## Three-dimensional perspective: finding the vanishing point

The formulas on the previous page tell us how to find an image given the artist's viewpoint. Now we want to reverse the process: given an image, we want to find the viewpoint. To start, consider the image in the applet. The subject is a wire model model of a cube with added diagonal struts on the top and bottom. In addition, the front face of the cube is parallel to the picture plane.

We will now make use of a fact you see everyday: the image of parallel lines meet at a point. (Think of railroad tracks meeting at the horizon.) In the applet, the line segments A'D', B'C', E'H', and F'G' are all parallel and perpendicular to the picture plane. If we extend these segments, they will meet at the principal vanishing point --- the point where all lines perpendicular to the picture plane meet. You can also see that F'H' and B'D' have their own vanishing point. You can click on the buttons to show these vanishing points. You can also move points A', D', E', and V to create different images.

We can use the principal vanishing point to find where an artist was standing by using the following theorem:

Vanishing Point Theorem:The vanishing point of a line headed away from the picture plane is the point where a parallel line extending from the artist's viewpoint intersects the picture plane.
(For a proof of this theorem, see Marc Frantz's site [2] or my paper "How to Correctly View a Flatland Painting".) This theorem tells us that the principal vanishing point (the vanishing point of all lines perpendicular to the canvas) is where a line from the artist perpendicular to the canvas intersects the canvas. In other words, the principal vanishing point is where the artist was looking.

Note that we already mentioned this result in the hallway applet. The point A was the principal vanishing point. If you looked at the image with your eye across from A, the image appeared more three-dimensional than from other angles. You can see other examples of principal vanishing points in classical artwork at these sites:

We know now where the artist was looking, but we do not know how far from the canvas the artist was standing. Our next page will solve this problem.

Mark Schlatter

Last Modifed: 8/5/2004