The true shape of a cut

Cut a solid with a plane and the cut is a polygon in space. It has an area, and lengths, and angles at its corners, and none of those is what the front view of the solid shows.

That is the reason a drawing carries section views, and the reason a course in descriptive geometry keeps going after the true-length drill. What is worth measuring is which of the section’s properties the standard view damages, because the damage is not uniform — one of them is recoverable from a single number and the rest are not.

The area is a cosine and the angles are not a cosine #

An orthographic projection along a direction $\hat{\mathbf{d}}$ multiplies the area of a plane patch by $|\hat{\mathbf{n}} \cdot \hat{\mathbf{d}}|$, where $\hat{\mathbf{n}}$ is the patch’s normal. One factor, the same for every part of the patch, and the same for every patch in the same plane.

So the section’s area in the front view is its true area times a single number, and if that number is known the true area follows. The drawing office does know it: the cutting plane’s orientation is given on the drawing, as the direction of the cutting-plane line and its arrows.

The angles behave completely differently, and the reason is the shape of the map rather than its size. Projecting a plane along a direction not perpendicular to it is, in the plane’s own coordinates, a scaling of one direction by the cosine and of the perpendicular direction by one — an anisotropic scaling. Anisotropic scalings do not preserve angles. Two directions that meet at $90°$ in the cut meet at something else in the picture, and the something else depends on how the pair sits relative to the direction being squashed.

So there is no correction factor for the angles. There is a correction factor for the area, and it does not help.

Why the angle error is not monotone in the foreshortening #

The pair of numbers in those two figures is worth pausing on, because the naïve expectation is wrong in a way that matters when reading a drawing.

At $8°$ the front view keeps only 14% of the area and gets every corner within $2.91°$. At $68°$ it keeps 86% of the area and bends a corner by $8.34°$. More foreshortening is not more angular error.

The reason is that an anisotropic scaling by factor $k$ takes an angle to

$$\arctan\!\left(\frac{k \tan\theta}{1}\right)$$

measured from the unsquashed direction — a map that fixes $0°$ and $90°$ exactly and moves everything between them, most strongly near $45°$. So the error depends on where the section’s own edges sit relative to the squashed direction, which is a fact about the shape of the cut and not about the severity of the projection.

The worst a corner can be bent #

The non-monotonicity is not a licence to give up on the question, because the map has a bound and the bound is one expression.

Maximising $\arctan(k\tan\theta) - \theta$ over $\theta$ gives $\tan\theta = 1/\sqrt k$, and since $\arctan\sqrt k + \arctan(1/\sqrt k) = 90°$, the largest displacement any single direction can suffer is $90° - 2\arctan\sqrt k$. A corner is two directions and they can be displaced in opposite senses, so

$$\text{worst corner error} \;\le\; 180° - 4\arctan\sqrt{k}.$$

Against the three figures: at $k = 0.8640$ the bound is 8.36° and the measured worst corner is 8.34 — the section’s edges happen to sit almost exactly at the worst orientation. At $k = 0.6886$ the bound is 21.3° and the measurement is 15.6, so that cut is at three quarters of its worst case. At $k = 0.1389$ the bound is 98° and the measurement is 2.91, which is three per cent of it: the edges of that section lie almost along the two directions the squash leaves alone.

So the three cases are not three severities of the same thing. They are one bound, evaluated at three values of $k$, sampled at three quite different places inside it — and the bound is monotone in $k$ even though the measurements are not. What is not monotone is where a particular cut’s edges happen to fall; what the projection can do to a corner is.

That turns the reader’s question into an answerable one. A section may be trusted for angles to within $180° - 4\arctan\sqrt k$, whatever its shape, and requiring that to be under two degrees needs $\sqrt k > 0.983$, or

$$k \;>\; 0.966 \qquad\Longleftrightarrow\qquad \text{the cut within } 15° \text{ of square-on.}$$

Fifteen degrees is the whole working range. Beyond it a corner may be wrong by more than two degrees, and whether it is depends on the cut’s own orientation, which the reader cannot see. That is a much smaller allowance than the area’s, which is exact at every angle once the cosine is known — and the gap between the two allowances is the reason a drawing carries a section view at all rather than a note saying by how much the front view is foreshortened.

It also gives a rule for reading a drawing that is not annotated. An unmarked cut whose foreshortening looks slight may be trusted for angles; one whose foreshortening looks severe may not, and one whose foreshortening looks moderate is the dangerous case, because that is where the bound is large enough to matter and small enough not to advertise itself. The 46° cut above, at three quarters of a 21° bound, is exactly that case: it does not look badly foreshortened and one of its corners is out by more than fifteen degrees.

Which is the same asymmetry a ruler on an isometric drawing finds between what a parallel drawing preserves and what it merely appears to, and the same reason a development is a separate drawing rather than a note on the first one: some properties survive with a correction factor and some do not survive at all, and no amount of care with the ones that do repairs the ones that do not.

One more consequence of the bound, and it is the one that explains a drawing-office habit. Since the allowance closes as $k \to 1$ and the area correction is exact at every $k$, a drawing that must communicate angles has to supply a view, while a drawing that must communicate area need only supply a number. That is why sections are drawn and foreshortening factors are written — and why an unannotated section is a drawing that has silently promised the harder of the two.

One thing the bound does not depend on is worth naming: the section’s own size. Both $k$ and the bound are angles, so a small cut and a large one in the same plane are misreported by the same angles and the same fraction of area. A drawing office cannot buy accuracy by drawing the section larger, only by drawing it from a different direction — which is the difference between a scale and a view, and is why the two are separate decisions on a sheet.

Which means a reader cannot look at a foreshortened section, judge that it is “not very foreshortened”, and conclude that its angles are nearly right. That inference is available for the area and not for the angles, and the two figures above are a counter-example in the direction people do not expect.

What the section view is #

A section view is an auxiliary view taken along the cutting plane’s normal, with the material in front of the plane removed.

The second half is a drawing convention about what to show. The first half is the whole of the geometry: look along the normal and the face draws at true shape, because $\hat{\mathbf{n}}\cdot\hat{\mathbf{d}} = 1$ and the anisotropic scaling becomes the identity.

So “true shape of the section” is not a phrase about accuracy. It is a statement that one particular projection direction returns the object’s own metric exactly, and every other one returns a picture from which the metric cannot be recovered without knowing which direction it was.

Two sections that draw the same picture #

The unrecoverability has a sharp form, and it is the same shape as the ambiguity the rest of this field keeps meeting.

Take any section and squash it by a cosine in one direction. The result is a perfectly good polygon, and it is the front view of the original cut. It is also the true shape of a different cut — one lying in a plane square on to the view, with the squashed outline as its actual shape.

So a foreshortened section and a genuine flat one are the same picture. Nothing in the front view distinguishes a $46°$ cut of area 2.8996 m² from a flat cut of area 1.9966 m², and the cutting-plane line on the drawing is what separates them. Remove the annotation and the drawing is ambiguous, in exactly the way three views leave a solid ambiguous and a parallel drawing leaves the near corner ambiguous.

Where the outline comes from #

The section polygon in these figures is computed rather than drawn, and the procedure is worth a paragraph because one step of it is a trap this fleet has fallen into twice.

A plane meets a box in a polygon whose vertices are where the plane crosses the box’s twelve edges. Finding them is a sign test and a linear interpolation per edge, which is straightforward. What is not straightforward is that they arrive in whatever order the edge list happens to be written in, and a polygon drawn in that order is a self-intersecting bow-tie — a shape that fits its viewBox, renders cleanly, contrasts well, and is the wrong region.

So the hits are ordered by angle about their own centroid, in the cutting plane’s own basis. Which is the only ordering available: the vertices are coplanar, so “clockwise” means something in that plane and nothing in the picture until a direction has been chosen.

The site has a gate for exactly this — every filled polygon in every figure is tested for self-intersection, 658 of them across the generator families — because the failure is invisible to every other check. It caught a shadow drawn as a bow-tie over the wrong quarter of a floor, and it is the reason a section computed this way can be trusted to be the region it claims.

An offset section is not one plane #

The convention has a case where the essay’s central claim quietly stops holding, and drawings use it constantly.

An offset section steps the cutting plane sideways part way through, so that one section view can pass through two features that do not lie on a common plane. The cutting-plane line on the drawing has a jog in it, and the section view is drawn as though the jog were not there.

Which means the section view is two projections, each true for its own half, presented as one picture. Each half is at true shape; the relationship between the halves is not a shape at all, because the two pieces have been slid together along the jog. Lengths within a half are true; a length measured across the jog is a length in no plane.

That is the same structure as a shadow that crosses a crease: two maps, each exact on its own side, glued along a line, with the joint carrying no meaning. Drawing conventions and light do it for the same reason, which is that a projection is defined relative to one surface, and two surfaces are two projections however the result is presented.

The circle, and the drawing office’s construction for it #

The commonest section in a machine drawing is a circular hole cut obliquely, and it is worth following because it is where the anisotropic scaling is most visible and where the drawing office’s own construction departs from the geometry.

A circle squashed by $\cos\theta$ in one direction is an ellipse with axis ratio $\cos\theta$. That is why a hole drawn in any view that is not square on to it is an ellipse, and why the ellipse’s minor axis points along the direction of the squash.

The construction traditionally used to draw that ellipse is the four-centre approximation — four circular arcs, tangent to the enclosing rhombus at the midpoints of its sides. It is fast with a compass, and it is not the ellipse.

Which is the fleet’s standing distinction between a construction that is a projection and one that stands in for one. The auxiliary view is exact at every step; the four-centre ellipse is a compass-and-straightedge convenience with a measurable error, and both are taught in the same course in the same tone.

What survives the cut, in each direction #

Collecting what the standard view of a section does and does not keep:

Lengths along the direction that is not squashed survive exactly. The projection keeps one direction in the cutting plane at full scale — the one perpendicular to the squash — so any length in the section that happens to run that way is at true length in the picture.

Every length not along it is short by a factor between $\cos\theta$ and 1. Which factor depends on the direction, and the direction is not readable off the picture.

Parallels survive. The map is affine, so parallel edges of the cut stay parallel in the front view. That is worth noting because it is the one thing a reader can rely on without knowing the cutting plane.

Ratios along a line survive. The midpoint of an edge of the section is the midpoint of its image, exactly — the affine invariant this whole field runs on.

Angles and areas do not survive, and only the area is recoverable.

The perspective version of the same question #

A photograph of a cut face is a different problem, and the difference is the one this site keeps returning to.

The front view’s damage is an affine map, so it is undone by an affine map — which needs one number if the plane’s orientation is known, and is not determined at all if it is not. A photograph’s damage is projective, so undoing it needs a homography, which is four correspondences or a vanishing line.

The trade is that the projective case leaves the evidence in the picture. A photographed circle draws as an ellipse whose centre is not the image of the circle’s centre, and that displacement is a measurable signature of the projection that a parallel drawing has none of.

One number, and the reason it is not enough #

The compact statement of the whole essay is a comparison of two counts.

Undoing the area needs one number — the cosine — and the drawing supplies it. Undoing the shape needs two: the cosine and the direction in the picture along which the squash was applied. The second is exactly what a foreshortened outline does not record, since the same outline is produced by a whole family of (direction, cosine) pairs acting on a whole family of cuts.

Draw the section in its own view and both are supplied at once, by construction. That is why the convention is a picture rather than an annotation, and why no amount of care with the front view substitutes for it.

The rule a draughtsman is following #

Everything above condenses into the convention as it is actually taught, and the point of the measurement is to say why the convention has the shape it has.

Dimension a feature in the view where it is true. A hole’s diameter is dimensioned in the view square on to it; a slot’s length in the view along it; a section’s shape in its own section view. The rule exists because the picture cannot say what it is foreshortening.

Annotate the cutting plane. The cutting-plane line and its arrows are the missing number: they name the direction, which makes the area recoverable and tells the reader that the shape they are looking at in the other views is not the shape.

And draw the section view rather than correcting the front view. The area could be corrected. The angles could not, so there is nothing to correct to — which is why the convention is a second picture rather than a note.