because the silhouettes do not determine the part. The result above is the reason for the apparatus rather than an objection to it.

What is worth keeping is the shape of the deficiency, because it says which additions help. The information a view is missing is position along its own line of sight, so any addition that supplies depth ordering — a hidden line, a section, a dimension into the page — is the right kind of addition, and any addition that refines the outline is not.

Where the missing information went #

A parallel projection is a projection through a centre with the centre taken to infinity, and the limit is genuinely a limit: photographs from further and further away with the lens lengthened to match converge to the parallel drawing.

Everything a projection through a finite centre knows about depth is carried by the rate at which things shrink — the perspective divide. Take the centre to infinity and the divide goes with it. What is left is an affine map, and an affine map is blind to position along the direction it is taken along in a way no amount of care can recover.

That is one statement with several familiar consequences on this site, and they are worth putting side by side because they are usually met separately:

• Which corner of a drawn box is nearer is not decided. A perspective drawing rules the reversed reading out at a rate inverse in the eye’s distance; a parallel one never rules it out at all.
• Moving the object does not change its drawing, so a picture cannot say where a thing is along the ray.
• And the result here: three views bound a solid between a hull and something considerably smaller.

A view is a shadow, and this is shape from silhouette #

The three views are not merely like shadows. A shadow cast by a light at infinity is an orthographic projection of the occluder onto the receiving plane, so the lamp taken to infinity and the eye taken to infinity are one operation — and a front view is what a part’s shadow looks like under a sun in front of it.

Read that way, the intersection of the three extrusions is the intersection of three shadow volumes, which is exactly the reconstruction a turntable scanner builds out of silhouettes. Its known limitation is the one measured above, and it has a sharp form:

No concavity ever appears in any silhouette. A dimple in a face is invisible from every direction, because the outline of the part is decided by its extreme points and a dimple has none. So the visual hull contains the convex parts exactly and fills every concavity, and the object can differ from it by any amount of material the silhouettes never see.

The count, and why three views is a choice rather than a theorem #

The three standard directions are chosen for a reason that is not about determination: they are the directions in which a rectilinear part’s faces are square on, so lengths in the views are true lengths and the drawing can be dimensioned directly.

The cost of that choice is visible in the sheet above. Its cells lie one to a column in every one of the three coordinate directions, which is possible because the three directions are mutually perpendicular and the grid is aligned with them. A fourth view along a diagonal cuts it down immediately, and this is worth being precise about: adding views shrinks the hull monotonically, because each new view can only forbid more cells, but no finite set of views ever reaches into a concavity.

So the sequence of hulls decreases and does not converge to the object. That is a stronger statement than “three is too few”, and it is the statement that separates silhouette information from depth information for good.

What survives, and what it is good for #

Nothing here says the three views are worthless, and the fleet’s habit is to measure what a construction does keep rather than to rank it against something else.

Three orthographic views along the coordinate directions preserve, exactly:

Every length parallel to a coordinate axis. This is the whole reason the convention exists. A dimension read off the front view is a true length in the world, because the projection is orthographic and the edge is parallel to the plane.

Every angle in a coordinate plane. A right angle in the front face is a right angle in the front view.

The convex outline in three directions, which is what the hull is built from.

What they destroy is the depth ordering — which of two features lies nearer along the line of sight — and everything that depends on it.

The same absence, in three sizes #

It is worth putting this beside the site’s other two statements of the same kind, because the three make a ladder and the middle one is usually the surprise.

One picture supplies every ratio and no size. A world 137 times larger, photographed from 137 times further away, gives the identical picture. The freedom is one number.

Two perspective pictures supply shape and no size. The second picture converts ratios in the plane into a full three-dimensional shape and adds no length at all. The freedom is still one number.

Three orthographic views supply a hull and not a solid. The freedom is not a number: it is every subset of the hull whose three shadows are the three given ones, which at $n = 6$ runs from 64 cells to 216.

The reason the third is a different kind of freedom is that the first two are freedoms of a map — a scale factor applied to everything — and this one is a freedom of a set. There is no parameter to be uncertain about. There is a family of solids, and its members are not versions of one another.

The direction is a choice, and the three standard ones are not special #

Nothing in the geometry privileges the coordinate directions. A view can be taken along any direction at all, and choosing the direction is the whole content of the auxiliary view — the drill that puts a line at true length by looking across it and turns it into a point by looking along it.

Which means the three-view convention is a decision with a stated benefit and a stated cost, rather than a description of how projection works. The benefit is that a rectilinear part’s own edges come out at true length. The cost is that a solid aligned with those same three directions can hide the most material from them, and the diagonal sheet is that cost written as a formula.

Reading a drawing after this #

The practical residue is small and worth stating plainly.

A three-view drawing is a set of constraints, not a description. Reading one is solving the constraints, and a reader who arrives at a solid has found a member of the family — usually the intended one, because the intended one is usually the simplest and because the notes and dimensions do the rest.

The intersection of the extrusions is worth computing when it matters. It is mechanical, it needs nothing but the three pictures, and it is the tightest bound available without additional information. If the hull is the part, the drawing determines the part.

And the gap is where the hidden lines are. A drawing office’s dashes are not decoration and are not redundancy: they are precisely the information the silhouettes cannot carry, which is why a drawing without them is ambiguous in a way that no amount of careful outlining repairs.