How Many Points It Takes

What is being measured

An orthogonal is an edge that, in the room a painting shows, runs straight back from you: a joint between floor tiles, the side of a ceiling beam, the top of a side wall. In a picture built by the rule Alberti wrote down in 1435, every orthogonal is drawn aimed at one point, the one he called the centric point. So a painting can be asked a simple question: of all the receding edges in it, how much of their length points at a single spot?

Each painting was read twice. Two readers, working apart and never shown a fit, a point, or any published claim about where a painting's point lies, marked every orthogonal they could see, roughly, on a coordinate grid. The mark was only a pointer. A program then found the painted edge under each mark, read its brightness every three pixels along its length, and fitted the line to the paint, so where a line lies was decided by the painting, not by the reader's hand. Each reader then looked at the placed lines and threw out any that had settled on an edge other than the one they meant. The readers were two separate instances of an AI model (Claude), each in its own context; that is a real limitation and it is taken up below.

A line counts as agreeing with a spot if it misses it by no more than one degree beyond its own measured doubt. That doubt was measured, not assumed: a short line cannot have its direction fixed as well as a long one, and how much worse was measured on a room drawn by computer with every orthogonal exact (a 30 px line carries about 0.45° of doubt, a 300 px line about 0.24°), then widened line by line by how much each placed line wobbled when its reader's mark was nudged.

The controls say what one point looks like

Three pictures went through the whole process whose lines truly do meet: the computer-drawn room, whose point is known to the pixel, and two photographs of long corridors taken with ordinary lenses. On the drawn room both readers' lines agreed on one spot for 100.0% of their length, and that spot fell within a pixel of the true one. The corridors scored 99.1% and 99.6%. A painting that scores near 99% is as single-pointed as a camera, as far as this method can tell; the green band on the chart below is that level.

Nineteen paintings, 1304 to 1511

Single paintings have been measured before, some of them by methods close to this one (Criminisi, Kemp and Kang on the Trinity, Simon on van Eyck, Raynaud on the Trinity as it stands on the wall, Field in front of the Lorenzetti panel itself); a search of the literature for this study found no measurement of a whole run of paintings on one scale, though that search did not reach every book. Across the eighteen measurable paintings, agreement rises with date: the rank correlation of share with year is 0.62, and in 20,000 random reshufflings of the dates a correlation that high or higher came up in 0.4% of them. That describes this set of paintings, which was chosen, not sampled: they are famous showpieces of perspective, and a different nineteen would give different numbers. What is more useful than the trend is where the exceptions are.

Before 1344: several points, or none

Duccio's Last Supper (1308–11) puts 57.5% of its receding length on one spot, and that spot is far outside the panel, because the best a single point can do here is catch lines that barely converge at all. The two ends of the tablecloth run almost parallel; switch to “as many as it takes” and the ceiling finds its own region above the table. That is close to what Panofsky described in 1927: in Duccio's interiors only the orthogonals of the central section of the ceiling converge, toward a spot he said is marked by a small lozenge in some panels and hidden “in the Last Supper by the nimbus of Christ”, while the side sections follow what he called a vanishing axis. Pietro Lorenzetti's Birth of the Virgin (1342) scores 61.4% and needs three points to reach 97.6%; its floor could not be traced in this reproduction, so what was measured are its walls and furniture, across a triptych whose side panel Panofsky already noted keeps a system of its own.

1344: Ambrogio Lorenzetti's floor

Panofsky wrote that in Ambrogio Lorenzetti's Annunciation of 1344 “the visible orthogonals of the ground plane are here for the first time all oriented toward a single point”. Measured, the pavement puts 92.3% of its receding length on one spot, about as much as Masaccio's Trinity fresco of the 1420s (92.9%), and the two readers, tracing it separately, came out at 91.3% and 93.1%. The spot lies on the twisted column that divides the panel. This measures only the orthogonals: J. V. Field, who examined the original, agrees they converge but found that the tile diagonals, which a correct construction also fixes, form a broken line, and nothing here tests that.

The 1420s: not yet a single rule

Masaccio's Trinity is the picture most often called the first exact one-point construction. Its coffered vault puts 92.9% on one spot, which lands at the very bottom of this reproduction, on the plain face of the ledge the donors kneel on, directly below the foot of the cross. That is short of the camera level, and in line with Dominique Raynaud's study of the fresco: “The concurrence of the vanishing lines in one point is only approximate”. Fra Angelico's Prado Annunciation, painted about the same time, scores 78.1% on only 9 edges.

The Netherlands in the 1430s: points stacked in a column

The three Netherlandish pictures of the 1430s all sit near two-thirds: the Campin workshop's Mérode Annunciation 61.8%, van Eyck's Arnolfini Portrait 70.2%, van der Weyden's Saint Luke 69.1%. None of them looks careless when you ask for as many points as it takes: the Arnolfini's floor finds one point, and the rest of the room needs two more, one above the other, all three close to a single vertical line. Gilles Simon reached the same shape in 2021 by a different method (a statistical line detector run on the paintings themselves), finding in the Arnolfini “four centric points, roughly evenly distributed along an axis”, and the same pattern in five van Eyck paintings. Fitted element by element here, the Arnolfini's floor scatters by only 0.58° about its own point and its ceiling by 1.19°.

The North catches up: Christus and Bouts

Which Netherlandish painting first used a single point for the whole room is an old question. Panofsky, citing G. J. Kern, put it “first by Dirk Bouts … or at the very earliest by Petrus Christus”, and Kern thought Christus's Frankfurt Madonna of 1457 already had “a single vanishing point for the entire space”. Measured, the Christus scores 85.8% on 11 edges (it is a dark reproduction, and its floor is a lozenge pattern with no orthogonals), with the spot on the Virgin's breast. Bouts's Last Supper (1464–68) is the tightest painting in the set: 99.7% of 76 edges on one point, which is to say indistinguishable from the photographs, with the point at the centre of the mantelpiece above Christ's head, where James Snyder's account (as quoted in the encyclopedias) places it.

Italy after 1450

Leonardo's Last Supper puts 100.0% on one point, and the point lands on Christ's right temple, which is where the museum at Santa Maria delle Grazie says it is, “at a height of four meters, by Christ's right temple”; neither reader was told that. Crivelli's Annunciation of 1486 scores 98.9% with its point far to the left, down the street, as the National Gallery notes it is. Piero's Flagellation scores 90.4%, and here the readers disagree: reader A traced the coffers of the rear ceiling bays, which scatter widely about the point, and reader B left them out, unable to tell painted ribs from incisions and cracks. Toggle the readers and watch it move from 84.0% to 100.0%; the floor alone scatters 0.37°. Raphael's School of Athens scores 87.4%, held down mostly by the block Heraclitus leans on, whose top edge both readers took for an orthogonal and which misses the room's point by about forty-eight degrees.

Two numbers, and why there are two

The study was designed around a different statistic: the scatter of the lines about one point, beyond what their measured doubt can explain, in degrees. It is shown under every painting above, in grey. When the full results came in, it was clear that it squares every miss, so a single object drawn to another point (Raphael's block, the parallel ends of Duccio's table) dominates the whole painting's figure: Raphael's floor scatters 0.74°, but the painting as a whole reads 8.65°. The share on one point was added then, after seeing the results, because one rogue object cannot hijack it. So was widening each line's allowance by its own measured doubt (without that, the share still ranks the paintings much the same way, and both versions are in the data). The order in which these choices were made is part of the result, and it is written down in the research notes.

What this cannot tell you

The readers are not art historians. They are two instances of one AI model. Their marks were independent, and their scores for the same painting mostly agree within a few points (the table shows both), but two instances of one model can share a blind spot in a way two people would not. Where a reader decided an edge was an orthogonal, the painting is measured on that decision; Raphael's block is a case where both may have been wrong.

The paintings are photographs. Frescoes photographed from the floor can be keystoned; a camera lens can bend straight lines slightly; restorations have repainted edges. The reproductions differ widely in quality, from clean museum scans to an old photograph of a Giotto fresco 852 pixels tall, and a low-resolution picture measures worse than its painter did. The photographs of corridors are the check on how much of this reaches the numbers, and they score 99%.

This is nineteen chosen paintings, most of them famous for their perspective. It says nothing about ordinary painting of the period.

Only orthogonals are measured. A perspective construction also fixes how quickly tile rows shrink and where tile diagonals meet. A painting can aim its orthogonals perfectly and space its rows wrongly (Field's finding for the Lorenzetti floor); none of that is tested here.

The placing program has a random step, and on 8% of lines a different random seed would have placed the line on a neighbouring edge or on none. The lines used are the ones the readers looked at and confirmed; the seed test is published with the rest.

curl -sO https://artwaste.land/checks/research/perspective-measured/verify.mjs
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Sources

• Erwin Panofsky, Perspective as Symbolic Form (1927), trans. Christopher S. Wood (Zone Books, 1991): the Lorenzetti sentence (p. 57), Duccio's ceilings and the lozenge (n. 42), Pietro Lorenzetti's side panel (n. 46), Bouts and Christus and Kern (p. 61, n. 53).
• Gilles Simon, “Jan van Eyck's Perspectival System Elucidated Through Computer Vision”, Proc. ACM on Computer Graphics and Interactive Techniques 4(2), 2021, doi:10.1145/3465623.
• J. V. Field, “Investigating illusion: Panofsky and others on perspective”, Archive for History of Exact Sciences 80 (2026), on the Lorenzetti floor.
• A. Criminisi, M. Kemp, S. B. Kang, “Reflections of Reality in Jan van Eyck and Robert Campin”, Historical Methods 37(3) (2004), 109–122, doi:10.3200/HMTS.37.3.109-122 (the Trinity's dominant orthogonals, Fig. 1).
• Dominique Raynaud, “Fact and Fiction Regarding Masaccio's Trinity Fresco”, in Studies on Binocular Vision (Springer, 2016), doi:10.1007/978-3-319-42721-8_4 (the abstract).
• Museo del Cenacolo Vinciano, “A powerful theatrical machine” (the Last Supper's vanishing point); National Gallery, London, on Crivelli's Annunciation; James Snyder on Bouts, as quoted in Wikipedia's article on the Altarpiece of the Holy Sacrament (the Grove text itself was not seen).
• Leon Battista Alberti, On Painting (1435), trans. John R. Spencer (1956), Book I, the centric point.
• Every reproduction is from Wikimedia Commons (public domain for the paintings; the two corridor photographs are by Dietmar Rabich and Mario Luna, CC BY-SA 4.0); the file behind each one is listed in the table above.

Beside this layer: The Floor That Builds Itself builds the construction these painters were aiming at; Widen the Window Until It's True measures another thing people say about paintings.