Two outlines are two curves

Multiview drawing has one method and it is the transfer line. A feature at some height in the front view is at that same height in the side view; a feature at some horizontal position in the front view is at that position in the top view. Every construction in a drawing office is that rule applied repeatedly, and the whole discipline of orthographic projection is built on it.

It is exact for a flat-faced solid, and it is exact for a reason that is worth saying: on a polyhedron, a silhouette feature is a vertex — a place on the object — and a vertex is the same place in every view.

A curved surface’s outline is not on the object twice #

The outline of a smooth surface is the image of its contour generator — the curve where the surface’s tangent plane contains the ray. On a sphere under a parallel projection that is the great circle perpendicular to the ray.

Change the ray and the great circle changes with it. Turn the view by one degree and the curve on the ball turns by one degree: a hundredth of a radius of arc, at every point of it.

So the front view’s outline and the top view’s outline are the images of two different curves on the same ball. They are perpendicular great circles, and two perpendicular great circles meet in exactly two points.

Those two points have a name: frontier points, the places where both views see the same bit of surface. They are the two ends of the axis perpendicular to both rays, which for a front view and a top view is the horizontal axis across the drawing.

And the transfer line joins two places √2 radii apart #

Take the front view’s outline at horizontal position $x$. The point of the ball it comes from is at $(x, \sqrt{R^2 - x^2}, 0)$ — on the great circle in the plane of the front view’s own contour.

The top view’s outline at the same $x$ comes from $(x, 0, \sqrt{R^2 - x^2})$.

Those are two different points of the ball, separated by

$$\sqrt{2\left(R^2 - x^2\right)}$$

which is zero at the two frontier points and $\sqrt2\,R$ in the middle.

A draughtsman transferring the top of a ball’s outline from the front view to the top view is connecting a point at the top of the ball to a point at the back of it. The construction still produces the right drawing — a circle, in both views, in the right place — because the ball is symmetric enough that the wrong points happen to be at the right coordinates.

Which is the uncomfortable part. The rule works and the reason it works is not the reason it is taught.

The control: on a box the two points are one point #

The finding needs its other half or it is a statement about transfer lines rather than about curved surfaces.

Take a box. Its silhouette in any view is a cycle of its own edges, so a silhouette feature is a vertex, and the vertex at a given $x$ in the front view is the same vertex at that $x$ in the top view. The gap is zero — at the arithmetic floor, for every vertex.

So the transfer rule has two regimes and nothing in a drawing announces which one is in force. A drawing of a bracket with a boss on it has both on the same sheet: the bracket’s corners transfer exactly and the boss’s outline does not.

A silhouette jumps and a contour slides #

The deeper difference is about how the outline behaves as the view changes, and it separates the two regimes cleanly.

On a polyhedron the silhouette is a set of edges, and it is the same set over a whole open range of directions. Rotate the view a little and nothing happens; rotate it past the plane of a face and the set jumps to a different one. The outline is piecewise constant.

On a smooth surface the contour generator moves continuously, and it moves as fast as the view does — a degree for a degree, on a ball.

That is why a curved surface’s outline is not a feature of the object at all. It is a feature of the object together with the direction it is being looked from — the same status as a shadow’s edge, and for the same reason.

The edge of a shadow is drawn on the object makes exactly this point in the light field: a shadow’s boundary is the terminator, which is a curve on the occluder that moves when the lamp does. Replace the lamp with an eye and the terminator becomes the contour generator, and every property transfers.

The gap for other angles, and for other surfaces #

Two generalisations of the $\sqrt2\,R$ turn it from a fact about the standard views of a ball into a rule, and the second of them dissolves the two regimes into one.

Other angles. Two views at angle $\gamma$ have contour generators whose planes meet at $\gamma$, so the two points a transfer joins are ends of a chord subtending that angle:

$$\text{gap} \;=\; 2R\sin\frac{\gamma}{2}.$$

At $90°$ that is $\sqrt2\,R$, which is the measured value; at $180°$ — a view and its opposite — it is $2R$, the whole diameter, which is right, since the front and back views of a ball see opposite hemispheres and share only the silhouette itself. And for a small $\gamma$ it is $R\gamma$, which is the essay’s “a degree for a degree” written as a length.

Other surfaces, and this is the part that matters for a drawing office. Rotating the view by $\mathrm d\gamma$ slides the contour generator across the surface by

$$\frac{\mathrm d\gamma}{\kappa},$$

where $\kappa$ is the normal curvature in the direction of the slide. On a ball $\kappa = 1/R$ and the slide is $R\,\mathrm d\gamma$, which recovers everything above. But it says something the ball cannot: the flatter the surface, the further the contour generator moves.

Taken to the limit that is the polyhedron. A flat face has $\kappa = 0$, so the slide is infinite — which is exactly the jump the essay describes, the silhouette staying on one set of edges and then switching to another. The two regimes are not two behaviours; they are $1/\kappa$ at its two ends, and a polyhedron is the case where the sliding rate has diverged and the motion has become discontinuous.

The practical reading inverts the intuition, and it is worth stating plainly because a draughtsman’s instinct runs the other way. The transfer rule is worst on the flattest curved surfaces. A 20 mm boss transfers points 28 mm apart on its own surface; a fairing with a two-metre radius of curvature transfers points nearly three metres apart, if it is that large. The small tight feature is nearly honest and the broad gentle one is not — and a broad gentle dome is precisely the shape a drawing office would transfer without hesitating, because it looks almost flat and flat things transfer exactly.

So the boundary between the two regimes is not “curved or not”. It is a comparison between the surface’s radius of curvature and the size of the patch being transferred: while $R\gamma$ is small against the feature’s own extent the transfer is nearly right, and when it exceeds it the two views are looking at different parts of the object. For adjacent standard views, $\gamma = 90°$, that condition is simply radius of curvature small against the feature — which is a bolt head and not a shell.

Which is why a shell, a fairing and a tank are drawn with sections and station lines rather than with transfers, and a bracket with a boss on it is not. The drawing office’s practice already encodes the inequality; what the arithmetic adds is that the quantity being compared is a curvature and not a shape.

Where the outline’s own shape comes from #

One more piece is worth having because it connects this rung to the one before it.

The image of the contour generator is the outline, and for a quadric it is a conic computable in one line — three matrix products and no sampling. So the drawn curve has a closed form even though the curve on the object it comes from moves with every change of view.

That is a clean separation and it is why both essays are possible. The outline is a property of the object and the projection together, computable without ever asking which points produced it; the contour generator is the answer to that second question, and nothing about the drawn conic depends on it.

Which is also the trap. A one-line formula for the outline makes it easy to treat the outline as the object’s own edge, since nothing in the arithmetic mentions that a curve on the surface was involved. The formula is a statement about the image, and the image is all it is a statement about.

What the drawing office does about it #

Nothing, and it is right not to.

The transfer rule is used to draw outlines, not to identify points, and for that purpose the identity of the point does not matter — what matters is that the outline in one view lands where the outline in another view says it should. For a surface of revolution about an axis in the drawing’s own plane, that always works, because the symmetry supplies the coordinates the rule assumes.

Where it breaks is a surface with no such symmetry, and there the office’s answer is to draw more of the object: section lines, cutting planes, a table of ordinates. A hull’s lines plan is a drawing that has given up on transfer altogether and puts a family of sections in its place.

So the practical statement is narrow and true. Transfer between views is exact for edges and vertices; for a curved surface’s outline it is a construction that produces the right picture and identifies no point on the object. A draughtsman using it to locate a feature — the highest point of a fillet, say — is doing something that has no meaning.

What this costs a reconstruction #

The drawing office can live with it. A reconstruction cannot, and the consequence is worth following one step because it is where the finding stops being about drawings.

Anything that builds a shape out of outlines — a silhouette-based reconstruction, a lathe operator reading two views, a program fitting a surface to a drawing — is treating the outline as a set of points on the object. It is not, and triangulating two views’ outlines as if corresponding points produces a surface that is systematically outside the true one, touching it only at the frontier points.

That is the visual hull, and the metrology field prices it: more views shave the excess away, at a rate, and there is a second error term that no number of views reaches at all. The bias measured here is the two-view case of the first term, and the fact that it is $\sqrt2 R$ rather than something small is why the visual hull needs many views rather than a few.

There is a second consequence and it is about what a picture of a quadric determines. An outline fixes the cone of rays that graze the surface, which is a strong constraint — but it fixes nothing about which points of the surface produced it, and the two facts are easy to conflate. The first is why one photograph of a ball gives its direction exactly; the second is why it gives no point on the ball at all.

Why √2 and not something else #

The number is worth a paragraph because its size is the whole reason this matters.

The two contour generators are perpendicular great circles. At the middle of the transfer — $x = 0$ — the front view’s point is at the top of the ball and the top view’s is at the back of it, and those are a quarter turn apart on a sphere of radius $R$. The chord across a quarter turn is $\sqrt2\,R$.

So the gap is not a small correction; it is the largest separation two points of the same great-circle radius can have without being antipodal. On a ball fifty millimetres across, the transfer line at the middle of the outline connects points thirty-five millimetres apart.

That is what stops the finding being a technicality. A discrepancy of a per cent could be argued away as a drafting tolerance; a factor that reaches $\sqrt2$ of the object’s own radius is a statement that the two views are looking at different parts of the thing.

The two facts a reader should keep separate #

There are two statements in this essay and they are easy to run together, so it is worth setting them out apart.

The first is about correspondence: the outline in one view and the outline in another are images of different curves on the object, so a point of one does not correspond to a point of the other. That is what the transfer rule quietly assumes and what fails.

The second is about position: the outline in each view is nevertheless exactly where it should be, because each is the correct image of its own contour generator. Nothing on the drawing is misplaced.

A reader who takes only the first away concludes that multiview drawings of curved parts are unreliable, which is false. A reader who takes only the second concludes that nothing has happened, which misses the point. The pair of them is the finding: the picture is right and the reading is wrong, and the reading is the one every textbook teaches without noticing that it has two cases.

That shape recurs on this site and is worth recognising. The taught cube constructions produce perfectly good drawings of solids that are not cubes; the equal-steps depth method produces a perfectly good picture of a row that is not equally spaced. In each of them the drawing is a drawing of something and the mistake is in what it is taken to be a drawing of.

Two rays that are not perpendicular #

The standard views are perpendicular, which is where the quarter turn comes from. The frontier points move if they are not.

For two rays at an angle $\theta$, the two contour generators are great circles whose planes are at $\theta$, so they still meet in exactly two points — the ends of the axis perpendicular to both — and the maximum gap is the chord across $\theta$ rather than across a right angle. As $\theta$ goes to nothing the two views coincide and so do the two curves.

frontierPoints refuses two views from the same direction, which is that limit. The refusal is the honest form of the statement: at zero separation there is no pair of curves to have frontier points between, and returning “every point” would be returning a fact about a degenerate case dressed as a result.

What this does not settle #

It does not treat a general surface. A ball’s contour generator is a great circle and the arithmetic is clean; a fairing’s is a curve found by solving, and the gap has no closed form — though the qualitative statement survives, because it comes from the contour generator moving rather than from the surface being round.

It does not treat the case where the surface is a developable, whose contour generator is a pair of its own generators — straight lines, which transfer more nearly like edges and still slide continuously as the view turns.

It does not treat perspective. Under a camera the contour generator is the section by the eye’s polar plane rather than a diametral circle, and it moves with the eye’s position as well as its direction.

And it does not say a multiview drawing of a curved part is wrong. Every line on it is where it should be. What is wrong is a particular way of reading it, and the reading is one nobody states because the rule that licenses it has never been separated into its two regimes.

A construction that produces the right picture is not thereby producing the right correspondence. On a flat-faced solid those two are the same thing, and every method learned on flat-faced solids inherits an assumption it does not mention.