6. Subclassifications of projections

For each main kind of projection, there is a different question to ask. We'll assume the object is box-like and that one face is designated the top and another as the front.

(I) Orthographic projections. First ask: ``Is the object shown face-on (a ''principal<sup>1</sup>view``), or corner-on (''axonometric``)?''

(In other words, is a face parallel to the viewplane, or is some corner closer to the viewplane than any other corner is? If the object is lined up with the $x,y,z$-axes of R$^ 3$, then in terms of a normal N to the viewplane we are asking whether only one coordinate of N is nonzero, or all three. We don't consider ``edge-on'', which would be the case where an edge is parallel to the viewplane but no face is, or equivalently, exactly two of the coordinates of N are nonzero.)

If the object is shown face-on, we ask: ``Are we looking towards the front, a side, or the top?'' As in architecture, these principal views are called respectively, a ``front elevation'', a ``side elevation'', and a ``plan''. If all three are given, we have ``multiview orthographic''.

If the object is shown corner-on (axonometric projection), then we ask: ``At the image of a corner, how many different angles are there?''

If only one angle (i.e., three angles of $120 ^\circ$ each), the projection is isometric, if two (so exactly two are the same), dimetric, and if three (all different), trimetric. If the object is lined up with the $x,y,z$-axes of R$^ 3$, you can tell the subtype by seeing how many different numbers are involved in the three coordinates of N--one, two, or three.

(II) Oblique projections. Here we'll assume that one face is shown undistorted, which means it's parallel to the viewplane--again, this could be called a ``principal view''. We ask: What is the ``foreshortening ratio''? Consider a line segment perpendicular to the viewplane. The foreshortening ratio is the length of its image divided by its original length.

If the foreshortening ratio is $\frac 12$, there is a special name: cabinet projection. If the ratio is 1, the projection is a cavalier projection.

Otherwise, there is no special name. Furthermore, it doesn't make any difference how the object is turned compared to the viewing direction V; only the foreshortening ratio matters, in this terminology.

If the viewplane is the $x,y$-plane and the viewing direction is V$= (a,b,c)$, or equivalently, $(a/c,b/c,1)$, then the perpendicular line segment $(0,0,0)$ to $(0,0,1)$ projects to $(-a/c,-b/c)$, and so the foreshortening ratio is
$\vert(-a/c,-b/c)\vert/1 = \sqrt{a ^ 2 + b ^ 2}/\vert c\vert$.
In terms of trigonometry, if V makes an angle $\theta$ with the $x,y$-plane, then the foreshortening ratio is ``adjacent over opposite'', which is $\cot\;\theta$.

(III) Perspective projections. Ask: Of the three families of parallel edges of the box, how many are not parallel to the viewplane (so that their images are nonparallel and meet at a finite point)?

If one, it is a one-point projection; if two, it is a two-point projection, and if all three, it is a three-point projection. Each has a distinctive look.

In terms of the viewplane normal N, these are equivalent to asking how many of the coordinates are nonzero--one, two, or three?