Fig. 2.3 Earliest known illustration of a camera obscura. Engraving from R. Gemma Frisius, De radio astronomico et geometrico liber, 1545. The legend translates: "Observing solar eclipse of 24 January 1544."

The optical projection underlying perspective is illustrated by the device known as the camera obscura ("dark chamber" in Latin, a term coined by Kepler) illustrated in Figures 2.3 and 2.4. Although the issue is shrouded in uncertainty, there is some evidence that the device was invented by Alberti (Pastore and Rosen, 1984).³ It is no more than a box, or a room, with a relatively small hole in it, called pinhole a the box is to serve its purpose as a camera obscura, light should not enter it except through the pinhole. The side of the box opposite the pinhole is called the picture plane. If the picture plane is painted white and all the other sides are lined with light-absorbing black velvet, we can be sure that all the light that falls on the picture plane has traveled in a straight line from an object outside the box through the pinhole and that none of it has been reflected from the walls. So, moving into geometry, a camera obscura creates an image x of an object point v by ensuring that one and only one ray of light, called a projecting ray, coming from v hits the picture plane at x after passing through the pinhole. Unless the camera obscura is a room in which a spectator can stand and look at the picture plane (in which case the picture will be both large and very faint), we must devise a way of showing the picture it takes. There are two ways to do that: Either replace the wall of the picture plane with a piece of ground glass and view the image from outside, or replace it with a photosensitive plate that can be developed into a photograph. In the latter case, we will have a pinhole camera, that is, a photographic camera with a pinhole for a lens.⁴

Fig. 2.4 Main features of central projection.

A camera obscura does not correspond exactly to Alberti's window, for it inverts right for left and top for bottom(as Figure 2.3 shows). To understand the basis of perspective as discovered by Alberti,⁵ consider Figure 2.5: The three-dimensional scene, represented by one of its points, M, projects through the plane on which the picture is to be painted, P, to a point in space, O, called the center of projection, representing the eye of the painter. For a geometric abstraction such as this, we can say that the optical information about the three-dimensional scene is exactly replicated by the light passing through the picture plane.⁶
Because central projection is a geometric abstraction, it can be misleading to link it too closely with the perceptual processes of vision. The eye may resemble a camera, or a camera obscura, but the passive response of film to light does not remotely resemble active neural processing that takes place after light hits the back of the eye. It is better, therefore, to first comprehend perspective as a purely geometric procedure for the projection of a three-dimensional scene on the two-dimensional picture plane before one tries to understand its perceptual effects on viewers. One should avoid thinking of central projection as a method that mimics what we see.

		Fig. 2.5 Main features of central projection

We will see later (Chapter 4) that there is a sense in which a perspective representation mimics what a person sees from a certain point of view, but we will not discuss why a perspective picture looks compellingly three-dimensional until Chapter 4). The traditional representation of Figure 2.1, which places an eye at the center of projection and calls the center of projection the "vantage point"⁷ may be a convenient pedagogical device.
The formulation of perspective as central projection to a plane is still somewhat more general than Alberti's window, which implies that only objects behind the window can be projected onto the picture plane (see Figure 2.5)). Indeed, in the large majority of the cases, perspective is restricted to the region beyond the window. The geometric evidence for this point can be found in the size of the depictions of known objects. The geometry of perspective implies that the painting of an object which is in front of the picture plane will be larger-than-life; since Renaissance painters very rarely painted larger-than-life figures, most figures must be at the picture plane or behind it.⁸

 

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³ The first casual reference [to the Camera Obscura] is by Aristotle (Problems, ca 330 BCE), who questions how the sun can make a circular image when it shines through a square hole. Euclid's Optics (ca 300 BCE), presupposes the camera obscura as a demonstration that light travels in straight lines.

⁴ The pinhole camera and fascinating experiments using it are described in Pirenne (1970). (See also Hammond, 1981, and Kitao, 1980.)

⁵ http://www.mcah.columbia.edu/arthumanities/pdfs/arthum_raphael_reader.pdf for excerpts from On Painting.

⁶ [REVISIT] Alberti developed this conceptualization in a form that anticipates the computer representation of objects by plane sectors: "Each plane contains within itself its pyramid of colours and shadings. Since bodies are covered with planes, all the planes of a body seen at one glance will make a pyramid packed with as many smaller pyramids as there are planes. ... When [artists] fill the outlined regions with color, they should seek only to present the form of things seen on the surface [of the picture] as if it were of transparent glass. Thus the visual pyramid could pass through it, set at a definite distance with definite lighting, a definite centre of projection and in a definite location with respect to the observer." If the artist (or photographer) replicated the pattern of light in the plane of the picture, an eye located at the point of projection would receive exactly the same information from the picture as from the depicted scene, with no distortion (to the extent that the pupil of the eye approximates a point in space). This equivalence is not limited by the size of the window represented by the picture frame, or the angle of slant chosen for the picture plane. The geometry of perspective projection is therefore accurate and complete for any flat picture plane, regardless of size or location.

⁷ The vantage point is sometimes also called the "station point." Some, however (Todorovic, 2004) prefer to reserve the term for the point at ground level directly below the vantage point.

⁸ "You must make the foremost figure in the picture less than the size of nature in proportion to the number of braccia at which you place it from the front line... " (Leonardo da Vinci, 1970, 538, p. 324).