degrees of freedom — where it appears
Named by 37 essays across 13 fields — each of them below, with the objects they name alongside it.
A point is a line over there
Knowing where a mark sits in one photograph does not say where it sits in the other. It says which line it is on, which removes one of the two unknowns and leaves one — and the five lines drawn here meet at a point to within a thirtieth of a billionth of a pixel.
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.
The near plane can be any plane
Rewrite one row of a projection matrix and the near plane stops being perpendicular to the axis and becomes whatever plane is asked for. Every x and every y is untouched — it is the same projection of the same scene from the same eye — and the depth order is wrecked, which is a clean separation of the two things a projection matrix does.
Seven marks, three answers
Seven correspondences leave a two-dimensional nullspace, and the requirement that a fundamental matrix be singular is a cubic in the mix — one or three real roots. Here it has three, and all three satisfy every one of the seven marks to 8.9 × 10⁻⁹ pixels. The eighth mark, withheld, separates them by more than an order of magnitude.
The marks name the place, not the height
Run the camera-recovery round trip on an anamorph — hand it the floor marks and ask for the eye back — and it returns the spot on the floor to eleven decimal places with nothing assumed at all. It does not return the height. What the marks fix is the product of the eye's height with the design's aspect ratio, and no amount of looking at the floor separates the two.
The lines a surface leaves alone
Only the plane draws every straight line straight, which is easy to measure and which gets read as though a curved surface bent everything. It does not. Every curved picture surface here keeps a one-parameter family of world lines exactly straight, and the family is a curve on the sphere of lines rather than a region of it — so a grid of samples finds none of it, which is what the first version of this measurement reported.
Two matches are enough
A general fundamental matrix has seven free numbers and needs eight correspondences. A mirror pair's has two, and two correspondences fix it — with a straightedge, on a print, by drawing the line from each mark to its reflection and marking where the two cross. Given the same sixteen marks read to four tenths of a pixel, the constrained fit lands 4.8 times closer to the truth than the eight-point algorithm.
The same person, twice on one panel
A panel showing one figure at four moments is geometrically the least strange thing in this field — one camera, one floor, and every pair of copies meeting the horizon to 8.5 × 10⁻¹⁴ px. What the picture withholds is the order, and four copies admit twenty-four readings, and a reading convention supplies 4.6 bits from outside the marks. Enlarge one figure by six per cent and the horizon test that passed the panel catches it at 36 px.
The two points a picture hides
The stratification rung says a photograph of a plane is projective, becomes affine when the vanishing line is known, and becomes metric with one more fact. The one more fact has a size and a shape: it is two points, complex conjugates on the vanishing line, and a rectification built from those two and nothing else returns every world angle to a tenth of a picodegree and every ratio of lengths to five parts in a quadrillion — and refuses to name a length.
Two mirrors make one turn
Reflect a point in two mirrors meeting at 36° and the images arrive nine at a time, every one of them on a single circle about the line where the mirrors meet, to 4 × 10⁻¹⁶ m. The rule taught for the count — three hundred and sixty over the angle, less one — is right at six of nine angles tried and wrong at the rest, because it is a rule about angles that divide a half turn and it is quoted for angles that divide a whole one.
A picture in bands
A register picture stacks its scene in horizontal bands, each with its own ground line and every figure drawn at one height. The feet line and the heads line of any pair are then parallel to the arithmetic floor — 0° against 15.4° in a photograph of the same figures — so the picture has no horizon anywhere in it, and what a reader recovers is an ordering with no metre attached.
What a flat map leaves alone
A projectivity of the plane is eight numbers in a matrix, and reading them tells a reader nothing. What it does is decided by its fixed points, and there are exactly three cases: three isolated fixed points, or a whole line of them with one point off it, or a whole line of them with the point fallen onto it. The middle case has five numbers instead of eight and every point slides along a line, and it is what most of the maps this site builds turn out to be.
What two parallel views leave free
Two perspective pictures give shape and no size. Two parallel pictures do not even give shape: the metric upgrade has six unknowns, two views supply six equations, and the system comes out rank five — a one-parameter family of solids that redraw both pictures to 7e-16 m. A third view closes it, and the mirror image survives every count.
The pond with its trees laid flat
An Egyptian garden pond is drawn in plan with its trees rotated outward about the bank they stand on. A rotation is an isometry, so every length in that drawing is exactly the length it is in the garden — zero error, not a small one. What is spent is the angle between any two faces, which reads 180° across every hinge and is 90° in the garden, and the four walls admit sixteen assemblies, so a reader supplies four bits to fold it back up.
The wedge recovered with the camera
Admit the glass into the model and the fit finally has something left over. A two-degree wedge is invisible to a reading confined inside eighteen degrees of the axis — the fit calls the whole displacement a rotation of the camera and is right to — and by fifty-five degrees it comes back to 0.07 of a degree. What the picture does not separate is the ordinary glasses — assuming an index of 1.50 for a true 1.52 costs one per cent in the angle and 0.07 pixels of residual.
The dimetric the set square draws
An orthographic direction has two parameters and produces three axis scales, so the achievable triples are a surface rather than a list. The drawing office's dimetric — one axis at 1 in 8, the other at 7 in 8 — has the right three scales exactly and the wrong two angles, and the picture it makes is an oblique projection of a cube rather than an orthographic one.
Two mirrors are three cameras
A photograph with two mirrors in it holds three views of the scene from three places, at baselines of 2.90, 3.10 and 2.26 metres. Two of the three pairwise geometries are mirror pairs and are skew; the third is a rotation by twice the angle between the mirrors, and it is not new evidence — five numbers read off the print rebuild it to 9.4 × 10⁻¹⁴ pixels, where a general three-view arrangement needs eighteen.
What one oblique drawing shows
A parallel projection is a linear map from three dimensions to two, so it has a direction it throws away. Slide any point along that direction and its mark does not move — which makes every parallel drawing the drawing of a three-parameter family of solids, and the depth scale a convention chooses is one direction through the family rather than a boundary of it.
A tiring panel keeps its order, not its direction
Let a painter's creep grow from one strip to the next as the panel is worked, and the strips' habits carry the order they were drawn in — but only as a line, never as a direction: tiring from the floor to the book and steadying from the book to the floor put the same drift on every strip. Four strips find the order a quarter of the time against a twelfth by chance; six find it nineteen times in twenty. And the order costs the reading its refusal: once it is free, two steady hands fit one tiring hand nearly as well as a tiring hand does.
Five marks and the sixth
Five points determine a conic exactly — five coefficients up to scale, five equations, nothing left over — so a fit through five marks on a photograph is not a fit at all. The sixth mark, withheld, lands on the curve to 1.9e-13 px. And the moment a sixth mark is used, the arithmetic changes character completely: it becomes a least-squares problem, and the residual starts telling you something the five could never say.
A strip keeps its ratio, not the end it began
A painter dividing a strip into bays tires as they go, and each bay comes out a little larger than the last. Divide the strip from its far edge instead of its near one and the tiring runs the other way — but the rows record none of it: a strip divided from the far edge by a tiring hand is, to the last digits, a strip divided from the near edge by a steadying one. A whole picture recovers every strip's direction anyway, because one hand shared one rate of tiring across strips of different splay.
Two circles, one picture
A photographed circle leaves its own pose ambiguous, and not a little: two congruent circles in planes 23.6° apart draw the same conic to 1.1e-16 on normalised coefficients, both of them in front of the camera. On top of that the distance is free, so the family is two discrete poses each with one continuous parameter — and a plane one degree from either draws a conic 2.5e-4 away, which is what makes the agreement a measurement.
The drawing that gives the solid back
One parallel view of a general point determines nothing: two equations, three unknowns, and the kernel is free. What closes it is not a second view but the correspondence — knowing which drawn edge runs along which world axis — and with it the whole solid comes back out of one drawing, exactly.
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.
No solid casts an aspective figure
Fitting the best single rigid view to an aspective figure — head and legs in profile, eye and shoulders turned square — misses its own marks by 2.6% of the drawn height, and no yaw does better than 3.0% in a full sweep. A genuine single-view drawing of the same body fits to 7.6e-13 pixels, and the five rotations recovered from the marks alone match the convention's own list to 0.0e+0°.
A strip's scatter points to the end drawn last
The size of a tiring painter's bays cannot say which edge of a strip they started from, because a creep read backwards is a creep. The scatter of the bays can: a hand that worsens as it goes leaves its largest slips on the rows it drew last. One six-bay strip names its edge three times in four when the slip triples, a four-bay strip carries no evidence at all, and a workshop that divided every strip from one edge is named nineteen times in twenty from six strips — with no creep in the picture.
A stepped hand passes for a tiring one on a short strip
A painter who steps dividers from each row to the next leaves an error that accumulates — a walk from the edge begun at, with no fatigue in it. The fit of habit and creep takes about four-fifths of that walk before any reading of the scatter starts, because a walk is slow and so are the two numbers. What is left names the starting edge nearly as well as fatigue does, and on a six-bay strip it cannot say which of the two mechanisms left it: the verdict needs twenty-four bays on one strip, or sixteen six-bay strips by one hand.
A stepped hand's walk is in the creeps it was given
Fit each strip of a picture alone and a stepping hand's walk vanishes into a creep of that strip's own. Fit the picture once, with one habit and one creep for the hand, and the walks come back as a disagreement between strips that their small residuals cannot explain. Read against a line set once, eight six-bay strips name the hand ninety-five times in a hundred; the residuals of the same strips name it eighty-six, and needed sixteen strips for what the creeps do in eight.
The bundle keeps the needles turned, if the wall has three rows
Six cameras in two groups at sixty degrees either side of a wall measured its flatness four and a half times better than six on a narrow arc, with their poses known. Found from the same pictures in one bundle adjustment, they still do: thirty-nine marks cost them three per cent. Ten marks in two rows cost them seventy-eight, and the wall's likeliest error becomes a twist — but a third row of marks takes most of that back, and fifteen marks in three rows beat fifty in two.
A line is a space of its own
Most of what is said about projective geometry in pictures is said about the plane. One dimension down there is a smaller object with a complete theory: a point of a line is one ratio, a map of a line is three numbers, three pairs fix it, and the cross-ratio is not merely an invariant but the only one — which is a claim that can be made to fail.
Four marks before anything is said
A reader fitting a correct perspective to a row of transversals has three numbers to choose, so three transversals fit whatever they are and the fourth is the first that can disagree. Below that count a pavement is unfalsifiable, and a great many painted pavements are below it.
The map a row of posts is
Walking one bay further down a row is a map of the drawn line to itself, and which map it is settles everything about how the spacings behave. It is parabolic — one fixed point, counted twice, and that point is the vanishing point — which is why the drawn posts crowd toward it and never arrive. Doubling a distance instead gives a hyperbolic map whose multiplier a straightedge can read.
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 drawing does not run out of lines
Every post supplies a line to every lamp, so a drawing of five posts offers ten lines to two lamps and twenty to four — the unknowns and the constraints grow together and two posts fix any number of lights. What runs out is the partition, whose margin falls from 251 pixels to six as the share of lines assigned correctly falls from all to just over half.
Dividing to a point off the board
A wall turned forty degrees to the view has its vanishing point 0.65 canvas widths past the edge of the paper, and the construction that aims every course at it without ever reaching it is exact to 1e-13 px. Putting the vertex where the sheet ends instead costs 20.9 px, which on this wall is 300 mm of masonry, and nothing in the drawing says so.
Desargues read the other way
The theorem's converse is not a second theorem. Exchange points and lines in the ten-point configuration and every one of its thirty incidences holds to 5 × 10⁻¹⁶, the axis becomes a point, the three axis points become lines through it — and what has been written down is the converse, read off the drawing rather than proved.
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.
DemonstrationResidualVanishing pointConditioningDrawing conventionIdentifiabilityFalsifiabilityFree parameterHorizonleast squaresModel errorreconstruction ambiguity