Formula 1.B.

Looking Glass (1994)

Now what if there is foreshortening of the height? The "Looking Glass" on the left is done with the foreshortening of the height but with the artist-observer in the same standpoint as in Figure 1. Nothing else of Formula 1.A. changing, what you first need to do is plot the vanishing point for the vertical lines of the 2 wall planes (yz & yx). That is to say, foreshortening of height means just that the lines of the vertical planes (in the y direction) do this time converge on a vanishing point (in the case of "Looking Glass", below the horizon line). This vanishing point for the lines of y direction is to be plotted any where on the line (but below the horizon line in the present case) marking the intersection of the wall planes (i.e. the y axis itself). (Figure 5) This line marking the intersection cuts, as in Figure 2, the horizon right in half and its intersection with the horizon line is where the vanishing point for the diagonals of the floor plane xz is located. The further you plot the vanishing point for the y-axis lines away from the horizon line, the lesser the foreshortening of the height; the closer, the greater the foreshortening. In the "Looking Glass", it is plotted rather close (as is also the case in the figures below), resulting in great foreshortening from head to toes of the person.

Figure 5

|<---- a ---->|<---- a ---->| *--------------*--------------* (horizon line) | | | | | | | * <- vanishing point for lines of y axis

Remember that the vanishing point (*) for diagonals of the floor plane xz
lies still at the middle of the horizon line

Secondly -- the crucial step -- you need to find the vanishing point for the diagonals of the vertical wall plane (say) yz. Here is the formula I've discovered: the vanishing point for the diagonals of the vertical wall plane yz is found by drawing a straight line (1) that is perpendicular to the straight line connecting the vanishing point for lines of z axis with the vanishing point for lines of y axis and (2) which passes through the point of intersection between all 3 planes of space (plane xz, yz, and xy; this point is the 0 coordinate for all 3 x, y, and z axes). This straight line will be the first diagonal for the plane yz and its intersection with the straight line connecting the vanishing point for lines of z axis with the vanishing point for lines of y axis will be the vanishing point for the diagonals of the plane yz. (Figure 6; the diagonals of the vertical wall plane yz are marked blue.)

Figure 6

The space can now be carved out into cubes of equal size (1 m x 1 m x 1 m): i.e. the 3 dimensional Cartesian coordinate in foreshortening including the foreshortening of height.

L. C. Chin. 2004.