Geometric Photo Manipulation - Projections

Analytic geometry of the sort usually found in multivariable calculus enters our discussion as we imagine displaying a three dimensional object on a two-dimensional computer screen or photographic print. Figure 2 depicts the two most common ways of accomplishing such a projection.

           (a) Perspective projection                                                  (b) Orthographic projection

Figure 2

• In perspective projection [Figure 2(a)], we imagine that the eye of an observer is at a point that is neither a point of the object (the cube) nor a point of a given plane , where the scene will be projected. Then any point of the object is projected to the point where the line through and meets the plane . The final step is to coordinatize this plane so as to identify its points with the pixels of the output image.

• In orthographic projection [Figure 2(b)], we imagine that the eye of the observer is at infinity, and the observer views the scene along a fixed direction vector . This time, each point of the scene is projected to the point where the line through with direction vector meets a given plane (a plane with normal vector ).

(a) Perspective projection                                      (b) Orthographic projection

Figure 3

As one can see from Figure 3, a cube drawn with a perspective projection has back edges that are smaller in size than the front edges, suggestive of the greater distance from the eye. In contrast, the orthographic projection yields front and back edges of the same size. Both projection methods are fundamental in computer graphics, and I think a discussion of them can provide a compelling application of certain topics from calculus and linear algebra.