A curved surface made of straight lines

A cooling tower curves in two directions at once and is built out of straight bars. Both statements are true of the same surface at the same time, and the second is not an approximation to the first — the bars are exactly straight, and the surface they sweep out has negative Gaussian curvature everywhere.

The surface is a hyperboloid of one sheet, and it carries two families of straight lines rather than one. Through every point of it two of them pass, one from each family, crossing at an angle the surface’s proportions decide. That is why the shuttering of such a tower is a lattice: the straight members are the surface, laid in two directions.

So a photograph of one contains a large number of exactly straight lines that bound nothing flat. This essay measures their straightness in the picture, finds where they are all heading, and shows that the answer is one conic rather than two — then puts a sphere and an ellipsoid beside the tower as the cases where none of it happens.

The rulings, written down #

The two families are not found by searching. They have a closed form, and having it is what makes everything below a computation rather than a fit.

Write the surface as

$$\frac{x^{2}}{a^{2}} + \frac{z^{2}}{c^{2}} - \frac{y^{2}}{b^{2}} = 1,$$

and take the two curves

$$L_{\pm}(\theta, t) = \bigl(a(\cos\theta \mp t\sin\theta),\; b\,t,\; c(\sin\theta \pm t\cos\theta)\bigr).$$

Each is linear in $t$, so each is a straight line for every fixed $\theta$. Substituting into the surface equation gives $1 + t^{2} - t^{2} = 1$, identically, for both signs and every $\theta$ and every $t$. That is a two-line proof that the surface carries two one-parameter families of lines, and it is checked here at ninety points of both families before anything is drawn.

Two things about that form are worth reading rather than passing over. The two families differ only by a sign, which is why they are congruent to each other: reflecting the surface in a plane through its axis exchanges them. And $\theta$ parameterises the waist circle, so each family is indexed by where its line crosses the narrowest part — which is exactly how such a tower is set out on site.

The surface is doubly ruled, which is a small club. A plane is doubly ruled trivially. A hyperbolic paraboloid is the other quadric in it, and nothing else in three dimensions is: a theorem of Cayley’s says the only doubly ruled surfaces are those three. Singly ruled surfaces are far more common — a cone, a cylinder, and every developable surface a sheet of metal can be rolled into — and they are a different kind of object, because a developable can be flattened without stretching and a hyperboloid cannot.

The measurement, and the curve that is not straight #

That the drawn generators are straight is not a surprise: a projection through a centre carries a straight line to a straight line, so the claim is a consequence of a fact this collection has used since its first page. What makes it worth measuring is what the measurement is against.

The parallel is the control and it is chosen to make the comparison airtight. It lies on the same surface, is drawn by the same code, is projected by the same camera, is sampled the same number of times, and is measured against its chord in the same way. The only difference between the two readings is what curve was fed in. A residual of 1.4e-13 pixels beside one of 120 is therefore a statement about the geometry rather than about the arithmetic, and if the projection had a defect in it that bent straight lines, the generator’s profile would show it.

That control is doing more work than it looks. The obvious alternative — measure the generator, find a tiny number, declare success — is a measurement whose instrument has never been shown to be capable of returning anything else, which is the failure this collection has shipped several times and now watches for. Here the same instrument returns 120 pixels on a curve of the same length on the same surface.

The parallel is also the curve a reader is most likely to confuse with a generator on a real photograph, because the tower’s horizontal ribs are parallels and its diagonal members are generators. In the picture the ribs bow and the diagonals do not, which is a way of telling them apart with a straightedge and no other information.

Where the generators are heading #

A family of straight lines in the world has a family of directions, and every direction has a vanishing point — the place in the picture where parallel lines meet. The generators of one family are not parallel to each other, so they do not share a vanishing point; each has its own. The question is what the set of them looks like.

The derivation is short. A generator’s direction is the derivative of $L_{\pm}$ with respect to $t$, and substituting a pure direction into the surface’s quadratic form kills the constant term, leaving the asymptotic equation — the same quadratic form set to zero rather than to one. That equation is a cone of directions, which is a conic on the plane at infinity, and the camera maps the plane at infinity to the picture by a plane homography. A conic mapped by a homography is a conic. So the vanishing points of the generators lie on the image of the asymptotic conic, exactly, and the image is one matrix product away from the surface and the camera.

Nothing is fitted. The conic in that figure is not a curve found by least squares through forty marks; it is computed and the marks are then checked against it. That is the same discipline as computing a quadric’s outline from the dual quadric rather than tracing it, and it has the same benefit: the residual measures the geometry rather than the fit.

One conic, not one each #

The natural expectation is that two families of lines give two conics. They do not, and the reason is worth stating carefully because it is the essay’s best finding.

Both families satisfy the same asymptotic equation. The plus and minus rulings differ by a sign in their parameterisation, but their direction vectors are two different solutions of one homogeneous quadratic, not solutions of two different ones — the asymptotic cone is a single object belonging to the surface, and each family traces one of the two ways round it. So the forty vanishing points lie on one conic, and a conic fitted to the plus family alone agrees with a conic fitted to the minus family alone to 4.8e-18 on normalised coefficients, which is well past any level a difference could hide at.

That has a consequence for reading a photograph. A conic through five points is determined exactly, with nothing left over, so five vanishing points from either family — or three from one and two from the other — determine the whole thing. A reader marking the diagonal members of a real tower does not have to know which family each belongs to, which matters because in a photograph the two are hard to tell apart.

The control on that claim is not the sphere and the ellipsoid below; it is a curve on the same surface. The tangents of the tower’s own parallels are 715 pixels off the same conic at the nearest, against the rulings’ 1.6e-12. The parallels are curves on the surface and their tangent directions are not asymptotic directions, so they have no business on that conic and are measurably not on it.

The controls: two surfaces with no lines on them #

The claim “a hyperboloid is ruled” is only worth something if there are surfaces that are not, and if the difference shows up as a number rather than as a search that failed to find anything.

The test is three lines of algebra rather than a search, which is what makes it a control rather than a failure to find something. A line on the surface through a point of it must lie in the tangent plane there. Restrict the quadratic form to that plane and two terms vanish for free: the constant, because the point is on the surface, and the linear term, because the plane is tangent. What is left is a binary quadratic, and a binary quadratic factors into two real linear forms exactly when its discriminant is positive. So the surface carries two real lines through the point when the discriminant is positive, one when it is zero, and none when it is negative.

The three readings are 2.57, -4.00 and -1.00, and the two negative ones are not small negative numbers that a better search might turn positive. They are the algebra saying there is nothing there.

The sign of that discriminant is the sign of the Gaussian curvature turned round. A surface with negative curvature at a point is saddle-shaped there and has two asymptotic directions; a surface with positive curvature is bowl-shaped and has none. So “ruled” and “saddle-shaped everywhere” are close relatives, and the reason a sphere carries no straight line is the reason it has no saddle point.

An outline is a conic whichever surface it belongs to #

The most useful half of that control is the half that reads negative, and it deserves its own figure because it is the misreading most available to somebody looking at a photograph.

Every quadric’s outline is a conic, because the outline is the section of the surface by the eye’s polar plane and the section of a quadric by a plane is a conic. A sphere gives one, an ellipsoid gives one, and a hyperboloid gives one. So a conic in the picture is no evidence at all that the thing photographed carries straight lines, and the elegance of the outline is not the elegance of the ruling.

That distinction has a practical form. Somebody reading a photograph of a curved building has two conics available to them — the outline, which is a conic belonging to the pair surface and eye, and the vanishing conic, which is a conic belonging to the surface alone. The first moves when the camera moves and the second does not, which is a test a reader with two photographs of the same object can run.

The silhouette is not one of the straight lines #

One more misreading, and it is the one the picture itself invites.

The name does not help. “Contour generator” and “generator” are two different objects that share a word: the first is the curve where the surface turns away from the eye, and the second is a straight line lying in the surface. On a tower photographed from the side the contour generator runs down the visible edge of the shape and looks, at a glance, exactly like the outermost straight member.

Sixty-seven degrees is not a near miss. The two curves cross at a large angle, and the distinction between them is the distinction between a curve that belongs to the object and a curve that belongs to the object and the observer together — which is why a hollow the outline never reaches leaves no trace in any photograph and why one picture of a ball recovers less than it seems to.

There is a straightening at the far end of the family that is worth naming without measuring it here. Stretch the surface along its axis and the two families of generators turn towards each other; in the limit they coincide, the angle between them goes to zero, and the surface is a cylinder — a ruled surface with one family instead of two, and the honest degenerate end of the arrangement. The tangent-plane discriminant follows the two families down and reaches zero exactly where the two lines become one.

What a reader with one photograph can do with it #

The three curves above are computed from the surface and the camera. A reader has neither, so it is worth separating what survives that restriction.

The straightness does, and it is the easiest thing on the list. Lay a straightedge along a diagonal member in the print. If it follows, the member is a straight line in the world; if it bows, the member is a rib. That is a decision made with no camera, no calibration and no measurement, and it is available because a projection carries a line to a line and destroys almost everything else — the one surviving property being exactly the one the test uses.

The vanishing conic survives too, and it is the more valuable half. Mark five diagonal members, take each one’s vanishing point by extending it against its neighbours in the same family, and five points determine the conic exactly — or, if the members are easier to lay tangents against than to intersect, five tangents determine the same curve by the dual algebra. What comes back is the image of the surface’s asymptotic cone, and from that the surface’s proportions follow: the ratio of the waist radius to the height scale is what decides how open the cone is, and the drawn conic is that cone seen from the camera’s attitude.

What does not survive is anything about the outline. A reader can mark the silhouette perfectly and learn only about the pair of surface and eye, which is why a photographed tower’s silhouette is a poor thing to fit a shape to and its diagonal members are a good one.

Two neighbouring cases sharpen the contrast. A ball drawn in an axonometric projection is not drawn round, because the outline is a conic whose shape depends on the projection and not on the sphere. And the difference between a drawing of a curved plate and its development is exactly the difference a hyperboloid does not have: a developable can be unrolled flat and this surface cannot, so there is no flat pattern for a cooling tower and its straight members are not a flattening but a structure.

What the measurement does not settle #

Three limits, in the order a reader would meet them.

The straightness result is a statement about an exact pinhole. A real lens bends straight lines, most at the frame’s edge, and a photograph of a tower filling a wide frame will show generators that are measurably not straight for reasons that have nothing to do with the surface. Everything here would have to be preceded by a distortion correction, which is a separate problem with its own machinery.

The vanishing conic is exact and it is not easy to get at. Twelve of the forty vanishing points here are near enough to the picture to draw at all, and the rest are further out than the diagram; on a real photograph each one has to be found by intersecting a bundle of marked lines, and the ones far from the frame are located badly by short bundles of nearly-parallel edges. So the conic is determined in principle by five of them and determined well by rather more, and by the ones nearest the picture.

And none of it recovers the surface. The vanishing conic is the image of the asymptotic cone, which is a statement about the surface’s directions — its proportions and its attitude — and carries nothing about where it is or how large. That is the ordinary condition of a single view, and it is the same one length a photograph never supplies turning up in a new place.

One machine, three readings #

What this arrangement is an instance of is worth naming at the end, because the same three matrix products have now been asked three questions.

The dual quadric gives the outline, which is a conic belonging to the surface and the eye. The quadratic form restricted to a tangent plane gives the ruling test, which is a discriminant belonging to the surface and a point of it. And the same form set to zero rather than one gives the asymptotic conic, whose image is a conic belonging to the surface and the camera’s attitude and nothing else. Three conics in one picture of one object, each answering a different question, and none of them interchangeable with the others.

That is the shape this collection keeps finding, and the reason it draws figures rather than states results. A photograph of a cooling tower has all three curves in it. Nothing about the photograph says which is which, and the difference between them is the difference between a fact about the building, a fact about the camera, and a fact about both.