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Chapter II: The Elements of Perspective (page 6)

The Elements of Perspective

Proposition 3: The perspectival images of parallel lines that are neither parallel nor perpendicular to the picture plane converge onto a vanishing point at some location away from the principal vanishing point (and which is not necessarily within the confines the picture). The location of the vanishing point is determined by the angle of its parallels to the eye.


Fig. 2.10 A Toronto skyscraper

Proposition 3 may be exemplified by a slanted picture plane. If the picture plane is not vertical, the images of all vertical lines converge onto the one and only vertical vanishing point (Figure 2.10). Figure 2.11 illustrates this property as it occurs in a picture of a box projected onto a picture plane tilted sharply downward. Next, consider a set of horizontal lines. They too converge onto a vanishing point. To specify the location of this vanishing point, we must first define the horizon line of the picture (see Figure 2.12), which is the line defined by the intersection the picture plane (which need not be vertical) and a horizontal plane that contains the center of projection. Any two horizontal lines that are parallel to each other intersect the horizon line at the same point, which is their horizontal vanishing point.

It can be confusing at first to realize that when one looks at a picture like Figure 2.11, one cannot tell whether the box was slanted and the picture plane was vertical, or the box was upright and the picture plane slanted (as we described it to be).11 Had we described it as the picture of a slanted box projected onto a vertical picture plane, we would have had to relabel the vanishing points, for the box would have neither horizontal nor vertical surfaces and the vanishing points of the cube would therefore not lie on the horizon line.


LEFT Fig. 2.11 Vanishing points.  
RIGHT Fig. 2.12 Definition of the horizon line.

This point can be further clarified by noting that in Figure 2.2 one projecting ray is singled out and given a name its own: It is the principal ray, that is perpendicular to the picture plane. It is along this line that one measures the distance between the center of projection and the picture plane. The principal ray is the projecting ray that runs perpendicularly from the center of projection into the picture, whatever the slant of the picture plane. If the canvas (or film plane) is rotated upward to encompass a tall tower, the principal ray is correspondingly rotated upward.12 In Figure 2.4, the principal ray intersects the picture plane at the horizon. This is true only in the special case when the picture plane is vertical.13

Having described various approaches to perspective construction, we are now ready to consider the reconstruction of a scene from its perspective representation. This scene inference problem is known as the inverse projection problem. In common with many inverse problems, it cannot be solved without some assumptions. To solve the inverse projection problem, we need only one assumption: that we know the shape of one object in the scene. Thus, if we know that a particular floor-tile is square, that tells us that it has both right angles and a length/width ratio of one, defining the projective shape of the entire floor grid. These considerations lead to the final proposition:

11 This, of course, is not a difficulty with pictures like Figure 2.10, because we know that most skyscrapers are vertical.

12 Alberti (1966, p. 48) called the principal ray the "prince of rays," a play on the common etymology of the words principale and principe, which derive from the Latin princeps, meaning "first."

13 We have chosen not to use the term horizon for the the locus of the vanishing points of lines orthogonal to the picture plane and parallel to each other, which some call the "horizon line" for that particular orientation in space

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