Looking Through the Glass

Projective geometry allows us, as its name suggests, to project a three-dimensional world onto a two-dimensional canvas. A perspective projection often includes objects called vanishing points, which are the images of projective ideal points; the geometry of these points frequently allows us to either create images or to reconstruct scenes from existing images. We give a particular example of using a pair of vanishing points to locate the position of the artist Canaletto as he painted the Clock Tower in the Piazza San Marco. However, because mappings from three-dimensional space to a two-dimensional plane are not invertible, we can also use perspective and projective techniques to create and analyze illusions (e.g., anamorphic art, impossible figures, the dolly zoom, and the Ames room). Moving beyond constructive (e.g., ruler and compass) projective geometry into analytical projective geometry via homogeneous coordinates allows us to create and analyze digital perspective images. The ubiquity of digital images in the present day allows us to ask whether we can use two (or many) images of the same object to reconstruct that object in part or in entirety. Such a question leads us into the emerging field of multiple view geometry, straddling projective geometry, algebraic geometry, and computer vision.