No view draws a curved plate true

Descriptive geometry’s first drill is an auxiliary view: choose a direction of sight and the drawing changes what it is good for. Look perpendicular to a line and it draws at true length; look along it and it draws as a point; look perpendicular to a plane and that plane draws at true shape, so an angle on it can be measured with a protractor and an area with a planimeter.

Every one of those is a statement about a plane, and the reason it works is that a plane has one normal.

The area a parallel drawing gives a patch #

Under a parallel projection along $\hat d$, a patch of surface with unit normal $\hat n$ is drawn at $|\hat n \cdot \hat d|$ times its true area. That is the whole of the local relationship: a cosine, and nothing else.

So a view is true where $|\hat n \cdot \hat d| = 1$, which happens where the normal is along the ray. A flat plate has one normal, so one direction makes it true everywhere.

A curved plate has a field of normals, and one direction cannot be along all of them.

The search, and why it is a search #

The best direction could be written down. For a symmetric patch it is the middle normal, and the residual follows in a line of trigonometry.

bestTrueShapeView searches the sphere instead — a hundred and eighty by ninety directions, evaluating the worst area ratio over the patch for each — and then the closed form is asserted against what the search found.

That order matters. Writing the answer down and quoting it is asserting; searching and finding the same thing is measuring, and the difference is whether a reader is being told the conclusion or shown the ground it stands on. On a flat plate the search returns a residual of exactly zero, which is the control that says the search is capable of finding a true view when there is one.

A bent plate: 1 − cos w #

A cylinder patch of half-angle $w$ has normals sweeping an arc of that half-angle. Point the ray down the middle of the arc and the worst ratio is at the two edges, where the normal is $w$ off the ray:

$$\text{worst} \;=\; 1 - \cos w$$

At half a radian — a plate bent through fifty-seven degrees — that is twelve per cent of area, at the plate’s edges, in the best view there is.

A dished plate: 1 − cos²w, and the factor is 1 + cos w #

A spherical cap of half-angle $w$ has normals filling a cap, not an arc. The worst direction is the corner of the parameter square, where the normal is at $\arccos(\cos^2 w)$ from the axis, so

$$\text{worst} \;=\; 1 - \cos^2 w$$

which is worse than the cylinder’s by exactly $1 + \cos w$ — near two for a shallow patch, falling toward one as the patch wraps further.

That factor is the whole difference between the two curved cases stated as a number, and its origin is a count. A cylinder’s normals need one parameter to describe; a cap’s need two. The residual is the departure of a cosine over the set of normals, and a two-dimensional set reaches further.

The family a developable has, and the dish does not #

Area alone gives the two curved cases the same character: neither has a true view, one is worse than the other, and a reader could reasonably stop there.

They are not the same character, and what separates them is not an area at all.

A cylinder’s generators are all parallel. So a view perpendicular to them draws every generator at the same scale — spread zero, at the arithmetic floor — and a single ruler laid on the drawing reads any length along that family, anywhere on the plate.

A cap has no family of parallel curves on it. There is nothing for a ruler to read.

That is the same shape of finding the curved field made about picture surfaces: every surface keeps some family of lines straight, and which family it is, is the surface’s signature. Here the question is not which lines stay straight but which lengths stay comparable, and the answer separates the developable from the doubly curved where the area measure does not.

It is also the isometric ruler’s question moved from three axes to a surface. That essay asks what a scale laid on an axonometric drawing measures and finds that each axis has its own factor and a direction between them has none. This asks the same of a curved plate, and finds that a developable has one direction with a factor and a dish has none.

Why the count of parameters is the right thing to count #

The factor $1 + \cos w$ has a derivation and it also has an explanation, and the explanation is the transferable half.

The residual is how far $|\hat n \cdot \hat d|$ departs from one over the set of normals the patch has. So the question is not how curved the patch is but how large that set is, and the two curved patches differ in its dimension: an arc for the cylinder, a cap for the sphere.

A one-dimensional set of directions has a middle and two ends. A two-dimensional set has a middle and a boundary curve, and its furthest point from the middle is a corner rather than an end — which is where $\arccos(\cos^2 w)$ comes from, and why it exceeds $w$.

That is the same accounting the axis-scale identity does in the parallel field, where the three axis scales are not free because they are the images of three perpendicular unit vectors and lie on a sphere. Here the normals lie on a sphere too, and how much of it they cover is what the residual measures.

The measure, and one it is not #

The quantity searched over is the worst area ratio, and two other measures were available. Which one is chosen changes what the answer means, so it is worth saying why this one.

An average area ratio over the patch would be smaller and would be about the wrong thing: a drawing is used to take a measurement at a place, and a measurement at the worst place is what the drawing has to support. An average is a statement about the picture and a worst case is a statement about what can be read off it.

A length ratio in a stated direction is the other candidate, and it is what the last section of this essay uses — but it needs a direction to be stated, and a patch with no distinguished family has none. Area needs nothing chosen, which is why it is the measure that can be compared across all three patches.

So the two measures are answering two questions, and the reason both appear is that the first separates the flat case from the curved ones and the second separates the two curved ones from each other. Either alone gives a two-way answer to a three-way question.

The trichotomy, and why it is three rather than two #

Put the three together and the field has a classification it did not have.

A plane has a view that is true. Everything on it — every length, every angle, every area — comes off the drawing with one scale.

A developable has a view that is true along one family. Lengths along the generators come off with one scale and nothing else does.

A doubly curved patch has neither. No direction, no family, no ruler.

The middle case is the one that earns the classification. If the answer were “planes are true and curves are not”, the drawing office’s practice of putting a cylinder on a drawing and dimensioning along its generators would be a convention nobody had checked. It is exact, and the exactness is a property of the surface’s developability rather than of the drawing.

What this changes about a multiview drawing #

The three standard views of a part are three orthographic projections along three perpendicular directions, and the drill above says each of them shows some plane at true shape.

For a part made of flat faces that is a complete account: three views do not fix the solid, but each face parallel to one of the three planes comes off its own view exactly, and a face parallel to none of them gets an auxiliary view of its own.

For a part with a curved surface on it there is no view to add. An auxiliary view is a choice of direction, and this essay’s finding is that no direction works — so the fourth view a draughtsman would reach for does not exist, and the office’s answer is to stop drawing views and start drawing something else.

That is a real gap in the method rather than a shortcoming of any particular drawing, and it is worth naming as one. The auxiliary view is presented in every textbook as a general repair for foreshortening, and it is a repair for exactly one kind of surface.

And it is not the same as the development #

There is a sentence that sounds like this essay’s conclusion and is a different claim, so it is worth separating.

A developable unrolls. Its development is a flat pattern that keeps every length on the surface, exactly, because the flattening is an isometry — which is what makes sheet metal work possible at all.

That is not a view. No direction of sight produces the development, because a development is not a projection: it cuts the surface and lays it out, moving different parts of it by different amounts. The drawing and the development are two different flat pictures of one plate and they keep different things, which is the next rung and is the reason this one stops here.

The refusal, and what it says about the classification #

bestTrueShapeView refuses a patch that is not one of the three named kinds. That looks like housekeeping and it is doing something.

The three patches are not a library. They are the three cases of a classification by how many parameters the normal field needs — zero, one, two — and a fourth patch would either be one of the three or would be a saddle, whose normals also need two and which behaves like the cap for this purpose.

So the refusal is the classification enforced. Adding a fourth kind to the file would be a claim that the trichotomy is a list rather than a count, and the refusal makes that claim have to be made deliberately.

What a draughtsman is actually doing #

The practice this measures is unremarkable and worth stating, because the measurement gives it a reason.

A curved part on a drawing gets two pictures: a view, which shows where it is and how it sits, and a development, which is what gets cut. Nobody dimensions the curved surface off the view.

The convention is old and the usual explanation is that the view “would be foreshortened”, which is true of a plane too and does not distinguish the cases. The real reason is that a plane’s foreshortening is a single number that an auxiliary view removes, a developable’s is a single number along one family and a variable everywhere else, and a dish’s is a variable in every direction.

What this does not settle #

It does not treat a patch that is not symmetric. The best direction for a lopsided patch is not its middle normal, and the search would find something the closed forms here do not describe.

It does not treat a patch cut from a surface that is not one of the three — a torus, a fairing, anything whose curvature changes sign — where the normals cover a region with holes in it and the worst case is somewhere the search would have to find rather than the closed forms.

It does not treat perspective. A camera’s foreshortening varies over a plane as well as over a curve, so the classification collapses — there is no true-shape view of anything, and the field that says so is viewing.

And it does not say a curved part cannot be measured off a drawing. It says which direction to measure in and which drawing to use, which is what the office already does and now has a number for.

Nor does it say anything about how the plate got its shape. This site computes what a drawing does to a surface, and the surface’s own geometry — what unrolls and what does not, and what a stretch costs when it cannot be avoided — belongs to the curved field and is measured there rather than assumed here.

A classification is worth more when its middle case has to be earned. Two cases are usually a dichotomy somebody wanted; three, with the middle one distinguished by a property neither of the others has, is a count.