0. The setup

There are three ingredients: A viewplane, an object, and a viewpoint, which can be at infinity. The viewplane is often taken to be the $x,y$-plane.

There are several versions:

#1: A right-handed coordinate system with the viewplane thought of as horizontal. This is the usual picture from calculus. The viewpoint is regarded as being in the positive $z$-direction.

#2: A right-handed coordinate system with the viewplane thought of as vertical, as it would be if the $x,y$-plane is a display screen. The viewpoint is regarded as being in the positive $z$-direction.

#3: A left-handed coordinate system with the viewplane thought of as vertical. The viewpoint is regarded as being in the negative $z$-direction.

Notes.

(a)

The first two versions are exactly the same mathematically! The same computer program would give correct output for each.

(b)

It doesn't really matter whether the object is on the same side of the viewplane as the viewpoint, or on the opposite side, or straddling it. A formula valid for one case works for the other; it's just that points of the object may have positive $z$-values in one case and negative in another.

(c)

With a left-handed coordinate system, the algebraic computation of cross products stays the same, while the geometric meaning of the cross product obeys the left-hand rule instead of the right-hand rule.

(d)

If the viewpoint is at infinity, we usually think of it as being on one side of the viewplane or the other, even though in P$_ 3$ it doesn't really make a difference. Otherwise, we couldn't talk later about hidden lines and faces.

(e)

Although we usually talk about ``an object'', of course the object could be a whole complicated scene with many parts.

(f)

For the present, we ignore the issue of hidden lines and faces, which is quite complicated when you analyze it in detail. Solids, then, can be wire-frame figures, where only edges are given and faces are not filled in.

(g)

The viewpoint and viewplane are often called the center of projection and the projection plane.