4-Point Perspective Grid
Two-point perspective with the top and bottom of the world added: the horizon stays straight, the verticals bow.
What a four-point grid is
Stand at the base of a tower and look straight ahead, level, at its middle. The top of the tower is far above your line of sight and the base is far below it, and if you could draw everything you see without turning your head, both would converge: the vertical edges bow inward toward a point in the sky and a point in the ground. That is four-point perspective: the two ordinary horizon vanishing points of two-point perspective, plus a zenith and a nadir, with the camera held level so the horizon itself stays straight.
The grid here is the site's curvilinear projection opened at the framing that name usually means: level camera, 180° angle of view, the corner of the ground squares turned 45° so that one set of horizontals goes off to the left vanishing point and the other to the right, with nothing converging in the centre. The dashed horizon runs straight across the middle. The two dots on it are VP1 and VP2; the dots at the top and bottom of the disc are the zenith and nadir. Every vertical edge in the scene is an arc between those two.
Why the horizon stays straight
A curvilinear grid puts the world on a sphere, and on a sphere the only lines that stay straight are the ones through the centre of view. With the camera level, the horizon passes through that centre, so it is drawn straight; every other horizontal line is an arc that leans toward the zenith above the horizon and toward the nadir below it. Tilt the camera even a little and the horizon becomes an arc too, which is the moment four-point stops being four-point and becomes the general fisheye view on the 5-point page.
Drawing with it
Block the scene in two-point first, as if the picture plane were flat: floor plan, corner, the two vanishing points on the horizon. Then, instead of drawing the verticals as parallel lines, run each one as an arc from the zenith dot to the nadir dot through the corner you have placed. Near the centre of the disc these arcs are nearly straight; the further a vertical is from the centre, the more it bows, and the shorter it gets. Horizontal lines that would be parallel to the picture plane in two-point (the front face of a box seen straight on) are the ones that bow most noticeably here, curving away from the horizon like the lines of latitude on a globe.
For the arcs themselves, trace against the grid or export it: a transparent PNG for Procreate, Clip Studio, Photoshop or Krita, or an SVG when you want the curves as real vector paths. The curvilinear perspective lesson builds a room this way, and one-, two- and three-point perspective explained is the linear starting point this grid extends.
Questions
What is 4-point perspective?
Two-point perspective with the vertical direction added at both ends. The camera is level, so the horizon is straight and horizontal lines converge to two points on it, as in two-point; but the picture surface is curved, so vertical lines are arcs that meet at a zenith above and a nadir below. Four vanishing points in all.
How is it different from 3-point perspective?
Three-point tilts a flat picture plane so verticals converge one way, up or down. Four-point keeps the camera level and curves the picture surface instead, so verticals converge both ways at once. A tall building seen from its base with the ground visible is the classic case: three-point can show the top or the bottom, four-point shows both.
How is it different from 5-point?
Only by where you look. Turn the same curvilinear view so one set of horizontal lines runs straight at you and a fifth vanishing point appears in the centre; that is five-point. Turn it 45° so both sets go off to the sides and the centre point disappears; that is four-point. The grid here opens at the 45° framing.
When would I use a 4-point grid?
Tall interiors and street canyons where you want floor and ceiling, or ground and rooftops, in one drawing without tilting the camera; panoramas assembled from a fixed eye level; and any scene wider than about 100° where a flat grid would stretch the sides.
Can I make the verticals less curved?
Yes: lower the angle of view. At 150° the bow is gentle and the central third of the picture is almost linear; at 180° the verticals are full semicircles from zenith to nadir. Pick the angle by how much of the room or street you need to fit.