If you take a square and look at it from some point in space it looks like a quadrilateral. What are the possible shapes of this quadrilateral?
For 2point perspective, the angles at the respective vanishing points can be determined by laying out a second line parallel to the horizon (i.e., parallel to the line determined by the two vanishing points), and laying off equally spaced segments, as this secondary line is considered to be at a given distance from the eye, such as the view of the bottom of a proscenium arch, that is, the edge of a stage, to take an example. Then make a triangle using one vanishing point and an adjacent pair of these proscenium points. Do the same with the other vanishing point and a different pair of these points such that the two triangles intersect. The intersection will be one such perspective view of the square.
When viewed at other angles, a 1point perspective can be used. Ultimately it produces a trapezoid (trapezium outside the U.S.) as the perspective view of the square, resulting from the triangle formed being isosceles and the second (and third actually) vanishing points being infinitely far to the right and left.
The spherical excess referred to in the first post are in fact achieved in these views as indeed the flat picture must be viewed by the eye in perspective as well. The flat picture is just a crosssection in space of the pyramid formed by the eye and the square.

Posted by Charlie
on 20091107 10:20:39 