Introduction

The philosophers of the Renaissance considered mathematical investigation integral to some of their theories.  Such a combination of mathematics with natural philosophy was known as the “mixed sciences”, and this included the study of optics.  The term “optics” is Greek in origin, and was introduced in the sixteenth century as part of the Renaissance determination to return to the Greek origins of science.  The medieval name is the Latin term perspectiva.  Unlike its contemporary counterpart, it was a complete science of vision, encompassing not only the nature and behaviour of light, but the anatomy and functioning of the human eye as well. 

 Initially, it was believed that sight was due to the active emission of “eye beams”, but it was later discovered to be the reception of light by the eye.  In any case, the geometrical methods used to describe the way we see remain unchanged.  Ignoring problems of physics and focusing on geometry, the theory of perspective describes how to project a three-dimensional object onto a two-dimensional surface.  Prior to a more detailed discussion, it is necessary to understand what is meant by “cone” or “pyramid of vision”.   Consider Fig. 1 shown below. 

Fig. 1 – The pyramid or cone of sight is defined by the cube
and the centre of rotation O of the eye of the spectator.

The centre of projection O is the centre of rotation of the eye, and the convergence of all light rays from the object.  As a representation of these rays, straight lines are drawn from each point of the cube to point O.  The intersection of each of these lines with the surface of projection FGHI forms the correct perspective of the cube in two dimensions.  For instance, the point A is projected along the line AaO and a is its projection on the surface FGHI.  Similarly, the corners B, C and D are projected on b, c and d respectively, such that the projection of the top face of the cube is abcd.  Note that the visual angles subtended by the projections of these points on the surface FGHI  are the same as those subtended by the cube.  For example, the angle AOB is the same as the angle aOb.

Specifically, the perspective projection is the intersection of a plane with the pyramid or cone of sight.  It should be noted that the view of the object in perspective is only valid for one particular viewing distance and location.  When viewed in perspective, the object is said to be “degraded”.