The plane of focus a tilted lens tilts

What a lens is doing drew three rays through a thin lens and found every point of an object brought to a point of the image. The focus that is a slab, not a plane found that a lens focused at one distance renders a range of distances acceptably sharp, a slab of space whose thickness the aperture controls. The flat scene that comes back curved found that a real lens images a flat object onto a curved surface, and the four cosines at the edge of a photograph followed what happens to the light at the edges of the field.

All of those kept the lens square to its axis and the object plane square to it as well. This essay tilts them. A camera pointed down a road at a low angle, a copy camera photographing a page held at a slant, a surveyor’s camera in an aeroplane looking at the ground obliquely: in each, the plane of interest is not square to the lens, and an ordinary camera can make only a strip of it sharp. The remedy is a geometric fact about thin lenses so simple it is easily missed — a lens maps planes to planes — and a rule about how those planes are arranged, which an Austrian army officer, Theodor Scheimpflug, patented in 1904 for correcting aerial photographs.

Planes into planes #

The thin-lens equation, $1/u + 1/v = 1/f$, says where a point at distance u in front of the lens is imaged, and the magnification v/u says how far from the axis. Take the points of a plane tilted to the axis: each sits at a different distance and is imaged at a different distance behind. It is not obvious that the images lie on a plane, but they do. The thin-lens map from object space to image space is a projective transformation, the kind of map that takes straight lines to straight lines and planes to planes; the four traced points in the figure fall exactly on one line.

The image plane is tilted, but by less than the object plane. And the two have a simple relation to each other and to the lens: extended, the object plane, the plane of the lens and the image plane all pass through one line. In the figure that line is 150 units below the axis, and all three planes meet there. That is the Scheimpflug rule.

The reason it must hold can be seen without algebra. Where the object plane crosses the lens plane, its points are at zero distance from the lens, and a point at zero distance is imaged at itself: a ray through the lens at that point has not yet been bent. So the line where the object plane meets the lens plane is part of its own image, which means the image plane must contain it too. The three planes share it because the lens leaves it alone.

A map that stretches depth twice as fast #

The projective map that makes all this work has a feature that matters for what a tilted image looks like. Sideways, a thin lens scales the scene by the magnification m. Along the axis it scales distances by $m^2$, because the image distance changes faster than the object distance: move an object a millimetre closer and its image moves $m^2$ millimetres further away. Lateral and longitudinal magnifications agree only at unit magnification.

That is why the image plane leans less than the object plane when the magnification is small, and it is also why the image of a tilted subject, though sharp, is not a scaled copy of it. A square drawn on a tilted table top is imaged as a trapezium, foreshortened along the direction of the tilt and with its near edge larger than its far edge: the perspective an eye would see from the camera’s position, with no correction. Scheimpflug’s own purpose was to remove exactly that. Aerial photographs taken obliquely from balloons and aircraft show the ground in perspective, and to turn them into maps he projected each negative through an enlarging lens onto an easel tilted to undo the perspective. With the negative, the lens and the easel tilted to satisfy his rule, the whole rectified image was sharp on the easel at once.

The mismatch between $m$ and $m^2$ is also why no lens can image a volume perfectly at any magnification but one. Abbe’s sine condition, which the condition a lens must meet found sets what a lens must do off the axis, and the corresponding condition along the axis cannot both hold except at unit magnification. A real lens is corrected to image one plane well. The Scheimpflug rule follows that plane wherever the lens’s tilt sends it, as an idealisation good to the paraxial order a thin lens is good to.

How much to lean #

The tilts are related through the magnification. If the object plane leans by θ from square and the image plane by θ′, then tan θ′ = m tan θ, with m the ratio of image distance to object distance at the axis. For a distant subject the magnification is small, and even a steeply tilted subject — a road receding almost along the line of sight — needs only a modest tilt of the image plane, sixteen degrees for a road leaning at eighty. At life size the two planes lean equally. Close up, at higher magnification, the image plane leans more than the subject does, and a small tilt of the subject needs a large one behind the lens.

In practice it is easier to tilt the lens than the sensor, and the same rule says what to do. With the sensor fixed square to the camera’s body, tilting the lens by a few degrees tilts the plane of sharp focus in front of it by a great deal, because for a distant subject the lean is magnified by 1/m. Large-format view cameras, with their lens and film held on separate standards joined by bellows, let a photographer tilt either; tilt lenses for smaller cameras offer a few degrees of lens tilt, which is all a landscape needs.

Shifting instead of tilting #

There is a companion movement on the same cameras that uses the rule in reverse. Photographing a tall building from the street, a camera tilted upward to take in the top puts its sensor at an angle to the building’s face, and the building’s vertical edges converge in the picture as they rise: the face is a tilted plane, imaged in perspective. Keeping the sensor parallel to the face removes the convergence, because a plane parallel to the sensor is imaged without perspective distortion, at a single magnification. But then the building’s top falls off the edge of the frame.

The remedy is to slide the lens upward without tilting it, a shift, so that the part of the lens’s image circle that falls on the sensor includes the top of the building while the sensor stays parallel to the face. The subject plane, the lens plane and the sensor plane are then all parallel, meeting in the line at infinity, and the Scheimpflug rule is satisfied trivially: no tilt is needed because nothing is tilted. Architectural photographers combine the two movements — a shift to keep verticals straight, a small tilt to swing the plane of focus along a pavement or a courtyard — and the lenses that allow both need image circles much larger than the sensor they cover.

A hinge for every focus #

The Scheimpflug rule says that the three planes meet in a line, but with a tilted lens, refocusing moves the sensor relative to the lens and so moves the line. Photographers working with view cameras found that the result was confusing: the plane of focus did not move back and forth when they focused, as it does in an ordinary camera, but tipped. The rule that untangles it was written down in the 1990s by Harold Merklinger, a Canadian engineer who photographed with view cameras. With the lens tilted by α, every plane of focus, for every position of the sensor, passes through a second fixed line, the hinge: in the plane through the lens parallel to the sensor, at a distance $J = f/\sin\alpha$ from the lens.

The figure checks it for four focus settings. Each plane of focus is computed by mapping the sensor’s plane back through the tilted lens, and all four pass through the hinge 574 units below the lens. Focusing does not move the plane of focus; it rotates it about the hinge, from nearly horizontal, crossing the axis far away, to steep, crossing it near the lens. The practical order of work follows. The tilt sets the hinge: a five-degree tilt on a lens of ninety millimetres puts it about a metre below the lens, at ground level for a camera on a tripod. The focus then sets the angle of the plane about that line, until it lies along the ground.

Depth of field as a wedge #

An untilted lens’s depth of field, as the focus that is a slab drew it, is the region between two planes parallel to the plane of focus: the nearest and farthest planes whose images are blurred by no more than an acceptable amount. With a tilted lens the near and far limits are the planes of focus that correspond to the sensor moved slightly forward and slightly back, and by the hinge rule they pass through the same hinge. The zone of acceptable sharpness is no longer a slab but a wedge with its thin edge at the hinge, opening with distance from it.

That corrects a common belief, that tilting a lens increases depth of field. It does not; the aperture still sets how thick the zone is at a given distance. What the tilt does is lay the zone along the subject. A field of flowers running from a metre to infinity, which no aperture could make sharp with an untilted lens, lies within a thin wedge whose edge is at the photographer’s feet. A tall tree standing on that field, poking up out of the wedge, will be blurred at the top however the lens is stopped down. The tilt chooses which plane the depth is measured from, and the subject has to suit it.

A miniature made by tilting the wrong way #

The same wedge used the other way produces a well-known illusion. Tilt the lens against the scene instead of along it, so that the plane of focus cuts steeply across a view of a city or a railway yard taken from a height, and only a narrow band of the scene is sharp. Everything above and below the band blurs quickly. A photograph like that looks like a picture of a model, because a very shallow depth of field across a large scene is what a camera shows when photographing something small from close up: the blur gradient a viewer associates with a tabletop model at a magnification of a tenth. The brain reads the scene’s size from the depth of field and gets it wrong by a factor of a hundred.

The illusion is a reminder that depth of field carries information about scale, and that a tilted lens can supply a false value of it. It is also a reminder of what the Scheimpflug rule does not change: a lens cannot make sharp more than the wedge its aperture allows, and can only choose where the wedge points.

An eye photographed in section #

The rule’s most valuable uses are in measurement, where a tilted plane of interest is fixed by the instrument. An ophthalmologist examining the front of the eye shines a thin sheet of light into it from in front, which lights a cross-section through the cornea, the space behind it and the lens of the eye. To record the whole lit section sharp, from the front of the cornea to the back of the lens, the camera looks at the sheet from the side at an angle and its sensor is tilted according to the Scheimpflug rule, so that the oblique sheet is entirely in focus. Rotating the sheet and camera round the eye’s axis builds up the shape of the cornea and the depth of the chamber behind it, measurements used to plan cataract and refractive surgery and to detect diseases of the cornea before they affect sight.

Laser triangulation scanners that measure the shapes of objects use the same arrangement: a laser line projected onto the object and viewed at an angle by a camera whose sensor is tilted so the whole plane of the laser line is in focus. Light-sheet microscopes that image a tilted sheet inside a sample do too, sometimes correcting the tilt with a second tilted stage of optics, as the mirror that lights a tenth of a micrometre described for a sheet of light made by total reflection instead.

A laser beam as a ruler #

The most extreme use puts the tilted plane of interest kilometres long. A laser beam sent out into the air is a line, and the light it scatters from dust, aerosols or insects along its length comes back from every range at once. A telescope beside the laser, looking along the beam, sees the beam as a line receding into the distance, a plane of interest containing both the beam and the telescope’s axis. With the sensor tilted according to the Scheimpflug rule, every point on the beam is in focus at once, from a few metres to infinity, and each range lands at a different position along the sensor.

That turns the tilted image plane into a ruler. A point on the beam at range r is imaged at a height Bf/(r − f) on the sensor, with B the distance between the beam and the telescope’s axis, so the range that one pixel covers grows as the square of the range: two millimetres at ten metres, twenty-two centimetres at a hundred, twenty-two metres at a kilometre. Conventional lidar measures range by timing a short pulse’s echo, which needs a pulsed laser and fast electronics. A Scheimpflug lidar needs neither — a continuous diode laser and an ordinary camera — and has been used since the 2010s to count and identify flying insects by the flicker of their wingbeats along a beam over fields and forests, at ranges where its resolution is still fine.

Where the thin-lens geometry stops #

Every figure here uses an ideal thin lens, which maps planes to planes exactly. Real lenses are thick, with principal planes in place of a single lens plane, and the Scheimpflug and hinge rules hold with the lens plane replaced by the principal planes, which changes the details and not the shape. Real lenses also have the field curvature that the flat scene that comes back curved traced, so the image of a plane is slightly curved and a tilted plane slightly more so. A tilted lens is also used off its axis, where its image quality is worst, and lenses designed for tilting have larger image circles to allow for it. The depth-of-field wedge is drawn with the blur criterion of a sensor displaced by the aperture number times the allowed blur, which is the standard first approximation and not exact for wide apertures. The lidar figure ignores the lens’s aberrations and the finite width of the laser beam, which set the resolution at short range.

Still open: how far the tilted plane can be pushed in microscopy #

Imaging a tilted plane inside a thick specimen, without moving the specimen, is attractive for fast three-dimensional microscopy: a sheet of light swept through a sample can be followed by the detection optics if the image of the oblique sheet can be brought onto a sensor sharp. Oblique-plane microscopes do this by relaying the tilted image through a second and third objective lens, one of them tilted, and the arrangement loses light and resolution at large tilts. How steep a tilted plane can be imaged at the full resolution the first lens offers, and how much of the loss is fundamental to the geometry rather than to the particular lenses used, is being worked out with new designs for the tilted relay.

The habit worth carrying away is to ask what a map preserves before asking what it distorts. A thin lens takes planes to planes, so a tilted subject plane, the lens plane and its image plane always meet in one line, with tan θ′ = m tan θ; and tilting the lens by α makes every plane of focus swing about a hinge f/sin α away — 574 units for a 50 lens tilted 5° — carrying the depth of field with it as a wedge. A lens cannot make more of a scene sharp, but it can choose which plane through the scene is.