Hyperbola
Conic section analysis of the hyperbolic appearance of circles

Besides providing a uniform description of circles, ellipses, parabolas, and hyperbolas, conic sections can also be understood as a natural model of the geometry of perspective in the case where the scene being viewed consists of circles, or more generally an ellipse. The viewer is typically a camera or the human eye and the image of the scene a central projection onto an image plane, that is, all projection rays pass a fixed point O, the center. The lens plane is a plane parallel to the image plane at the lens O.

The image of a circle c is

(Special positions where the circle plane contains point O are omitted.)

These results can be understood if one recognizes that the projection process can be seen in two steps: 1) circle c and point O generate a cone which is 2) cut by the image plane, in order to generate the image.

One sees a hyperbola whenever catching sight of a portion of a circle cut by one's lens plane. The inability to see very much of the arms of the visible branch, combined with the complete absence of the second branch, makes it virtually impossible for the human visual system to recognize the connection with hyperbolas.
keywords:forbidden_words, oulipo:avoid_letter[t]