Looking Through the Glass

Abstract

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.

The description in section “Going Backward from Pictures to 3D” of the three steps for reconstructing three-dimensional objects from a collection of photographs was influenced by a talk by Joe Kileel at the Algebraic Vision session at the SIAM conference on Applied Algebraic Geometry, July 31 2017 in Atlanta GA. Joe had just finished his PhD from UC Berkeley and was headed for Princeton.