An angle on the ground

The one thing a single view cannot give is the collection’s central metrological fact: a photograph supplies every ratio in a scene and no size at all, so one length has to come from outside it.

An angle is not a length and is not a ratio of lengths. It is scale-free — double the whole world and every angle is unchanged — so the fact above says nothing about it, and the closure an angle needs is a different fact entirely.

What an angle actually needs #

Recovering an angle between two directions on a plane needs the plane’s metric structure: enough to say which directions are perpendicular and what the ratio of lengths along two different directions is. That is strictly less than knowing a size and strictly more than knowing the plane’s outline.

What one picture of a plane determines sets out the ladder: a projective structure from the picture alone, an affine one once the vanishing line is known, and a metric one once two more constraints are added. An angle lives on the top rung of that ladder, and a length lives on the rung above it — the one with a size in it.

So a reader may have one and not the other. A photograph of a floor with a known square tile and no ruler gives every angle exactly and no length at all. A photograph with a metre rule lying along one direction and nothing else gives lengths along that direction and no angles.

Three quantities, one wrong assumption #

The cleanest way to show that the two closures are different is to break one of them and watch what moves.

The rectifier here is built from four marks of a reference rectangle whose aspect ratio is assumed. Sweep that assumption away from the truth and measure three things off the resulting map: an angle between two directions on the ground, a length running into the picture, and a length running along the reference’s own width.

At a quarter wrong the angle is out by 11.5 degrees, the length into the picture by 20 per cent — exactly the fractional error in the assumed aspect — and the length across by 3 × 10⁻¹² per cent, which is the arithmetic floor.

The third of those is the informative one. A length measured along the direction the reference’s width runs is completely insensitive to the aspect, because the aspect only rescales the other axis. So a reader with a wrong aspect gets some lengths right and some wrong, and there is nothing in the picture to say which.

Where the metric structure comes from #

The assumed aspect above is one way to supply it. There are others, and it is worth listing them because they are what a real photograph offers.

A known rectangle. Four corners and a stated aspect. This is the version above and it is what a door, a window, a paving slab or a sheet of A4 supplies.

Two perpendicular directions. A right angle on the ground, plus the vanishing line, fixes the metric up to a scale — because a right angle in the world constrains the two directions’ vanishing points to be conjugate with respect to the imaged circular points.

A circle. The image of a circle on the plane is a conic — the conic a circle becomes settles which conic, and its intersection with the vanishing line gives the imaged circular points directly. The two points a picture hides is where the collection builds that construction, and it is the most economical of the three: one drawn circle, and every angle on the plane comes back.

A known angle that is not a right angle, which is the same construction with more arithmetic and no more information.

Why the circular points are the honest account #

The four routes above are one route in four costumes, and the costume worth wearing is the circular points, because it makes the count of unknowns visible.

Metric structure on a plane is exactly the knowledge of where the two circular points sit on the vanishing line. Those are two points on a line, so two numbers, and every constraint above supplies two: a rectangle’s aspect fixes both, a right angle plus a known ratio fixes both, a circle fixes both at once.

That is why an angle and a length are different closures. The circular points give angles and ratios of lengths in different directions — the whole of the plane’s shape. The remaining freedom is a single overall scale, and that is the one thing they do not touch and the one thing a reference length supplies.

Two numbers for the shape, one for the size, and a photograph gives none of the three.

What goes wrong when the rectangle is not one #

The most common closure in practice is a rectangle assumed to be a rectangle, and the assumption is wrong more often than a reader expects.

A paving slab is nominally 600 by 600 and is cut to a tolerance. A door is nominally rectangular and has settled. A tennis court is rectangular to a standard and a football pitch is not — it has a range of permitted dimensions, so a pitch’s aspect is not one number at all. A sheet of paper is reliable; a window is not.

The sweep above prices those. A one per cent error in the aspect moves the angle by about half a degree in this arrangement; five per cent moves it by two and a half. So a reader who wants a ground angle to a degree needs an aspect known to about two per cent, which excludes most windows and includes A4.

The angle that costs nothing #

There is one angle a reader can always have, and it is worth naming because it is the exception that makes the rest of the essay legible.

An angle between two directions that both lie along the vanishing line’s own conjugate directions — in practice, an angle at a corner both of whose arms run along the two axes the reference rectangle defines — is fixed by the rectangle’s corners without the aspect entering. A right angle assumed at the reference is a right angle recovered at the reference, trivially, because it is the input.

That is not a free measurement; it is the input read back out. The useful version of the point is that angles near the reference’s own directions are less sensitive to the aspect than angles at forty-five degrees to them, so a reader with a shaky aspect should measure the angles most nearly aligned with it and distrust the ones between.

The aspect’s effect on an angle, exactly #

The sweep prices a wrong aspect empirically. It has a closed form, and the closed form contains both of the practical rules the sections above arrive at by inspection.

Assuming an aspect $k$ times the true one rectifies the plane to a copy of itself stretched by $k$ along one axis. A direction at $\theta$ to the reference’s own width therefore comes back at

$$\theta' \;=\; \arctan\!\left(k\tan\theta\right),$$

and for $k = 1+\varepsilon$ that is $\theta' \approx \theta + \tfrac{1}{2}\varepsilon\sin 2\theta$. An angle is between two directions, so what a reader measures moves by

$$\Delta \;=\; \tfrac{1}{2}\varepsilon\left|\sin 2\theta_1 - \sin 2\theta_2\right| \;\le\; \varepsilon \ \text{radians} \;=\; 57.3\,\varepsilon \ \text{degrees}.$$

A one per cent error in the aspect can cost at most 0.57° of angle, five per cent at most 2.9°, and the sweep’s 0.5° and 2.5° sit just inside those bounds because its two arms are not quite at the worst orientations. At a quarter wrong the linear bound gives 14.3° and the exact arctangent gives less, which is the 11.5° measured — the expression is concave, so a large aspect error costs less than proportionally.

Both of the essay’s practical rules are readable off it directly. The insensitive angles are the ones aligned with the reference, because $\sin 2\theta$ vanishes at $0°$ and $90°$; the worst are at $45°$ to it, where $\sin 2\theta$ is one. And a reader wanting a degree needs $\varepsilon \le 1/57.3 = 1.7\%$, which is the two per cent the section above quotes and is why A4 qualifies and a window does not.

One further reading, and it is the one that connects this section to the ladder. The map $\tan\theta \mapsto k\tan\theta$ is a one-parameter group acting on directions, and what it does not move is the pair $\theta = 0, 90°$ — the two directions the reference itself defines. That fixed pair is the residual freedom the metric closure is meant to remove: getting the aspect wrong does not produce a plane with no metric structure, it produces one with the wrong metric structure, and every angle on it is self-consistent. Nothing downstream can detect it, which is why the closure has to be right rather than merely plausible.

Angles between planes, and why they are harder #

Everything above is an angle in the ground plane. An angle between two planes — a roof pitch, the splay of a wall, the tilt of a ramp — needs more, and the extra is worth stating so that the two are not confused.

A single plane’s metric structure is two numbers on its own vanishing line. Two planes have two vanishing lines and two pairs of circular points, and relating them requires knowing something about the camera as well — the absolute conic, in the account one conic calibrates the camera gives. So a dihedral angle from one photograph is a calibrated measurement where an angle in one plane is not.

That is why the ramp has its own horizon treats an inclined plane by finding its own vanishing line rather than by relating it to the ground’s: the second is available only with the camera in hand, and the first is available from the marks.

An angle is a cross-ratio, which is why this works at all #

There is a reason the recovery is possible at all, and it is worth setting beside the sensitivity numbers because it explains why an angle is closer to a projective quantity than a length is.

An angle is a cross-ratio establishes Laguerre’s formula: the angle between two lines is a logarithm of the cross-ratio of four things — the two lines and the two lines joining their intersection to the circular points. Since the cross-ratio survives any projection, an angle survives a projection provided the circular points are known.

That is the whole of this essay in one sentence, and it puts the closure exactly where it belongs. The projection does not destroy the angle; it destroys the reader’s knowledge of where the circular points went. Supply them and the angle comes back exactly, through a construction that is joins, meets and a logarithm.

It also explains the sensitivities. A cross-ratio is a smooth function of its four arguments, and moving two of them — the circular points, which is what a wrong aspect does — moves the answer smoothly and by an amount that depends on how far the other two are from them. Angles near the reference’s own directions have their arguments close together and move little; angles at forty-five degrees have them spread and move more.

Why an angle is the quantity to want on a bad photograph #

There is a practical recommendation buried in the stratification, and it is the opposite of what a reader usually reaches for.

A reference length in a photograph is often the least reliable thing in it: a scale bar laid down carelessly, a person of assumed height, a car of assumed length. A reference shape is often the most reliable: a sheet of paper, a standard road marking, a tennis court’s service box, a brick’s face. Shapes are standardised more tightly than sizes are placed.

So on a photograph with no trustworthy length and one trustworthy shape, angles and ratios of lengths in one direction are available and metric lengths are not. That is a genuinely useful position to be in — a plan can be recovered up to scale, its angles are right, its proportions are right, and the whole of it can be scaled later when one length turns up.

The plan hidden in the photograph recovers exactly that object, and reading it after this essay makes clear which of its numbers came from the picture and which from the closure.

The boundary, stated #

One plane, its vanishing line, and a shape closure in the plane.

The vanishing line has to be found before anything metric can be attempted, and finding it is an affine measurement that has its own error — a horizon read ten pixels wrong is an affine structure ten pixels wrong, and every angle downstream inherits it. The sweep above holds the vanishing line exact and varies only the shape closure, so it prices one contribution and not both.

The plane has to be a plane, with the same complaint the previous two essays make about floors that dish and lawns that slope.

And an angle recovered this way is an angle in the world plane, not an angle the camera sees. A reader wanting the angle two lines appear to make on the print already has it with a protractor, and it is a different quantity — the whole content of this essay is that the two differ, and by how much.

Two closures, and which photographs offer which #

It is worth ending the argument with a table in prose, because the essay’s practical content is a matching problem between what a picture contains and what a reader wants.

A photograph offers a shape closure if it contains an object of known proportions lying in the plane: a rectangle with a stated aspect, a circle, a right angle with a known ratio of sides. It offers a size closure if it contains an object of known length in the plane, in any orientation.

A great many photographs offer exactly one of the two. A road with standard lane markings offers both, since the markings have a stated length and a stated width. A brick wall offers a shape closure from the brick’s face and a size closure from the same brick, which is why masonry is such a convenient subject. A tiled floor of unknown tile size offers a shape closure — the tile is square — and no size at all, and gives every angle and no metre. An unmarked field with a metre rule dropped in it offers a size and no shape, and gives lengths along the rule’s direction and nothing else.

Matching the closure to the question is most of the practical skill, and it is invisible until the two are named separately. Reporting “one length has to come from outside the picture” and stopping there conceals that a reader wanting an angle needs a different thing entirely, and may already have it.

What is measured here #

Three numbers under one wrong assumption, and a control.

With the reference’s aspect exactly right, the recovered angle is exact to 10⁻⁸ degrees and the recovered length to 10⁻⁸ per cent. With it a quarter wrong, the angle is out by 11.5 degrees, a length running into the picture is out by 20.0 per cent — which is exactly the fractional error in the assumption, to 10⁻⁶ — and a length running along the reference’s own width is out by 3 × 10⁻¹² per cent, which is the arithmetic floor.

That last one is the control. A closure that moved every quantity would be a closure this essay could not separate from a size, and the fact that one of the three does not move at all is what says the plane’s shape and the plane’s size are two different facts.

The short version #

An angle on a plane needs no length, because an angle is scale-free. What it needs is the plane’s shape — the two circular points on its vanishing line, which is two numbers — and that is a different fact from the one length a size needs.

Break the shape closure and the three quantities separate: the angle moves by degrees, a length into the picture moves by exactly the assumption’s own error, and a length along the reference’s own direction does not move at all. A reader with a wrong aspect therefore has some measurements right and some wrong, with nothing in the picture to say which.