# Homogenous, Normalized Device Coords and clipping

Went on to here (http://termespheres.com/perspective.html) from the page Zbuffer mentions. Termes talks about varous forms of perspective (1 point perspective, 2 point perspective, etc). The 1, 2, and 3 point pespectives all seem to map straight lines to straight lines, and the 4 or more point perspectives seem to map straight lines to curves.

The one point perspective seems to me to be the “standard” perspective projection used in computer graphics, but I’m wondering if a given scene would look actually look different to a 1 point perspective if it were displayed in 2 or 3 point perspective.

In other words, are they mathematically equivalent? Will the vertices end up being transformed to the same position on the screen?

Well I knew I should not have posted an artist page on “Math and Algorithm” … In this context, x point perspective just means the artist uses x point(s) as vanishing points. Drawing only axis aligned cubes need 3 points perspective to look like a planar projection (like the one in OpenGL). More points are used for both endpoints of each axis, curving the straight lines from one to the other.

Neither of these are mathematically equivalent to a proper 3D to 2D projection, as for example the point position is chosen almost arbitrarily, so that it “looks good”.

Projective transformations like the perspective projection of 3D space onto a plane always map lines to lines, so the 4+ point mappings are not projections on a plane.

The 4-point image could be the projection on a cylinder and the 5-point projection could be the projection on a hemisphere.

The projective transformation can map directions (= infinitely far points) to “normal” points in 3d space and vice versa. So every set of parallel lines has either one point on the plane where they intersect, or they stay parallel in the projection (if the direction is again mapped to an infinite point).

In this sense the difference between 1, 2 and 3 point projection is only how many directions are parallel to the projection plane.