Shift lens — where it appears
Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.
The principal point is not the centre
Every textbook that computes a focal length from two vanishing points supplies the middle of the frame as the principal point. On a shifted or cropped picture that is wrong, and it costs one and a half per cent of the answer at a fifth of a frame's shift.
Along a line of constant depth the page is affine
Stepping a texture by a constant amount per pixel is wrong across a receding floor and exactly right along any line of it that stays at one depth — and on every plane those lines run parallel to its own vanishing line. Turn a 120 px span 1° away from that direction and it is 0.79 px out; roll the camera a hundredth of a degree and a floor drawn to 30 m is out by 0.69 px on its worst scanline.
Keystone correction spends the panel unevenly
Turn a projector fifteen degrees and keystone correction throws away 18 per cent of its panel — the number a specification quotes. It is the smaller half of the price. What remains is spread unevenly: the near edge of the corrected picture gets a panel pixel for every source pixel, the far edge three-quarters of one, so the far edge carries 58 per cent of the source's detail. A tipped projector keeps more than a turned one, and a lens shift that places the same picture loses nothing at all.
A level picture shows its rise on its horizon, and hides its slide
A shift lens moves the principal point off the middle of the frame, and the textbook focal length from two vanishing points pays for assuming it did not. On a level camera the payment is optional. The principal point lies on the horizon, so a rise is read straight off the picture and costs nothing; a slide along the horizon is read by nothing at all, because two vanishing points fix only a semicircle of possible cameras. What a sideways crop costs then depends on how the building is turned: a tenth of the frame trimmed off the side is 23 per cent of the focal length on a facade seen eleven degrees off square and under one per cent at forty-five.
A second rectangle fixes the column a level picture hides
Two vanishing points of a building put a level camera on a semicircle and leave it free to slide along it; a second rectangular object turned against the first puts it on a second semicircle, and the two cross once, at the camera. On a facade seen twelve degrees off square and slid a hundred pixels, a second object twenty degrees further already reads the focal length to 1.3 per cent where the building alone, even with its principal point known, manages 2.9 — and where the frame's middle column is wrong by a third. The reading does not care where the column was slid, and it is best when the second object stands at forty-five degrees to the camera, whose semicircle is flat over the true column.
A circle on the ground names the column and the focal length at once
The picture of a circle lying on the ground meets the horizon in two complex points, and for a level camera they are exactly the principal point's column plus or minus i times the focal length. So one circle does what two rectangles turned apart do, with no turn to choose: a manhole cover 1 m across, 8 m off, marked at 24 points to half a pixel, reads the column to 14 px and the focal length to 1.5 per cent, as two rectangles twenty degrees apart do — wherever it lies across the picture. It pays for distance as the square: the same cover at 24 m is out by 128 px. And the building beside it is worth its horizon and nothing more.
The centre of the picture is not the centre of the paper
A crop translates the image rectangle, so the picture's optical centre leaves the middle of the sheet and the focal length does not move — 81.3 pixels apart at a fifth of the picture, with the horizon at 62.5 per cent of the print. A reader who takes the paper's middle for the picture's stands 2.36 cm out of position, which is 4.9 degrees of the wrong direction.
Named alongside it
The objects these essays reach for when they reach for this one.
Principal pointFocal lengthHorizonVanishing pointIdentifiabilityStation pointPicture planeAbsolute conicCircular pointsCrop factorDepth divisionForeshortening