The corner sees an ellipse

The centre has an area established that a world point images as a disc whose centre is exactly the pinhole’s mark, and named three places where the exactness fails. The first of them is the corner of the frame, and it fails for a reason that has nothing to do with lens quality.

A circle seen from off its own normal is an ellipse. The pupil is a circle. A point in the corner of the frame sees it obliquely.

The one cosine that changes the shape #

At a field angle θ, the pupil’s apparent width along the direction of tilt is cos θ times its true width, and its height is unchanged. So the patch a defocused corner point makes is an ellipse of axis ratio cos θ, with its short axis pointing at the centre of the frame.

Measured on a 50 mm lens over a full-frame sensor: the corner is at 23.0 degrees and the axis ratio is 0.920. On APS-C with the same lens the corner is at 15.6 degrees and the ratio is 0.963.

That is a small number and it is exactly computable, which is what makes it worth having. It is not aberration, not decentring, not a manufacturing tolerance: it is what a mathematically perfect lens does, and it is the reason out-of-focus highlights in the corners of a photograph are lens-shaped ovals pointing inward before any barrel clipping is considered — the same shape wide-angle is not distortion finds in the scene rather than in the pupil.

The four cosines #

The illumination falls as cos⁴θ, and the standard presentation gives the exponent without saying what the four are. They are four separate geometric facts and separating them is the point of this section.

One for the pupil’s foreshortening. The corner sees a smaller hole, by cos θ.

One for the sensor’s tilt. The rays arrive at the sensor obliquely, so the same bundle spreads over 1/cos θ more sensor area — a factor of cos θ in the illuminance.

And two for the distance. The corner of the sensor is further from the pupil than the centre by 1/cos θ, and illuminance falls as the inverse square, giving cos²θ.

Multiply: cos⁴θ. On a full-frame corner with a 50 mm lens that is 0.480 stops — a factor of 0.717 in brightness — and on APS-C 0.216 stops.

The decomposition is checked rather than recited: the exponent is recovered by taking the logarithm of the computed illumination against the logarithm of the cosine, and comes back as 4 to a part in a million.

Why the two cancellations differ #

Worth pausing on, because this collection has already measured a case where the cosines cancel and this one is a case where they do not.

A wall does not get darker establishes that a uniformly lit surface has the same brightness in the picture however far away it is, because the inverse square is exactly cancelled by the number of surface points falling in one pixel. That is a statement about radiance, which is conserved along a ray.

The falloff here is not about radiance. It is about how much of the pupil a sensor point can see and how obliquely it sees it — a statement about solid angle at the sensor, which is not conserved and which shrinks toward the corner for the four reasons above.

So the two results are not in tension. Radiance from a surface is constant across the frame; the illuminance a sensor point receives from that radiance is not, and the difference is the four cosines.

What a real lens adds, and what it cannot remove #

Real lenses depart from cos⁴ in both directions, and it is worth being clear about which departures are design and which are geometry.

Mechanical vignetting adds to it. The barrel, the hood and the elements’ own apertures clip the pupil for off-axis rays, which cuts the pupil’s area further and produces the cat’s-eye shape. This is design-dependent, it improves on stopping down, and it is what the corners of a wide-open photograph mostly show.

Pupil aberration can subtract from it. A lens whose entrance pupil grows with field angle — which some retrofocus designs do — passes more light at the corner than the bare cosine predicts, and the measured falloff is gentler than cos⁴.

And the four cosines cannot be removed at all. They follow from the pupil being a plane figure seen obliquely, the sensor being a plane, and light falling off as the inverse square. A perfect lens has them. Correcting them in software is a brightness adjustment, not an optical one, and it costs signal-to-noise in exactly the corners that had least to spare.

What it does to the blur criterion #

The previous rung’s band was computed on the axis, and this rung says what the corner gets instead.

A criterion stated as a diameter is ambiguous for an ellipse. Along the tangential direction the patch is cos θ narrower and reaches the criterion sooner, so the band is slightly deeper; along the radial direction it is unchanged. So the depth of field in the corner is anisotropic — deeper for detail running one way than the other — by the axis ratio.

At 0.920 that is an eight per cent difference in patch width, which moves the band’s limits by a few per cent. It is smaller than the difference between two people’s criteria and it is not zero, and it is the reason a corner’s out-of-focus rendering has a direction to it.

Stating it as an area criterion instead makes the corner’s band uniformly slightly deeper, because the ellipse’s area is cos θ times the disc’s. Which criterion is right depends on what a reader is looking at, which is the same free choice the sharp band is a decision is about, met one level down.

The same foreshortening elsewhere in this collection #

The cosine is not new here and putting the instances together is worth doing, because it is one geometric fact wearing four costumes.

A ball at the edge of the frame is drawn as an ellipse stretched by 1/cos θ — the ball at the edge of the frame measures it, and it is the same cosine acting on a sphere in the scene rather than on the pupil.

A lamp lights less than half a ball, and the lit fraction is a cosine integral over the source.

A design cast onto a face at an angle receives it stretched, which is the anamorph field’s own stretch — one face, one scale reports it as two singular values, and those two numbers are the ellipse’s two axes.

Each of them is a circle or a disc seen from somewhere it is not facing. Recognising that is what lets the same chord-of-a-disc arithmetic serve a shadow’s penumbra, a defocused edge and a corner’s patch.

What the corner’s patch does to a measurement #

The previous rungs drew a boundary — centroids are safe, edges are not — and the ellipse sits on the safe side, with a caveat.

An elliptical patch is still symmetric about its own centre, so its centroid is still the pinhole’s mark, exactly. Every construction built on point positions survives the corner as completely as it survives the axis. That is worth saying because “the corner is different” invites the conclusion that corner measurements are biased, and they are not.

The caveat is the third departure the centre has an area names: once the barrel clips the pupil, the patch stops being symmetric and its centroid does move. So the safe statement is about the geometric ellipse and the unsafe one is about mechanical vignetting, and the two are separable by stopping down.

Which gives a calibration recipe with a reason behind it: photograph a dot target at a moderate aperture, where the pupil is unclipped everywhere and the patches are symmetric ellipses, rather than wide open, where the corners are clipped and the corner marks are biased inward.

Measuring it without a light meter #

A reader can check the axis ratio directly, and the measurement is easier than the falloff.

Photograph a field of small bright points — a night sky, a string of lights, water sparkle — well out of focus, at a wide aperture, and look at the shapes in the corners. Their long axes point at the centre of the frame and their aspect is cos θ before any clipping. Comparing a corner blob’s aspect against cos of the corner’s own field angle, computed from the focal length and the format, is a one-photograph check of the geometry.

If the corner blobs are cut off rather than merely oval, that is mechanical vignetting on top, and stopping down two stops should restore them to ellipses of the predicted aspect. Which is a nice separation: the ellipse is geometry and does not change with aperture, and the clipping is the barrel and does.

Why the falloff is quoted in stops #

A unit choice worth explaining, because it makes the numbers comparable with everything else a photographer has.

A stop is a factor of two, and a factor of two is the smallest brightness change most people notice in a large flat area. Quoting 0.480 stops rather than “a factor of 0.717” says immediately that the corner of a full-frame frame with a 50 mm lens is about half a stop down — noticeable in a clear sky, invisible in a busy scene, and comfortably inside what a raw converter’s vignetting correction removes.

It also makes the format comparison legible: 0.216 stops on APS-C, about 0.1 on a one-inch sensor, and past a stop on formats and focal lengths wide enough to put the corner past thirty degrees. The falloff is a function of the corner’s own field angle and of nothing else, which is why a wide lens on any format shows it and a long lens on any format does not.

That is a small instance of the pattern a focal length is not an angle is about: the quantity that governs is the angle, the quantity written on the lens is a length, and the format is the conversion between them.

The falloff and the criterion together #

One last combination, because a reader who has both rungs will want to know whether they interact.

They do, slightly, and in the direction that reduces the corner’s problem. A corner receives less light and its patch has less area, so the illuminance inside the patch — brightness per unit area — falls by cos⁴θ while the patch’s area falls by cos θ, and the total light in the patch falls by cos³θ rather than by cos⁴θ.

That matters for a measurement that locates a mark by its centroid, because the centroid’s precision goes as the patch’s brightness over the square root of the light in it. At 23 degrees the total light in a corner patch is 0.360 stops down, which costs about a tenth of the centroid precision — small enough to ignore for most work and large enough to explain why the corners of a calibration are consistently the noisiest part of it.

None of that is a bias. It is variance, and it is the honest form of “the corners are worse”: worse by a computable factor, in a way averaging removes, and without moving any mark.

The corner’s angle is a function of the magnification #

The closing remark below — that focusing closer moves the corner’s field angle — is worth turning into the expression it comes from, because it says that everything in this essay depends on a quantity the lens barrel does not carry.

The corner’s angle is set at the sensor, not at infinity: $\tan\theta = (d/2)/v$ with $d$ the sensor’s diagonal and $v$ the lens-to-sensor distance. And $v = f(1+m)$ with $m$ the magnification, so

$$\theta_{\text{corner}} = \arctan\frac{d}{2f\,(1+m)}.$$

For a 50 mm lens on full frame that is 23.39° focused at infinity, 23.04° at three metres — the figure this essay quotes — 20.74° at 0.4 m, and 12.2° at life size, where $v = 2f$.

The three quantities the essay computes all follow it. The axis ratio goes from 0.918 at infinity to 0.978 at 1:1. The falloff goes from 0.487 stops to 0.130. So a macro photograph has almost no $\cos^{4}$ vignetting, and a landscape taken with the same lens has four times as much — from one lens, one format and one aperture, with only the focus ring moved.

That is a genuinely useful thing to know and it also explains a coincidence. The same $v = f(1+m)$ produces the bellows factor: the effective $f$-number at magnification $m$ is $N(1+m)$, so life size costs two stops of light in the middle of the frame. The corner’s falloff improves by 0.36 stops over the same range. Both come from the sensor moving away from the lens, and the exchange is lopsided — two stops lost everywhere against a third of a stop recovered at the edges — which is why nobody notices the second.

It also sharpens what a published vignetting profile is a profile of. A measurement made at infinity focus overstates the falloff for close work by up to a third of a stop, and a correction applied blindly to a macro frame therefore brightens corners that were never that dark. The profile belongs to a focus distance as much as to a lens, and only one of the two is usually recorded. That is the same omission this whole row keeps meeting in different places: a quantity that depends on two things gets published as a property of one of them, and the one left out is the one a photographer changes most often.

It is also the reason the corner angle is quoted here as a property of the format and focal length together rather than of either alone. Twenty-three degrees is what a 50 mm lens sees at the corner of a full-frame sensor focused at three metres; the same lens focused at 0.4 metres puts the corner nearer twenty-one degrees, and the falloff falls by a tenth of a stop for that reason alone.

Where the cosine is measured from #

One detail that decides every number here: the field angle is measured at the entrance pupil, not at the sensor and not at the front element.

That matters because the corner’s angle is arctan(r / v₀) with r the distance from the axis on the sensor and v₀ the sensor’s distance behind the pupil — and v₀ is not the focal length except at infinity focus. Focused close, v₀ grows, the corner’s angle shrinks, and the falloff falls with it. A macro photograph has less corner falloff than a landscape taken with the same lens, which is a small and genuinely counterintuitive consequence of the same geometry.

It also says why the entrance pupil’s position has to be known before any of this can be computed, which is the hole a scene actually sees: the angle a corner subtends is measured from a point that is an image of the stop rather than any part of the glass, and getting it wrong by a centimetre changes the corner angle by a degree on a normal lens.

The short version #

A circular pupil seen from a field angle θ is an ellipse of axis ratio cos θ, so the patch a defocused corner point makes is elliptical with its short axis pointing at the frame’s centre — 0.920 at the corner of a full-frame picture with a 50 mm lens, 0.963 on APS-C.

The light through it falls as cos⁴θ: one cosine for the pupil’s foreshortening, one for the sensor’s tilt, two for the extra distance. That is 0.480 stops at a full-frame corner, and the exponent is recovered from the computation rather than asserted. All of it happens to a perfect lens; mechanical vignetting is what a real one adds on top, and it is the part that improves on stopping down.