Three-dimensional perspective: finding the viewing distance

While we can determine a point on the canvas which the artist focused on, we have not yet determined how far away from the canvas the artist stood. That is, we have not yet found the viewing distance. To do this, return to the example of the cube. The image on the right shows a top view of the cube. Note that since the diagonals on the cube are at a 45 degree angle to the picture plane, by the Vanishing Point Theorem their vanishing point W is where a line extending from O meets the picture plane at a 45 degree angle.

topfindd (8K)

Note that the right triangle formed by O, V, and W is similar to the isosceles right triangle formed by B, C, and D. Thus, the viewing distance d is equal to the distance on the canvas between V and W.

We now have a procedure for finding the ideal viewing location.

  1. Find the principal vanishing point V of lines perpendicular to the picture plane.
  2. Find the vanishing point W of lines parallel to the ground at a 45 degree angle to the picture plane. (Squares in the picture make these lines easier to find.)
  3. Place your eye on the line extending from V perpendicular to the canvas at a distance equal to the distance between V and W.
Assuming V and W can be found using information from the picture, this method will always work. Unfortunately, to use this method, we need to find the intersection of lines on the canvas. In Flatland, our canvas will be a line. We will need to develop a new method.

Mark Schlatter

Last Modifed: 8/5/2004