Perspective Projection Distortion - Mathematical Description

Mathematical Description

Mathematically, the difference between artificial perspective projection (perspective projection onto a flat surface) and natural perspective projection (perspective projection onto a spherical surface) is a distortion resulting from the small-angle approximation which results from projecting onto a flat surface. Essentially, the distance used to calculate a represented object's size is based not the actual distance from the viewer (in Figure M1, the viewer is "S"), but the perpendicular distance ("z") to the picture plane ("P").

Figure M1 illustrates why this occurs. The two lines, "x" are the same length and are the same distance, "z," away from the picture plane, "P." When the two lines are projected onto the picture plane, toward the viewer, "S," the size of the lines represented on the picture plane are identical, "y." This is the case even though it is clear that the left line "x" is actually further away from "S" than the right line "x."

An example is a view where one is standing facing north towards a road which runs perfectly east-west. In an artificial perspective projection, every car on the road would be drawn at the same size, even though it is clear in reality that the farther away from the center of the picture that a car is, the farther away from the viewer that car would be. However, this seeming incongruity is cancelled out if the perspective is viewed from the same point as the generated perspective since the portions of the image away from the center will be further away from the viewer's eye. Thus, the cars will appear appropriately smaller to the viewer even though they are not drawn as such.