The picture on the right shows the painting with labeled points showing the change in colors. We know that the points A', B', C', and D' indicate the corners. For now, we will assume that the artist's viewpoint is below and to the left of C'. We need to find the vanishing point for the parallel line segments A'B' and C'D'. That will give us the vertical position of the artist's viewpoint.
To do this, we will choose a different parametrization of lines in the Cartesian plane. Given a fixed time t0, let
A point on a line traveling according to this parametrization will reach the 'horizon" as seen by the artist when t = t0. If the line is given by y = mx + b and we find the image of the point on the picture line as defined by our equation for y', we find
While this is a complex formula, we only need to note two important facts. First, under this parametrization, y' is a linear function of t. Returning to our painting, assume we have our parametrization for points starting at A' and C' and traveling perpendicularly to the picture line. Under this parametrization, the points will reach the back edge of the square room at the same time. Thus on the picture line, the image of the point starting at A' will reach B' at the same time the image of the point starting at C' will reach D'. The applet below shows an animation of this parametrization. To see the nonlinear nature of the parametrization for points in the plane, move the artist's viewpoint closer to the origin.
Second, given any point following this parametrization, its image reaches its vanishing point when t = t0. With these two facts, we can prove the following:
Flatland Vanishing Point Theorem: Given any line parametrized as above, the image of a point on that line travels at constant speed. Moreover, given points with the same x-coordinate on lines parametrized as above, both points will reach their vanishing point at the same time.
We will now use this theorem to find the artist's viewpoint for a Flatland painting.