Where parallel lines meet

Two rails run parallel across a plain. In the photograph they converge, and if the plain is large enough they appear to touch. They do not touch; a train that reached the meeting point would find the rails as far apart as ever.

The convergence is real and the meeting is real too, in a sense that took a long time to make precise. The rails meet at a point that is not in the plane the rails lie on, and the picture shows that point at a definite pixel. What the picture cannot show is that the point is unreachable.

Adding the points that were missing #

The ordinary plane has an irritating exception in it. Two distinct lines meet in exactly one point — unless they are parallel, in which case they meet in none. Almost every statement in plane geometry has to carry that exception, and almost every construction has a degenerate case where a needed intersection does not exist.

The repair is to add the missing points. For each direction, add one new point, declared to lie on every line of that direction. Two parallel lines now meet, at the point belonging to their common direction. The exception disappears, and two distinct lines meet in exactly one point becomes true without qualification.

The new points are the points at infinity, one per direction. Collected together they form a new line, the line at infinity, and the plane with these additions is the projective plane.

Two things about this construction are worth being clear on.

It is not a statement about very large distances. The point at infinity belonging to a direction is not “a point very far away”; it is a point that no amount of travelling reaches, and the difference matters because a point very far away has a position and this one has only a direction.

And it is not a fiction adopted for convenience. Under projection the points at infinity behave exactly like the others: they have images, the images are ordinary pixels, and a construction can pass a line through them, intersect them, and put them into a cross-ratio with no special handling anywhere.

The horizon is the image of that line #

A camera looking at a ground plane projects the whole plane, including its line at infinity. The image of that line is the horizon.

That sentence contains the entire theory of the horizon, and several consequences that are usually taught as separate rules fall straight out of it.

The horizon is at eye level. The points at infinity of the ground are, from the eye’s point of view, in the directions that are exactly horizontal — the rays that never descend to meet the ground and never rise. Those rays leave the eye level, so their images lie at the height of the eye in the picture. This is why the horizon cuts every standing figure at the same fraction of its height, and why the rule needs a condition on the camera’s tilt that is almost never stated with it.

The horizon does not move when the camera moves horizontally. Points at infinity have direction and no position, so translating the eye sideways does not change which direction each one names.

Every horizontal direction has its vanishing point on the horizon. A vanishing point is the image of a point at infinity; the horizontal ones lie on the line at infinity of the ground; so their images lie on that line’s image. Any construction that puts a horizontal vanishing point off the horizon has made an error that this fact makes checkable.

A direction parallel to the picture plane has no vanishing point in the picture. Its ray from the eye never meets the picture plane, so the image of its point at infinity is itself at infinity. The lines stay parallel in the drawing. This is the whole of what “one-point perspective” means, and it is a property of the orientation and not of the method.

The departure is worth watching rather than asserting, because it is continuous and the drawing gives no sign of it. Turn the same family of ground lines toward the picture plane and the meeting point does not fade or blur; it walks along the horizon and keeps walking, and the fit that finds it stays exact the whole way.

Finding a vanishing point rather than choosing one #

A vanishing point in a taught construction is placed. The horizon is drawn, two points are marked on it where the composition wants them, and everything follows.

That is backwards for this site’s purposes, so the figures here do the opposite: the vanishing point is found from the drawn lines, by least squares, and the residual is reported.

The reason for least squares rather than an intersection is the useful part. Two lines always meet somewhere. Given any two drawn lines that are not parallel, an intersection exists, and it proves nothing whatever about whether the lines came from a parallel family. Three or more lines meet at a single point only if they really do share one, and the largest distance from the fitted point to any of the lines is the number that says so.

For the twelve edges of a projected box that residual runs at 10⁻¹³ px, which is arithmetic noise. For a hand-drawn set of edges it does not, and the size of the residual is the measurement that separates a picture that is a projection from a picture that resembles one.

Where the vanishing points actually are #

An observation that surprises people who have only met vanishing points in a construction: they are usually not in the picture.

For an ordinary photograph of a box — a 44° lens, the box at a comfortable angle — the two horizontal vanishing points are typically several picture-widths off to either side, and the vertical one is thousands of pixels above or below. The construction taught in books puts them on the page because a page is what the construction is drawn on, and putting them on the page forces the two horizontal directions much closer to the picture plane than any real object usually is.

That has a practical consequence for anyone drawing. A construction laid out so that both vanishing points fit comfortably on the sheet is a construction for a very wide lens, and it will look like one: the near corner of the box juts, the receding faces flatten fast, and the object reads as having been photographed from a foot away. It is not that the drawing is wrong. It is correct from a viewing distance nobody will use, and the drawing method chose that distance without mentioning it.

The projective plane has no special line #

Adding the points at infinity produces a plane in which the added line is indistinguishable from any other. That is a stronger statement than it sounds, and it is the reason projective geometry is a subject rather than a patch.

A projection can send the line at infinity anywhere. Photograph a ground plane obliquely and its points at infinity land on the horizon, which is an ordinary line across the middle of the frame. Photograph it from directly above and the horizon goes back to infinity, and the picture has no vanishing points at all — a parallel projection, arrived at as a limit.

So which line is “at infinity” depends on the projection, not on the plane. In the projective plane there is no distinguished line, and the distinction between parallel and intersecting lines — which feels fundamental — is an artefact of choosing one line to remove.

That has a directly useful form. Any statement about a picture that would be easier if some troublesome line were at infinity can be made so, by applying a homography that sends it there. Rectifying a photographed façade to a flat elevation is exactly this: find the horizon of the façade’s plane in the photograph, apply the homography that sends it to infinity, and the façade’s parallels become parallel again and its measurements become readable.

What that homography buys, and what it does not, is a ladder rather than a switch, and the same camera draws all of it.

What a vanishing point is not #

Three confusions are worth clearing, because each one produces drawings that look plausible.

It is not a point of the object. Nothing is there. A vanishing point is the image of a direction, and the object’s own extent ends far short of it. Edges are run toward it and stop.

It is not a property of the scene. The same box has different vanishing points in every photograph of it, because the vanishing point depends on the camera’s orientation. It depends on the direction of the world edges and on the camera, and on nothing else — not on distance, not on the object’s position in the frame, not on how large the object is.

There is no such thing as “the” vanishing point. Every direction has one. A box has three because it has three edge directions; a hexagonal prism has more; a curved surface has one for every tangent direction. Constructions that speak of the vanishing points as though a picture had two or three are describing the object, not the picture.

The count is about orientation, not method #

The last of those is worth its own paragraph, because “two-point perspective” is taught as a technique.

A box has three families of parallel edges. Each family has a vanishing point. Whether that vanishing point is at a finite place in the picture depends on whether the family is parallel to the picture plane — and that depends on how the box is turned and how the camera is aimed, not on which method the drawer used.

Turn the box so that one family is parallel to the picture plane and its vanishing point leaves; two remain; the drawing is called two-point. Turn it so that two families are parallel and only one remains; the drawing is called one-point. Tilt the camera so that even the verticals converge and all three are finite; three-point.

The names describe a count of finite vanishing points, and the count is a fact about the arrangement. One construction produces all three, and the transition between them is continuous: a vanishing point does not appear, it arrives from infinity as the corresponding edges rotate out of parallel with the picture plane.

Why this is where the site starts #

The points at infinity are the reason a picture can be checked at all.

Without them, a vanishing point is a place where lines seem to head, which is an observation about a drawing and cannot be tested. With them, it is the image of a specific point, computable from the camera in closed form, findable from the drawn lines by least squares, and comparable between the two. The comparison is a number.

Every measurement on this site that goes anywhere depends on that: the recovery of a camera from three vanishing points, the check that a shadow’s vanishing point sits on the horizon, the demonstration that the taught depth constructions are not projections. All of them are statements about points that are not in the scene and not reachable from it, and all of them are checkable to the last bits of a double.

Duality, and the second way to say all of this #

The projective plane treats points and lines symmetrically, and running the whole of this essay through that symmetry gives a second description that is sometimes the more useful one.

In the dual reading, a line is the primitive object and a point is what a pencil of lines has in common. The horizon stops being a line made of points at infinity and becomes the pencil of all horizontal directions. A vanishing point stops being a place and becomes the fact that a particular family of drawn lines is concurrent.

That reading matches what the measurement actually does. vanishingPointOf is handed a set of drawn segments and asked whether they are concurrent; the point it returns is a by-product, and the residual — how nearly concurrent they are — is the answer. A drawing whose edges are not concurrent has no vanishing point, and saying so is more honest than reporting the least-squares meet of a set of lines that do not meet.

Duality also disposes of a question that comes up whenever the horizon runs off the frame. If the camera looks steeply down, the horizon is far below the picture, and it is tempting to conclude the picture has no horizon. It has one; it is not in the frame. The pencil of horizontal directions still exists, the concurrency of any family of horizontal edges is still checkable from the drawn edges alone, and the fact that the meeting point is at x = 4,300 px on a 690 px canvas is a statement about where it is rather than about whether it is.

Curved lines and the limits of the idea #

Everything in this essay is about straight lines, and the reason is that straightness is what a projection preserves. It is worth marking where the machinery stops.

A curve has no single vanishing point. What it has, at each of its points, is a tangent direction, and each tangent direction has a vanishing point of its own. For a curve that flattens out toward a limiting direction — a road curving and then straightening — the tangents’ vanishing points converge to the vanishing point of the final direction, and the drawn curve heads for it. For a curve that keeps turning, they do not converge at all.

The one clean case is a curve lying in a plane. Its image is a conic, and the conic’s relation to the horizon of that plane says which kind: a circle on the ground entirely in front of the eye images to an ellipse, a circle that crosses behind gives a hyperbola, and one tangent to the eye-height plane gives a parabola. The three cases are distinguished by how the circle meets the plane’s line at infinity, which is the projective way of saying the same thing without mentioning the eye at all.

That classification is exact and it does not extend to the circle’s other properties. The centre in particular does not survive, for the same reason a midpoint does not: both are defined by a ratio, and a ratio is one of the things a projection destroys.

The vanishing point as a measuring instrument #

This essay finds a vanishing point from drawn lines and reports the residual, and the residual is presented here as evidence about the drawing. The expansion phase turned it into the first step of a measurement, and it is worth noting what changes.

Nothing about the recovery changes at all. Three or more image lines known to be parallel in the world, a least-squares intersection, and a residual that says whether they really do share a point. Two such points give the horizon, and the horizon plus the vertical vanishing point turn a cross-ratio along any standing object into that object’s height.

What changes is the status of the residual. In a drawing it is a verdict on the illustrator. In a photograph it is the dominant term in an error budget: the horizon’s position error moves the horizon crossing directly, and the horizon crossing is one of the four points the whole measurement runs on.

Which bundle of lines to use therefore stops being a convenience and becomes the largest decision in the procedure. A bundle running across the view gives a well-conditioned point. A bundle running along the view gives one far off the frame, where a small angular error in any line moves the point a great distance. And a bundle exactly parallel to the picture plane has its vanishing point at infinity and no fit at all — which is where this essay began.

Past the transition #

The vanishing point of a family of ground lines runs off the frame as the direction turns toward parallel with the picture plane, ceases to exist there, and returns from the other side. A construction whose sides diverge is past that transition.