ur perception of space is dominated by perspective, in the sense of a reduction of the projected size of objects with distance. One of the key jobs of the visual brain is to decode this size diminution as distance in the third dimension, or egocentric distance. If the eye were a pinhole cameras, the projection of the world onto the back plane would be in perfect linear perspective (and in perfect focus). The succession of images projected on the curved retina within the eye what Leonardo da Vinci termed natural perspective, a series of distorted projections that needs to be integrated over time in a representation in the brain as the eye moves around the scene. How the brain decodes the information in natural perspective into an accurate appreciation of the spatial layout has yet to be resolved.

Incorporating lens optics into the projection system introduces the potential for curvature in the projected image. Such internal curvature may consequently be a property of human perception at the extremes of the field, but this curvature would apply equally to the original scene and to its projection from the picture plane to the eye, so does not affect the external projection rules of geometric perspective. The key simplification in perspective construction is that the pictorial image is governed by linear projection through the point where the pupil is located, regardless of any optical distortions beyond that point.

Historically, space representation through perspective has engendered great conceptual effort. Perspective scene painting was a springboard of mathematical geometry even at the time of Plato and remained influential through the Hellenistic era, but did not re-emerge as an artistic technique until the 1300s. However, full mastery of analytic perspective took another six centuries to evolve. Accurate one-point perspective dominated the1400s, being first used by Masolino da Panicale and his pupil, Masaccio. Two-point perspective was initially described by Viator in 1505, although the two-point construction remained unknown throughout the Renaissance until 1650, becoming widely used in the 1700 and 1800s. Three-point and multi-point construction diagrams for mathematical treatises were attempted unsuccessfully by Piero della Francesca and Leonardo da Vinci in the late 1400s, but none appeared in art works until an isolated example by Tiepolo in 1744. The three-point construction seems to have been first introduced into the 20th century art by Georgia O’Keeffe in her New York Series in the mid 1920s. Far from springing into force during the early Renaissance, therefore, a full understanding of linear perspective was not achieved for 600 years. Interestingly, most of the conceptual advances in perspective construction were made by artists rather than geometers.

Linear perspective is the geometry of projection of the lines in a scene through a picture plane to a point in space (or center of projection) corresponding to the pupil of the viewing eye (Fig. 1). The picture plane would be the canvas on which the painter wishes to depict the scene. For correct perspective, the picture will generate the same arrangement of light rays at the eye as did the scene behind it. When viewed from this point in space, therefore, the picture will form exactly the same image on the retina as did the original scene. The different forms of perspective construction concern the rules that apply to specific structures, but all are subcases of the same optical transform.

The following pages address several principles of perspective.