Harmonic conjugate — where it appears
Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.
The diagonals find the middle
Three taught methods for spacing a receding row are all wrong, and the best of them misplaces a post by three and a half metres. There is a fourth, it needs no measurement and no vanishing point, and it is exact at every camera and every depth — because 'the diagonals of a rectangle cross at its centre' is a statement about which lines meet where, and that is the one kind of statement a projection cannot damage.
Two triangles and the line nobody drew
Desargues' theorem is the reason a hand-drawn shadow construction closes. An object and its shadow are two figures in perspective from the lamp; the theorem says their corresponding sides meet, pairwise, on one line — which is the ground line. So the closure a draughtsman treats as confirmation that the work is accurate is a theorem they cannot violate.
Four lines have a cross-ratio
The cross-ratio is introduced as a property of four points on a line. Its dual — four lines through a point carry the same number, and any transversal reads it back — is not a second theorem. It is why the first one is true: four rays from an eye are a pencil, every picture of them is a section of that pencil, and a quantity belonging to the pencil cannot depend on which section was taken.
The centre, got back out of the picture
The image of a circle's centre is not the centre of the image ellipse — this site measured the gap four phases ago and left the obvious question unanswered. It has an exact answer, it needs a straightedge and the horizon, and it is one sentence: the image of the centre is the pole of the vanishing line with respect to the image conic.
The bay repeated by a straightedge
Draw one bay, then repeat it by diagonals alone — no measurement, no scale, no arithmetic — and after twelve bays the constructed corners sit 1e-12 px from the corners the camera projects. It is exact because the operation being iterated is a homology of the picture, not because the draughtsman was careful, and that separates it from every construction in this site's `wrong` field.
A mirror that is not parallel to the wall
Carry the depth in front of the glass an equal depth behind it, square to the wall. That is exact for a mirror hung parallel to the wall and 1.26 metres — 107 pixels — out for one turned 20°. Two invariants survive the turn instead, and one of the two nearly did not survive being tested, because it had been written in a form that could not fail.
Seven is not a power of two
Halving a receding depth by diagonals is exact, and every book gives it. Halving reaches a half, a quarter, three eighths — and never a third, however many times it is spent, because no power of two is divisible by three. There is a construction that reaches every whole fraction, it costs three lines rather than a stack of quadrangles, and the extra ingredient is not a measurement.
The circle in the square wants a number
Every manual draws a circle in perspective by inscribing it in a square, crossing the diagonals, and marking four more points about seven tenths of the way out. Seven tenths is right — along the diagonal of the real square. Along the drawn one it is 54 mm off a circle two and a half metres across, and the drawn diagonal is the only diagonal on the paper.
The polar with a straightedge
Two secants through a point cut a conic at four places; the complete quadrangle they make has two more diagonal points; the line through those is the polar. Not one length, angle or midpoint is used, so the whole construction survives the projection that made the picture — and three unrelated pairs of secants land on the same line to 4.3e-13, while moving the point moves it by fifteen orders of magnitude more.
The bays that are not equal
The diagonal repeats a bay exactly and forever, and what it reaches is the whole multiples of that bay and nothing else. An arcade of 1:1.5:2 is 250 mm short of its boundary with no halving, exact after one, and an arcade of 1:1.333:2 is out of reach at every depth whatever. A measuring point lays out all three to 6e-14 px.
What a straightedge reaches on a receding line
Three marks on a receding line fix a coordinate, and everything a straightedge builds from them is a ratio of whole numbers. Three complete quadrangles put 1655 marks on the unit interval with no gap above 0.0038, a third costs two quadrangles where repeated halving never reaches one at all, and the same net built on a line that does not recede arrives at the identical 30 coordinates.
The quadrilateral that finds the middle
The harmonic conjugate is usually built from four points. Built instead from four lines — the dual construction, the same number of straightedge steps — it lands on the same fourth point to 1.2 × 10⁻¹⁵ of the range's own length, gives a cross-ratio of exactly −1, and refuses the midpoint, whose conjugate is at infinity.
Named alongside it
The objects these essays reach for when they reach for this one.
Cross-ratioVanishing pointComplete quadrangleProjective invariantStraightedge constructionDyadic rationalConicDemonstrationHorizonIncidenceline at infinityProjective duality