What is curvilinear perspective?

A way of drawing a very wide view in which straight edges image as curves, flat near their vanishing points and bowed near the middle of the frame. It corresponds to projecting the scene onto a curved surface, as a fisheye lens does, instead of onto a flat window.

What is four-point perspective?

A curvilinear grid with two vanishing points on the horizon, left and right, and a pair above and below for the vertical family, with curved guides between each pair. A fuller five-point system adds a centre point for depth; the four-point grid shares its horizon points with two-point perspective.

Why do straight lines curve in a fisheye photograph?

Because the lens projects onto a curved surface. A straight line in the world becomes an arc on that surface, and only a line through the centre of the picture, the direction the lens points, flattens back to a straight line.

What stays straight in curvilinear perspective?

Only lines that pass through the centre of vision. The horizon is straight when it runs through the middle of the frame and bows when it does not. Everything else curves, more the further it runs from the centre.

When should I use a curvilinear grid?

When the width of the view is the subject: an interior taken in at more than about 90 degrees, a fisheye photograph, a comic panel that wants vertigo. Inside about 60 degrees the curvature is barely visible and a rectilinear grid is simpler.

Who defined curvilinear perspective?

Albert Flocon and André Barre published a systematic curvilinear perspective in 1968, with an English edition in 1987, in which straight lines image as curves and the size of objects decreases with distance in every direction. Earlier artists had drawn curved projections; theirs was the first worked-out system.

Curvilinear Perspective: When Straight Lines Bend

Curvilinear perspective is what a very wide view looks like. Take in enough of a room at once, or look through a fisheye lens, and a straight edge stops imaging as a straight line. It bows, flat where it approaches its vanishing points and bulging near the middle of the frame, so that the walls of a corridor swell outward like the sides of a barrel. Only one kind of line stays straight: a line that passes through the centre of vision, the direction you are looking straight along. Everything else curves, and the curves are lawful, which is what makes the view drawable.

Where You Are Standing

The rectilinear views, one, two and three point, are what you get by projecting the world onto a flat window. They are exact, and they have a cost: as the angle of view widens, things near the edge of the window stretch, and past about 90 degrees the stretching becomes absurd. A flat picture cannot show 180 degrees at all. The eye and the fisheye lens do something different. They project onto a curved surface, so a wide angle fits, and the price is that straight lines bend.

The construction below is a four-point grid: two points on the horizon left and right, as in two-point perspective, and a pair above and below for the vertical family. Between each pair the guides are curves, not lines. Drag the top or bottom point closer and the vertical guides bow harder; drag the side points and the horizontal guides follow. There is no house in this view, because a body with straight edges would contradict the grid around it.

Drag a point (Left, Right, Top, Bottom) and the grid follows. Arrow keys move a selected point one step; shift, ten.

The horizon bows too. It comes back straight in exactly one case, when it runs through the middle of the frame, because a line through the centre of vision is the one line that cannot bend. Drag the side points up or down and watch the horizon curve away from the centre line and straighten as it returns to it.

What Stays Straight, and Why the Rest Curves

Picture the world projected onto the inside of a sphere with your eye at the centre. A straight line in the world becomes a great circle on the sphere, the way a straight flight path becomes an arc on a globe. Now flatten the sphere onto the paper. A great circle that passes through the point you are looking at flattens to a straight line; every other great circle flattens to a curve, and the further from the centre it passes, the more it curves. Two parallel lines in the world, a floor edge and a ceiling edge, are two great circles that meet at two opposite points on the sphere: one ahead of you and one behind. On the paper those are the two points of a pair, and the curves between them are the guides.

That is why the grid works in pairs. The left and right points are where the horizontal family meets ahead and behind; the top and bottom points are where the vertical family meets. A fuller system adds a fifth point at the centre for the depth family, and that is the five-point construction Albert Flocon and André Barre set out in 1968, in which straight lines image as curves and objects shrink with distance in every direction, not only into the depth. The four-point grid here is one construction within that family, chosen because it answers to the same two horizontal points as the level views, so a drawing can begin in two-point and widen into curves without moving its horizon points.

A fisheye photograph is the same geometry made by glass. Lines through the centre of the frame stay straight; lines near the edge bow; a doorway at the side of the picture curves like a barrel stave. Comic artists borrow the effect for vertigo and speed. Architectural photographers spend their careers avoiding it, with lenses that hold the projection flat at the cost of the angle they can take in.

The Exercise: a Corridor That Bows

Ruling curves by hand is slow, so this exercise starts from the printable curvilinear grid below, with tracing paper over it, a hard pencil and twenty minutes.

Find the horizon on the sheet: the darker curve, 89 millimetres (3.5 inches) from the bottom edge at the centre of the sheet on Letter, 85 on A4. It bows because it does not pass through the centre of the frame.
Draw the far end of the corridor as a small rectangle centred left to right, 30 millimetres wide and 20 high, its bottom edge on the horizon. Near the centre of the sheet the projection is nearly flat, so this small shape may be drawn straight.
Draw the floor edges from the bottom corners of the rectangle outward along the two curved guides that pass nearest them, all the way to the side edges of the sheet. They bow downward as they approach the sides.
Draw the ceiling edges from the top corners along their guides. They bow upward. The corridor now swells: narrow at the far end, fat at the sides of the sheet.
Add three door frames along the left wall, one near the centre, one halfway out, one at the edge. The nearest to the centre is nearly straight; the one at the edge bows like a barrel stave, following the curved vertical guides.
Check. Trace any edge with your finger. Does it flatten as it approaches its pair of points and bulge in the middle of its run? If a line is straight anywhere but through the centre of vision, it has escaped the projection.

Then draw the same corridor on the one-point grid and lay the two side by side. The one-point drawing shows perhaps 60 degrees of the corridor. The curved drawing shows twice that, and the price of the extra width is written into every bowed line.

The Mistakes That Give It Away

Curves everywhere. A line through the centre of vision stays straight. A drawing in which the horizon through the middle of the frame also bows has bent something the projection leaves alone.
Curves that bulge the wrong way. Every guide flattens towards its pair of points and bulges away from the centre in between: floor lines bow down, ceiling lines bow up, verticals at the sides bow outward. A wall that bows inward has been drawn from the wrong point.
A straight-edged box in the view. Anything with straight edges must curve with the grid. A doorway drawn with a ruler at the edge of a fisheye view reads as a mistake, not as a door.
Curvilinear used for a narrow view. Inside about 60 degrees the curvature is barely visible and a rectilinear view is simpler and honest. Use the curved grid when the width of the view is the point.
Points at unequal distances above and below. The top and bottom points describe the same vertical family and stand symmetric about the centre of vision. Pull one closer than the other and the view claims to be looking up and down at once.

The Printable Grid

The curvilinear sheet carries the curved guides of both families, the bowed horizon, and no straight guides at all. Its four points lie well outside the sheet, where the footer says. It is drawn from the same geometry as the construction above and the perspective grid tool. Print at 100 percent.

Curvilinear (four-point) perspective grid, Letter landscapeDownload PDF
Curvilinear (four-point) perspective grid, A4 landscapeDownload PDF

The tool cannot read the curved grid out of a photograph, because straight edges are what a reading rests on and a fisheye picture has none. Place the four points by hand along the bowed edges of the picture instead; two or three edges lining up fixes the rest.

Narrow the view and the curves straighten into two-point perspective; tilt it and the verticals gain a point of their own in three-point perspective. The line that bows in this view and stays straight in the others is the horizon line. The one rule behind all of it, and the grids for every view, are on the perspective overview.

Frequently Asked Questions

What is curvilinear perspective?: A way of drawing a very wide view in which straight edges image as curves, flat near their vanishing points and bowed near the middle of the frame. It corresponds to projecting the scene onto a curved surface, as a fisheye lens does, instead of onto a flat window.
What is four-point perspective?: A curvilinear grid with two vanishing points on the horizon, left and right, and a pair above and below for the vertical family, with curved guides between each pair. A fuller five-point system adds a centre point for depth; the four-point grid shares its horizon points with two-point perspective.
Why do straight lines curve in a fisheye photograph?: Because the lens projects onto a curved surface. A straight line in the world becomes an arc on that surface, and only a line through the centre of the picture, the direction the lens points, flattens back to a straight line.
What stays straight in curvilinear perspective?: Only lines that pass through the centre of vision. The horizon is straight when it runs through the middle of the frame and bows when it does not. Everything else curves, more the further it runs from the centre.
When should I use a curvilinear grid?: When the width of the view is the subject: an interior taken in at more than about 90 degrees, a fisheye photograph, a comic panel that wants vertigo. Inside about 60 degrees the curvature is barely visible and a rectilinear grid is simpler.
Who defined curvilinear perspective?: Albert Flocon and André Barre published a systematic curvilinear perspective in 1968, with an English edition in 1987, in which straight lines image as curves and the size of objects decreases with distance in every direction. Earlier artists had drawn curved projections; theirs was the first worked-out system.

Sources

Curvilinear Perspective: From Visual Space to the Constructed Image, Albert Flocon and André Barre, 1968; English edition University of California Press, 1987. The system in which straight lines image as curves and size decreases with distance in every direction.
Linear perspective, Encyclopaedia Britannica. Orthogonals, horizon line and vanishing point; the system dated to about 1415 and documented by Alberti in 1435.
Vanishing point, Encyclopaedia Britannica. The point at which receding parallel lines appear to meet.

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