Perspective, Silhouettes, and Taxi-Cab Geometry

In a drawing with one-point perspective, all lines perpendicular to the picture plane converge to a single point on the horizon called the vanishing point.  In this picture of some railroad tracks, for example, it would appear that the railroad tracks meet at some point far off in the distance, even though they are actually parallel.

one-point perspective of railroad tracks

Now imagine that you are looking down at some cubes on a level surface, and that you draw them using one-point perspective.  The top of each cube will not only appear larger than the bottom of each cube, but also be drawn further away from the vanishing point.

one-point perspective of some level cubes

  

Next, you create a silhouette of each cube by coloring inside each of the drawn outlines.  The resulting shapes are either irregular hexagons, pentagons, or quadrilaterals.

the silhouettes of some level cubes drawn with one-point perspective

Amazingly, the area of each silhouette can be determined by just four variables – the side length a of the top square of the drawn cube, the side length b of the bottom square of the drawn cube, the horizontal shift x of the cube from a cube directly over the vanishing point, and the vertical shift y of the cube from a cube directly over the vanishing point.

The center of the top square of a cube in the first quadrant would be drawn at (ax, ay) and its vertices at (a(x ± ½), a(y ± ½)).

Likewise, the center of the bottom square of a cube in the first quadrant would be drawn at (bx, by) and its vertices at (b(x ± ½), b(y ± ½)). 

Each silhouette is a hexagon (or a degenerate hexagon) that is the difference between the area of a rectangle with sides c = a(x + ½) – b(x – ½) and d = a(y + ½) – b(y – ½) and the areas of the two opposite triangles, one with sides c – a and d – b, and the other with sides c – b and d – a. 

In other words, the area of the silhouette is A = cd – ½(c – a)(d – b) – ½(c – b)(d – a), and letting p = ½(a² – b²) and q = ½(a² + b²), this simplifies to A = p(x + y) + q.  A similar argument can be made for the other quadrants to give the general formula of A = pd + q, where d = |x| + |y|. 

Surprisingly, d = |x| + |y| is also the equation for distances in taxicab geometry, a seemingly unrelated mathematical topic.  In taxicab geometry, segments are limited to being either vertical or horizontal, just like roads in a city block.  Since a taxicab cannot drive through city blocks along a hypotenuse, the distance it must travel to get to an intersection that is 4 blocks east and 3 blocks north is at least |3| + |4| = 7 blocks (whereas a flying bird not limited to the roads can do it in √(3² + 4²) = 5). 

taxicab geometry paths from (0, 0) to (4, 3)

In general, the distance a taxicab must travel to get to an intersection that is x blocks east or west and y blocks north or south is at least d = |x| + |y|.

Therefore, there is a surprising connection between one-point perspective drawings and taxicab geometry.  The area of the silhouette of a cube drawn in one-point perspective is A = pd + q, where d = |x| + |y|, the same distance a taxicab must at least travel to get to an intersection that is x blocks east or west and y blocks north or south.  This means that the area of the silhouette is a linear function of its own taxicab distance.