One-Point Perspective: Lines Straight Back Meet at One Dot

Steps 1–2: the horizon and one dot

Think of the picture as an upright sheet of glass with the room traced on it. The horizon is the level line on that glass at the height of your eye. On that line, straight in front of your eye, sits the vanishing point. Leon Battista Alberti set up a painting the same way in De pictura (1435): he began with a rectangle that served him "pro aperta finestra", as an open window, and marked a single "centricus punctus", a centric point.

1. Rule a level line across the sheet: the horizon, which stands for your eye level.

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2. Mark one dot on it: the vanishing point.

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Where you put the dot is where you put the viewer. A box drawn below the horizon shows its top; one drawn above it shows its underside. Alberti placed his centric point no higher above the bottom of the picture than the height of the people he meant to paint, so that viewers and painted things seem to stand on one floor. He pointed to churches, where the heads of people walking about stay at about one height while the feet of those farther away come up to about the knees of those nearer.

Why the lines meet at one dot

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Look down on the scene from above. Two edges run straight back, side by side. The farther along an edge a point lies, the closer the line from your eye to that point comes to pointing straight ahead. So it crosses the picture closer to the spot in front of your eye, and the crossings from both edges crowd toward that one spot.

Put your eye at the origin and the picture d units in front of it. A point X units to the side, Y units up and Z units ahead then lands on the picture at (d × X ÷ Z, d × Y ÷ Z). Along an edge that runs straight back, X and Y stay fixed while Z grows, so both numbers shrink toward 0. Every such edge, wherever it starts, heads for the same spot, (0, 0): the vanishing point.

An edge that faces you keeps the same Z all along, so its picture is just the edge scaled by d ÷ Z. Level edges stay level and upright edges stay upright, which is why the front of the box in step 3 is an ordinary rectangle.

The whole construction assumes one eye at one spot. Move your eye and the match breaks, which is the idea behind Anamorphic Art: The Smear in The Ambassadors Is a Skull.

Steps 3–5: a box that recedes

3. Draw the face of a box that faces you: a square with level and upright sides. Put it below the horizon and to the right of the dot.

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4. Join each corner of the rectangle to the dot with a light ruled line.

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5. Draw the back face: a smaller rectangle with level and upright sides and its corners on those four lines. Erase the edges the box hides.

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The box sits below the horizon, so you see its top. It sits to the right of the dot, so you see its left side. Any depth you choose in step 5 gives a box, but not always the box you meant.

Steps 6–7: how deep is the box?

A cube needs a depth equal to its width. The four lines alone cannot give you that. The right depth depends on one more number: how far your eye is from the picture.

6. Mark a second dot on the horizon, as far to the right of the vanishing point as your eye is from the picture, at the scale of your drawing. This is a distance point, 300 units out in the drawings here.

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7. Rule a line from the front bottom-left corner of the box to the distance point. Where it crosses the line from the front bottom-right corner, put the back corner of a square base, then finish the cube.

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The line in step 7 is the diagonal of the square base, which runs at 45° on the ground: one unit sideways for every unit back. Put that into the formula and its picture heads for (d, 0), on the horizon d units to the right of the vanishing point: the distance point. Move the distance point out to 450 units and the same base comes out shallower on the page, about 42 units deep instead of about 57. Alberti also tied his floor rows to a chosen distance between the viewer's eye and the picture, worked out in a separate small drawing.

Steps 8–10: tile the floor

8. Divide the front edge of the floor into equal tile widths, and join each mark to the vanishing point.

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9. Rule one diagonal from the left end of the front edge to the distance point. Through each place where it crosses a line from step 8, rule a level line; these are the rows of square tiles.

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10. Shade every other tile. Then check: a second diagonal, from the right end of the front edge to a distance point the same distance to the left, must pass through a tile corner in every row.

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Step 10 is Alberti's own test:

"Qui quidem quam recte descripti sint inditio erit, si una eademque recta continuata linea in picto pavimento coadiunctorum quadrangulorum diameter sit." (Leon Battista Alberti, De pictura, Book I, 1435; Latin text edited by Cecil Grayson, 1973)

The rows are right, he wrote, if one continuous straight line runs as the diagonal of a chain of joined squares on the painted floor. He also rejected a shortcut that set each new row back by a fixed fraction of the gap before it, because its first row was placed by chance and its result had no fixed spot for the eye.

A common construction mistake is to space the rows evenly. The rows then stay equally deep on the page as they go back, and the corners that should line up along a diagonal bend away from a straight line.

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Check it on a painting

The Ideal City is a panel in oil and tempera, 220 cm wide, from about 1480–1484, now in the Walters Art Museum (37.677). The museum's record credits a Florentine artist and Giuliano da Sangallo (about 1443–1516).

The overlay below assumes what this article assumes: one vanishing point for every edge that runs straight back, on a flat panel. Its lines come from the side faces of the two palaces, whose fronts are painted with level edges, so those side faces should run straight back. Before any fitting, a tolerance of 1% of the image width was written down, with a rule: every straight edge there at least 100 pixels long, with both ends clearly visible, goes in. Fifteen edges qualify (roof and cornice edges, string courses, the foot of the right-hand arcade).

Each edge was located on the museum's 1,798 × 651 pixel scan and extended. The fitted point is the one with the smallest sum of squared distances to all 15 lines.

What the overlay finds

All 15 lines pass within 16.2 pixels of that point, 0.9% of the width and inside the tolerance. The typical miss (root mean square) is about 7 pixels, or 0.4%. On the panel itself, the largest miss comes to about 2 cm. The lines, the point and every distance are in one-point-perspective-ideal-city-lines.csv (CSV; where this file comes from: source, version and license, JSON) Source data: Measured for this page on the Walters Art Museum's public-domain scan of The Ideal City (37.677), 1798 × 651 px.

The point sits 4 pixels left of the scan's center line and 38% of the way up from its bottom edge. It falls at the small doorway of the far building framed by the central arch.

Three things could bend these numbers: the lens and framing of the museum photograph, warping in the wooden panel, and later restoration. None of them was measured, but four level edges on the palace fronts lie within 0.4° of the scan's horizontal.

The two largest misses, about 15 and 16 pixels, both sit on the cornice above the right-hand arcade, close to the 18-pixel tolerance. A first trial fit used one edge of each band on the right-hand palace and two edges read partly across a statue or column in front of them. It put this cornice's upper edge just outside the tolerance, by less than a hundredth of a pixel, and the CSV keeps that result. With those two readings corrected, the same 12 edges all pass inside.

That the painter ruled every one of these edges to one point, and that the spread comes from painting over the ruled lines, is this site's interpretive hypothesis, not something the overlay proves. The site's rules for painting overlays are on the checking page.

A printed plate can leave the construction in view. In plate 10 of the second part of Hans Vredeman de Vries's Perspective, etched by Bartholomeus Willemsz. Dolendo after his design, dotted lines run back toward one point on a dotted horizon line that the plate labels "Orison".

When one dot is not enough

One dot handles only edges that run straight back. Turn a box on the floor so that none of its edges does, and its level edges aim at two dots on the horizon instead, one on each side, while its upright edges stay upright. That is two-point perspective. The dot straight in front of you is still the vanishing point for any edge that runs straight back; the turned box just has no such edge.

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The formula says where the two dots go. With d = 300, an edge turned 40° away from straight back heads for a point d × tan 40° ≈ 252 units to one side, and the edges at right angles to it head for d ÷ tan 40° ≈ 358 units to the other. Only edges parallel to the picture keep their direction. Tip the picture up or down, as when you look up at a tower, and upright edges stop being parallel to it, so they run toward a third dot.