There are several fictional versions of life in a two-dimensional world. The most famous and influential is Abbott's Flatland [3] first published in 1884. In Abbott's world, figures move about on a horizontal two-dimensional surface. (Due to complex issues surrounding class, there is also a ban on the use of color - a fact we will conviently ignore when considering paintings.)
Our two-dimensional world will more closely resemble Dewdney's The Planiverse [4] in that we will think of its inhabitants as occupying a vertically oriented sheet. Unlike in Flatland, we will think of the artist (and observers) as standing upright and facing a painting.
In this two-dimensional world, it is important to note that a painting is constructed on a one-dimensional line segment. To paint such a canvas, an artist would simply apply color to portions of the line segment.
As an example, consider the applet on the right. In it, we set up a few assumptions that we will use throughout our discussion. First, our two-dimensional world is the Cartesian plane. Second, the artist's viewpoint is somewhere on the negative x-axis and the canvas lies somewhere on the y-axis. (We will also call the y-axis the picture line - it is the line onto which an image will be projected.) The subject of the painting will lie in the portion of the Cartesian plane where x is positive.
In our applet, our artist is painting two colored line segments (these might form the side of a house). To do a proper perspective painting, the artist envisions the line between her viewpoint and a point and then paints that point on the canvas where the line intersects the picture line. In the applet, you can move the artist's viewpoint as well as the endpoints of the segments to change the projected image. Note that the only image an observer of the painting would see is the color on the y-axis.
Given such a painting (and some information about the subject), we want to know how an observer of the painting can determine where the artist's viewpoint was. Standing in that point will give the observer the strongest illusion of two-dimensionality.