As before, we will impose a coordinate system on our world. Now, our viewpoint will be on the negative x-axis, while the y-axis will be the canvas (or picture line). Given a point P at (x,y) in the plane (where x is positive) and an artist's viewpoint O at (-d,0), we can determine the image of the point P' at (0,y') by using similar triangles:
Note that this formula is similar to the formulas we derived for the three-dimensional case.
Next, we want to understand how images of parallel lines converge to a point in two-dimensional perspective.
In the applet on the right, we can see what happens to the images of points on parallel lines. Click "Move the Line" - a moving line will define four points on two sets of parallel lines. The images of these points will be projected onto the y-axis as green dots. As the line moves farther and farther away, the images in each pair will move closer and closer to each other. Eventually, they will reach the vanishing point for that set of parallel lines. (Of course, the artist cannot see the lines in their entirety. In a two dimensional world, eventually one line would block the sight of the other. But if the lines were invisible, but not the points, the points would appear to converge to a single point.)
In the applet, you can move the artist's viewpoint as well as adjust the slope of the set of purple lines. A good exploration to follow is to determine what slopes of the purple lines
We can actually compute the vanishing point of a line by parametrizing travel on the line. If we assume that the point on the line is given by (x,y) where
then using our equation for y' above, we find that
and taking the limit as t goes to infinity, we find that
In other words, the vanishing point for the image of a moving point on a line is the slope of the line times d. Geometrically, the vanishing point for the image of a moving point on a line is where a line with the same slope emanating from the artist's viewpoint intersects the picture line. (This is the two-dimensional analog of the Vanishing Point Theorem.) You can see this line by clicking on "Show Vanishing Point" in the applet.
In particular, the principal vanishing point V for all horizontal lines is at y' = 0. Significantly, this is the point where a normal to the picture line contains the artist's viewpoint. For an observer, this means that if we can locate the image of horizontal lines in our painting and then find their vanishing point V, we know the vertical position of the artist's viewpoint.
In addition, if we can find the vanishing point of a line with slope 1 or -1 (both lines which make a 45 degree angle with the canvas), we can use the same procedure we did in the three-dimensional world to find the vanishing point W. Once we know the distance between V and W, we know the viewing distance (or the horizontal position of the artist's viewpoint).