One parameter between two surfaces

The curved field on this site compares six picture surfaces and treats each as a fixed object: the plane, the cylinder, the sphere unrolled, and three fisheyes. That framing hides something, because two of those six are the ends of a one-parameter family and the interesting members are in between.

The family has a name in the software that implements it, and one setting of it is what most wide architectural pictures are made on. What it has never had is a number saying why that setting.

The construction #

Take a cylinder of unit radius about the vertical through the eye, as the cylindrical panorama does. A cylindrical picture projects each direction onto that cylinder from the eye, at the axis. This family projects from a point $d$ behind the axis instead, along the view direction, and then unrolls.

Writing $\theta$ for the azimuth of a direction and $r$ for its horizontal distance from the axis, the map is

$$u = (d+1)\,\frac{\sin\theta}{d + \cos\theta}, \qquad v = (d+1)\,\frac{y/r}{d + \cos\theta}$$

At $d = 0$ that is $u = \tan\theta$, which is the flat plane exactly — the same $x/z$ every photograph is. As $d$ grows the azimuth flattens toward the cylinder’s even spacing without ever reaching it.

The one member with a closed form worth naming is $d = 1$, where the azimuthal half becomes $2\tan(\theta/2)$. That is the stereographic map of the horizon circle, and it is the entire geometric argument usually given for the value everybody uses.

What every member keeps #

One property runs through the whole family and it is the reason the family exists.

Every member images a vertical world line as a vertical straight line. The azimuth is constant along a vertical, and the map’s first coordinate depends on nothing else, so $u$ is constant and the image is a vertical segment. Measured over five members at $d = 0, 0.5, 1, 2, 4$, the worst departure from straight is under $10^{-16}$ of the chord — the arithmetic floor, at every member.

That is why architectural photographers reach for it. A wide flat picture of a building keeps its verticals too, but only by paying at the edges; a cylindrical picture keeps them and bends every horizontal; and this family keeps them at every setting while offering a choice about everything else.

And only the member at $d = 0$ keeps a general line straight, which is the price. At $d = 0.5$ a general straight line bows by 4.81% of its own chord, at $d = 1$ by 7.21%, at $d = 2$ by 9.61%, at $d = 4$ by 11.54%. The bend grows monotonically with the parameter and there is no member but the plane where it is zero — which is Beltrami’s theorem again, arriving inside a family rather than across one.

The battery #

The site’s curved field measures a surface on three things: how far it bends a straight line, how far it moves a right angle, and how far the area scale runs. Run across the family, all three produce a table with a shape.

The bend column is monotonic. The area column is nearly monotonic, falling steeply and then flattening.

The angle column is not monotonic at all. It falls from 48.06° at the plane to 16.84° at $d = 1$, and then rises again to 23.13° at $d = 2$ and 32.71° at $d = 4$. It has a minimum, and the minimum is close to the value everybody uses.

Why the angle column is the one to read #

Three columns and only one of them has a minimum, which invites the question of why that one carries the argument.

The bend column cannot have an interior minimum. Straightness is a property the plane has and nothing else has, so any departure from $d = 0$ costs bend and every further departure costs more. A family containing the plane has its straightness optimum at an end by construction, and an end is not a choice — it is the refusal to choose.

The area column falls steeply and flattens. Its best value is at the far end of the family, which is a surface nobody uses for a picture of a building, so the column argues for a value the practice rejects. That is a signal that area is not what the practice is optimising, and it is worth saying rather than quietly dropping the column.

The angle column is the only one whose best value is interior. Both ends are bad — the plane is 48.06° at the corners of a 130° field, the far end 32.71° — and the middle is not. So if the convention is defensible on any of the three quantities the site measures, it is defensible on this one, and the measurement is what turns that “if” into a number.

Where the minimum is #

Scanning the parameter finely rather than at five marks puts the minimum of the worst angular error at

$$d = 1.04$$

at 16.82°, over a 130° by 50° field. The minimum of the worst anisotropy — the other half of conformality, which this site insists on measuring separately after the cylinder passed an angle-only test three phases ago — sits at $d = 0.98$.

So the conventional value is within four per cent of both minima, and the two minima are not the same point. That is a real defence of the convention and a qualified one, and both halves are worth stating.

The defence: the value in universal use is the one that minimises the worst angular distortion over the field, and the minimum is genuine rather than a plateau — the curve rises by a third of its value by $d = 2$.

The qualification: the two criteria disagree, so a picture chosen for shape fidelity and one chosen for isotropy are not the same picture, and neither is exactly at $d = 1$. The usual argument for the value — that the azimuthal half becomes stereographic there — picks out $d = 1$ exactly and is a statement about half the map. It happens to land near the minimum of a property of the whole map, and the near-coincidence is not an explanation.

The minimum moves with the field #

That last caveat is worth a measurement of its own rather than a sentence, because it decides how much the result is worth.

The battery samples a rectangular fan of directions, and the numbers in the table are worst cases over that fan. Widen the fan and the worst case is taken further off axis, where the family’s members differ more; narrow it and they converge toward the plane.

So the minimising parameter is a function of the field, and quoting it without the field is quoting half a result. Over the 130° by 50° fan used here it is 1.04. That is the number this essay claims, and the figure prints the field beside it for exactly that reason.

What does not move is the shape of the curve. There is a minimum at every field width tried, it is interior rather than at an end, and it is near one. A family whose angular error fell monotonically would have no defensible interior value at all, and this one does.

What the reader can check #

Everything above is a measurement on a map, and the map is short enough that a reader can check the two claims that matter with a calculator.

Verticals. Take a vertical world line at azimuth 34°. Its directions all share that azimuth, the map’s $u$ depends only on the azimuth, so every point of the image has the same $u$. That is a two-line argument and it holds at every $d$; the figure’s number, under $10^{-16}$ of the chord, is the arithmetic confirming it rather than the evidence for it.

The plane at zero. Set $d = 0$ in the map. The first coordinate becomes $\sin\theta / \cos\theta = \tan\theta$, which is $x/z$; the second becomes $(y/r)/\cos\theta$, which is $y/z$. So the member at zero is the pinhole exactly, not approximately — the family contains the flat picture rather than approaching it.

The claims that cannot be checked by hand are the three columns, because each is a worst case over a fan of directions with a Jacobian differenced at every one of them. That asymmetry is deliberate: the two structural facts are arguments and the three numbers are measurements, and this site does not let a measurement stand in for an argument or the other way round.

What it is between #

Naming the ends correctly matters, because the family is often described as running from rectilinear to cylindrical and it does not.

At $d = 0$ it is the flat plane, exactly. As $d \to \infty$ the azimuthal map tends to $\sin\theta$ rather than to $\theta$, which is an orthographic azimuth rather than a cylindrical one. So the far end of the family is not the cylinder; it is a different surface again, and the family passes near the cylinder without containing it.

That matters for the reading of the area column. The area range falls steeply through the family — ×17.80 at the plane, ×2.65 at $d = 1$ — and then stops falling and turns back up slightly by $d = 4$, which a family running to the cylinder would not do. The turn is the far end being a compressive map rather than an even one.

The vertical is not free either #

One more measurement, because the family’s headline property has a limit that its users meet.

Verticals are straight at every $d$, and they are not evenly spaced. The vertical coordinate carries the same $(d + 1)/(d + \cos\theta)$ factor as the horizontal one, so a vertical line far off axis is drawn longer than one at the centre — by that factor, which at $d = 1$ and 65° off axis is about 1.4.

So a row of identical columns across a wide Panini picture is drawn straight, upright, parallel and increasingly tall toward the edges. That is the family’s characteristic look and it is not an artefact: it is the surface reporting that the edge columns subtend more of the picture, which they do.

A flat picture has the same effect and larger. A cylindrical picture has it too. There is no surface in which a row of equal verticals is drawn equal, straight and evenly spaced at once, which is the family’s trade stated in the form a photographer meets it.

Where it sits in the family of answers #

Three ways of making a very wide picture have now been measured on this site, and the family is the third of them.

The compass recipe chooses stereographic and accepts the area cost — every angle exact, every shape right, and the sizes gone. The cube map chooses six planes and accepts the seams — every line straight inside a face, and a kink where two faces meet. The Panini family chooses neither and takes a little of each cost.

What separates the three is not quality. It is which of the three impossible properties each declines to give up:

• the compass recipe keeps shape and gives up size;
• the cube map keeps straightness and gives up continuity;
• the family keeps verticals and gives up a little of everything else.

That third one is the odd member, because “keeps verticals” is not one of the three properties the field’s battery measures. It is a property of a subset of lines rather than of the map, and a family selected for it is a family selected for what a building looks like rather than for what a projection is. Naming that honestly is most of what this essay adds: the convention is not a compromise between two general goods, it is an optimisation of one specific line family with a defensible parameter attached.

The short version #

The projection most wide architectural pictures are made on is one member of a one-parameter family running from the flat plane through a compromise toward an orthographic azimuth. Every member draws a vertical world line as a vertical straight line, to the arithmetic floor. Only the member at zero — the plane — draws a general line straight, and the bend grows monotonically with the parameter.

The worst angular error over the field does not grow monotonically: it has an interior minimum, at $d = 1.04$ over a 130° field, and the conventional value of 1 sits within four per cent of it. The worst anisotropy has its own minimum at 0.98, so the two criteria do not agree exactly.

That is a defence of the convention, and it is a better one than the usual argument, which names an exact property of half the map and says nothing about the other half.