by The first law of perspective is that things get smaller as they get further away. Linear perspective, the system artists have used since the fifteenth century, follows that law along the line of sight and then breaks it at the edges. Curvilinear perspective is the repair.
Where linear perspective fails
Stand facing the middle of a long wall. The part directly ahead is closest to you, so it should look largest. The parts far to your left and right are further away, so they should look smaller. Draw this with a flat picture plane and one-point perspective and the opposite happens: the wall is the same height all the way across, and if you widen the view, the ends actually get taller.
The reason is the flat picture plane. Rays from the edges of a wide scene hit a flat surface at a steep angle and spread out. Your eye is not flat. Neither is a fisheye lens. They record the scene on a curved surface, where the ends of the wall do get smaller, and the wall's straight top edge becomes a curve.
What curvilinear perspective does
Instead of a flat window, imagine the picture as the inside of a sphere around your head, then flatten that sphere into a disc. Every direction you can look becomes a point on the disc; the centre is straight ahead, the rim is 90° to the side (for a 180° view) or directly behind you (for 360°).
Straight lines in the world become arcs. Each family of parallel lines still meets at vanishing points, but now there are two per axis, one at each end: left and right on the horizon, zenith and nadir on the vertical axis, and centre and (in six-point) the point behind you. Only lines that pass through the exact centre of view stay straight.
Five-point and six-point
Five-point is the 180° fisheye. Four vanishing points sit on the rim of the circle (left, right, up, down) and the fifth is in the centre. It is the most useful version: a whole room, a cockpit, a crowd around you.
Six-point shows the full sphere. The rim of the disc is now the single point directly behind you, and the 90° circle (dashed on the grid) sits halfway out. It is how a 360° panorama is drawn, and it is rare in finished illustration, but it makes the logic of the system obvious.
Four-point, which you will also see mentioned, keeps a straight horizon and curves only the verticals. It is a convenient hybrid for tall scenes and is easy to approximate on the grid by keeping the camera level.
How to draw with it
Start with the circle and its horizon. Mark the left and right vanishing points on the rim and the centre point. Any horizontal line running away from you is an arc from the left point through your chosen position to the right point; the closer the line passes to the centre, the flatter the arc. Verticals are arcs from the top of the circle to the bottom. Floors and ceilings are the family of arcs in between.
It is much easier with the grid underneath. Set the angle, rotate until the room sits how you want, export a transparent PNG, and draw over the arcs. The one-metre reference cube tells you how much a box bends at that position in the frame.
When to use it
Use curvilinear perspective when the field of view is wider than about 90°, when you want the dreamy, enveloping feel of a fisheye, or when a linear grid is stretching the edges of your drawing into nonsense. Use linear perspective for everything else; it is simpler, and at normal lens lengths the difference is invisible.
Open the curvilinear perspective grid to try it, or read 1-point vs 2-point vs 3-point for the linear side of the family.
Printable versions: the fisheye 5-point grid and the 6-point 360° grid.
Questions
Is curvilinear perspective more correct than linear?
It is closer to how a wide view actually reaches the eye, which is why fisheye lenses and our peripheral vision both curve lines. For normal fields of view (under about 60°) linear perspective is an excellent approximation and much easier to draw, so both are correct within their range.
How many vanishing points does curvilinear perspective have?
Five for a 180° view: left, right, up, down on the rim, and one in the centre. Six for a 360° view, adding the point directly behind the viewer. Four-point is a hybrid with a straight horizon and curved verticals.
Which artists use it?
M. C. Escher used it constantly. Comic artists such as Kim Jung Gi and many manga artists use it for wide interiors. Concept artists use it for cockpit and establishing shots. It is also the basis of every 360° panorama.