Principal point — where it appears
Named by 42 essays across 10 fields — each of them below, with the objects they name alongside it.
Straight lines that are not
Everybody says the edges of a wide-angle frame bow. Nothing is special about the edge. A radial map moves every point along its own radius, so the only line it leaves straight is one through the principal point, and the bend of every other is decided by how far it passes from that one place.
The divide is postponed, not avoided
A renderer does not divide by depth. It multiplies by a four-by-four matrix that carries the depth in a fourth coordinate and divides later, and the postponement is not an optimisation — it is what makes clipping and texture interpolation possible at all. The matrix and this site's pinhole put every point on the same pixel to five parts in a hundred trillion.
A lens destroys the invariant
The cross-ratio is the one thing a projection preserves, and nearly everything checkable about a photograph is checked with it. A lens returns it one and a third per cent out where the pinhole is exact to fifteen digits — and the height error that follows tracks a quantity nobody would guess.
Recovering the camera from the picture it drew
Draw a box from a known camera, forget the camera, and get it back from the twelve drawn edges alone. Agreement to one part in 10¹⁵ is a statement about the geometry, because the only thing that crossed between the two halves was a list of line segments.
Fitting a lens from straightness alone
No calibration target, no known scene, no camera. Only the knowledge that some edges in the picture were straight — and the coefficient comes back to fifteen digits. Then it comes back with a companion, and the two are correlated at −0.997.
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.
A pixel is not a point
Where the sample sits inside a pixel is a convention, and getting it wrong shifts every mark by half a pixel in each axis. What that costs can be measured by recovering the camera from the picture — the answer is a principal point exactly 0.707 px from the truth with the focal length untouched, and the other half-pixel mistake does precisely the reverse.
The third point put where it looks right
Three-point perspective is taught as two vanishing points plus a third placed by judgement. The third is not free: two points and the centre of the picture fix it exactly. It survives being placed by eye because in the layout a book draws it belongs thousands of pixels off the paper, where 400 px of error costs less than a degree — and in a photograph taken looking up at a tower the same 400 px costs ten.
The distance point is the viewing distance, drawn
There is exactly one place in the whole classical apparatus where the distance from the eye to the picture appears as a length on the page, and it is the offset from the centric point to the distance point. Everything this site exists to compute was drawn in the fifteenth century, on the horizon, and nobody said what it was.
The eye is a place, not a point
Rotate a camera about the wrong point and the sky still stitches perfectly while the foreground slides. The misregistration falls as one over the distance, exactly — which is what says the fault is the pivot and not the lens.
The pixel that is not square
A camera with two focal lengths is a real thing — anamorphic cinema optics, non-square photosites, a stretched video format. Hand the round trip of camera recovery a picture from one and it returns a focal length 49.6% out, a principal point far from the truth, three independent estimates agreeing to 1e-16, and bundle residuals at the noise floor. Every alarm the site has stays silent.
Both vanishing points on the paper
Putting the two vanishing points on the sheet is presented as a composition rule. It is a statement about the reader: with the two points one page-width apart the picture is a 90° view, correct from 80 mm, and a reader holding it at arm's length is shown a room five times as deep as the one drawn. The layout that is honest at arm's length puts both points four and a half pages off the sheet.
Which way the drawn circle leans
Two rules are given for the direction a drawn circle's short axis runs in: along the axis of the cylinder, and pointing at the centre of vision. On the optical axis both are exactly right. At the edge of an ordinary frame the first is out by three and a half degrees and the second by seventy-nine. The statement neither of them is comes out of differencing the projection, and it matches the drawn ellipse to a hundredth of a degree.
Four numbers and a window
A projection matrix is built from six numbers and one of them is not a number at all. Four sides carry the focal length and the principal point; the near and far planes move nothing a reader can see; and the bottom row, (0, 0, 1, 0), is the only place the depth divides — set it to (0, 0, 0, 1) and the same machine draws a parallel projection.
Turning the cameras inwards
A stereo pair made by rotating two cameras toward a common point puts the same world point at different heights in the two pictures — up to thirty pixels here, on a frame of four hundred. Two eyes level with each other see every point at the same height, so a pair with vertical difference is a pair of pictures of no scene at all.
The lens a pavement can hide
A photographed pavement reads as a correct drawing up to a radial coefficient of about four tenths — a lens strong enough to bow a straight edge across the page by nearly six pixels and to print as twenty per cent distortion at the frame's corner. The reason is that a pavement sits near the principal point, which is the one part of the frame a radial map barely touches.
The plane is a choice
A projection has a centre and a surface, and they move independently. Keep the eye and turn the picture plane and every point of any scene lands where one 3×3 matrix says, to 2.5e-13 px. Move the eye instead and the matrix fitted to four points is exact at those four and out by 32.0 px everywhere else. The first is a homography of the picture; the second is parallax, and nothing about the picture can undo it.
A tile is an off-centre frustum
Rendering a picture in tiles is exact, and the way to do it is one line of arithmetic: a tile's sides are the whole frustum's sides read at the tile's own pixel bounds. Aiming the camera at each tile instead is defensible at every step and is a different picture, out by about a tenth of a tile whatever the tile size.
Straightening does not move the eye
Correct a photograph's converging verticals and what comes out agrees with a level camera at the same point — one the correction was never shown — to 3e-13 px, with the verticals parallel to 0e+0°. The cross-ratio of four points along a ground line reads 1.3333 before and after, so the corrected picture measures exactly what the original measured, from exactly where the original was taken and nowhere else.
A tilted sensor is not a distortion
Tilt a sensor 3° out of square with its lens and every point of the picture moves — up to 7.5 px on the frame drawn here — yet every straight line stays straight to 10⁻¹³ px and the cross-ratio survives to 10⁻¹⁶. The picture is an ordinary pinhole picture whose principal point has moved 22.30 px. A calibration that frees its principal point absorbs it exactly; one that holds the principal point and reaches for tangential distortion terms leaves 1.87 px, and used as a correction it bends straight rows by 4 px.
The lines that calibrate a lens
One straight edge through the centre of a picture says nothing about a lens's distortion, and one 180 px from the centre determines k₁ to 9.0 × 10⁻⁴ — the precision rises in proportion to the offset. But distance from the centre is not enough. Crowd three edges on one side and, the moment the distortion centre is also unknown, the coefficient is ten times worse, because a bend on one side looks like a moved centre; put one edge across the centre and it barely changes.
Focusing is a zoom
A 50 mm lens focused at half a metre is not a 50 mm camera. It stands 55.56 mm from the sensor, its picture is a pinhole picture at that distance, and it covers 35.9° where the same lens at infinity covers 39.6°. Recover the camera from the picture and it reports 55.56 mm. Read the picture with the engraved 50 mm instead and a right angle comes back as 96.0°.
A wedge of glass turns the camera behind it
A pane with parallel faces moves every point and no direction, so the camera recovered through a window is the camera that took the picture. Tilt one face 2° and every direction turns, by 1.04° on the axis and 1.67° forty degrees off it. The best rotation of the frame, 1.14°, still leaves 0.94 px, and no homography does much better, so the picture is no longer a projection from the camera's centre. The camera recovered from three vanishing points through the same glass is turned 3.37° — three times as far — because vanishing points lie where the glass bends most.
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.
One conic calibrates the camera
A focal length is usually recovered from two perpendicular vanishing points by an orthocentre construction with a square root in it. There is a second derivation with no construction and no square root — two vanishing points of perpendicular directions must be conjugate with respect to one conic in the picture — and the two agree to the last bit. They are not two methods. The conic is what a calibrated camera is.
The response is at the ends and the information is not
A radial map bows a straight edge by an amount that grows as the square of the distance along it, so 93 per cent of an edge's response to the coefficient lies in its outer quarters. Spending the marks there is 16 per cent worse than spreading them evenly, because two clusters say nothing a shifted, tilted line could not say. What identifies the coefficient is a curvature, which needs three places — both ends and the middle, which beats an even spread by 11 per cent.
Rectification is a family, not an operation
Turn both pictures of a pair so their epipolar lines become shared rows. A turn about the line between the eyes and a focal length are left free, and every choice puts all 44 matches on common rows to a tenth of a trillionth of a pixel and every point back where it was. What the choices disagree about is the pixels — one stretches its pictures unevenly by 1.77, another by 4.86.
A dolly zoom is a step and a zoom, and they meet at one depth
Step 1.5 m toward a subject 5 m away while shortening the lens to hold its size. Every mark moves along the line from the centre of the picture, to a ten-trillionth of a degree — outward if nearer than the subject, inward toward a limit if further, and not at all on the subject's own plane. The step and the zoom each move everything one way; the dolly zoom is where they cancel.
No design separates the two coefficients
Separating a squared term from a fourth-power one was supposed to need marks at radii far apart, which is a statement about where edges are placed. It is not: one straight edge already runs from 20 px to 326. Spreading ninety-six marks over three edges or sixteen changes the answer by 23 per cent, spreading the offsets makes it 19 per cent worse, and the correlation stays at −0.98 whatever is done. What a plumb-line calibration determines is one number, to 1.52 thousandths, and which number depends on the model.
A known target sharpens the fit and does not separate it
Printed circles of stated size were supposed to break the −0.98 correlation between a lens's two radial coefficients, because a circle puts every mark at one radius and no straight edge can. They do not: every design of circles leaves the pair 0.979 to 0.9997 correlated, and for circles of known size the figure is exactly the cosine between r³ and r⁵. What a known target buys is precision — 3.8 times the straight edges' at the same budget — and only if its size in the picture is known to about a thousandth.
Three-point, laid out with a straightedge
Recovering a camera from a drawing is the familiar direction. The other direction — stand somewhere, measure a room, and lay the picture out — had never been taken in three-point, because the third axis needs a measuring point on a line nobody draws. With it, every corner lands where the camera puts it to three parts in ten million million of a pixel, and nothing anywhere is judged.
A dolly zoom off the axis keeps a line, not a plane
Step toward a subject along a track that is not quite the line of sight, and zoom to hold its size, and the plane that stood still in the classic shot stops standing still. The step now spreads from a point beside the centre while the zoom still shrinks toward the centre, and the two cancel only along one row of the picture, one depth per column. The subject itself slides by f·d·sin ψ over its distance — a pixel once the track is a quarter of a degree out — and turning to follow it holds the subject at the price of bending the rest of its plane, while shifting the frame instead holds the whole plane exactly.
A drawing has three horizons
The triangle three vanishing points form is not a scaffold. Each of its sides is the vanishing line of one of the three coordinate planes — the floor's horizon and two walls' — and its orthocentre is the centre of the picture. So the horizon a perspective drawing is built on is one side of a triangle, and it is special only because the ground is where things stand.
A rectification's free shift is free only near the pair
Sliding a rectified pair's principal points apart adds a constant to every disparity, and a constant changes nothing about where points are — on paper. A whole-pixel reading is not paper. The same slide moves every depth the reading can report, and for a wall ten metres away the choice between the best shift and the worst is 1.71 metres. Near the pair it is millimetres. A matcher with half a pixel of its own error erases the choice, and pays more than the worst shift did.
The arc every eye stands on
Four drawn corners known to be a rectangle fix the horizon of their plane and nothing else. The eye that drew them has to see the two vanishing points at a right angle, so it lies on the circle those points are a diameter of — and every point of that arc reconstructs a genuine rectangle, with right angles to five parts in ten million million of a degree, and a different proportion.
The proportion is the assumption
Read the proportions of a rectangle out of a photograph of it and the answer is a function of where the centre of the picture is assumed to be. Sweeping that assumption across the horizon takes one drawn quadrilateral from one part in fourteen to slightly wider than square, every reconstruction a genuine rectangle, and only a fiftieth of the sweep within five per cent of the truth.
The quadrilateral no rectangle casts
The relation that reads a camera out of a drawn rectangle has a minus sign in it, and the minus sign is a refusal: two vanishing points on the same side of the assumed centre give the square root of a positive number, and no camera makes that quadrilateral out of a rectangle. Watching the refusal arrive shows what it is worth — one corner has to travel most of the picture's width before it fires.
Perpendicular is a pairing
On a horizon, the vanishing point of a direction and the vanishing point of the direction at right angles to it are joined by a map that is its own inverse. Such a map has two degrees of freedom rather than three, so two pairs determine it — and its two imaginary fixed points are the focal length and the centre of the picture, handed back from two rectangles on one floor with nothing assumed.
The horizon has a pole
Every other duality in this collection is a matrix somebody picked. A calibrated camera fixes one nobody picked, and under it the horizon of a plane and the vanishing point of that plane's normal are pole and polar — to 2.7 × 10⁻¹² pixels. Run backwards, those two marks give the focal length with no known length, no right angle and no square anywhere in the scene.
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.
The hook is the centre, and the eye is not
Dürer's string frame projects from a ring of iron driven into a wall, so its centre of projection has a position anybody can measure with a tape. A hook 120 centimetres behind a frame 56 centimetres wide makes a drawing correct from 34.3 centimetres shown 160 millimetres across, with its principal point 161 px off the middle of the sheet. The gridded veil projects from a head instead, and 30 millimetres of head moves its marks by 13.27 millimetres.
The triangle a camera cannot move
Three mutually perpendicular directions give three vanishing points, and that triangle is self-polar with respect to the image of the absolute conic — to 3.1e-13 px, with no length and no angle anywhere in the statement. Turn one direction two degrees out of square and the polars miss their sides by 65.8 px. The statistic this collection has been printing as evidence for the same claim, meanwhile, is an identity that cannot fail.
Named alongside it
The objects these essays reach for when they reach for this one.
Focal lengthVanishing pointCamera calibrationDemonstrationHorizonResidualConditioningHomographyPicture planeRadial distortionStation pointCamera matrix