CHAPTR 5 Two-Point Perspective In addition to one-point perespective, another common perspective drawing technique is two-point perspective, illustrated in Figure 5.1. Unless otherwise stated, we will use the term twopoint perspective to refer to a picture that is set up in such a way that the picture plane is perpendicular to the plane of the ground (outdoors) or the floor (indoors), and as a consequence of this, only two vanishing points on the horizon line are needed to render buildings or other objects whose adjacent vertical walls or sides are perpenducular to each other. This is a typical situation when dealing with architectural subjects, both indoors and outdoors. Figure 5.1. A simple example of two-point perspective. Suppose the rectangular box in Figure 5.1 represents some kind of building, but we don t know anything about its size or proportions. Can we say anything about the correct location of the viewer? It turns out that we can. First, we can tell by looking that the picture plane must be perpendicular to the plane of the ground that is, vertical because the images of the vertical lines of the building are parallel to one another 69
70 Chapter 5 in the picture, and hence do not converge to a vanishing point. This can only happen if the picture plane is parallel to the vertical lines of the building, and hence perpendicular to the plane of the ground. Since our line of sight to any point on the horizon must be level, the viewer s eye is the same height as the horizon line in the picture. Thus the viewer s eye lies in a horizontal plane (the eye-level plane) H containing the horizon line (see Figure 5.2). Since the picture plane is vertical, H is perpendicular to it. It is convenient to think of H as a half-plane existing in the room where the viewer is to view the painting, as in Figure 5.2. The question is, exactly where in the plane H should the viewer s eye be located? picture plane Figure 5.2. The eye-level plane H. The correct viewpoint for the painting is somewhere in this plane, but where? the eye-level plane H??? If the picture were the result of a window-taping experiment, then the building would still be located beyond the window, and we could find out all sorts of things about it. ven though it s not a window, we can think of the painting as the projection of a building that was once behind the canvas. At this stage we don t yet know how the building would be situated to make such a projection. Figure 5.3 shows two possible cases. We can narrow down our choices for the viewpoint by recalling an important fact about perspective on windows. When the viewer s eye is at the correct viewpoint, the line (of sight) from to any vanishing point V on the window must be parallel to the actual line in the real world whose image has V as its vanishing point. Thus, regardless of how the building was oriented, the lines and in Figure 5.3 must be parallel to the corresponding building edges; since adjoining building edges are perpendicular, and are also perpendicular to each other. Figure 5.3 shows two examples of all possible cases.
Two-Point Perspective 71 top of building top of building parallel parallel parallel parallel top edge of picture plane the eye-level plane H Figure 5.3. Two of many possible locations for the viewpoint. Because the edges of the building form a right angle, the lines of sight to the vanishing points must form a right angle at the point. This brings up a question: What is the set of all points in the eye-level halfplane H such that and are perpendicular? It turns out that this set is a semicircle whose endpoints are and (see Figure 5.4). H top view Figure 5.4. The viewpoint must lie on a horizontal semicircle in the halfplane H. Theorem 5.1. The viewpoint for a standard two-point perspective painting (drawing, photograph) with vanishing points and lies on a semicircle with endpoints and. The plane of the semicircle is perpendicular to the picture plane. Proof. Consider a possible viewpoint in the half-plane H, as on the left of Figure 5.5. Since is a possible viewpoint, the lines and are perpendicular. Let M be the midpoint of and, so that the two segments M and M have the same length r. Let
72 Chapter 5 s denote the length of M. We are done if we show that s = r, for that will mean that all possible viewpoints are r units away from M, and therefore lie on a semicircle. Now and are adjacent sides of a rectangle, so draw the entire rectangle F, as indicated on the right of Figure 5.5. It s a well-known fact from geometry that the diagonals of a rectangle have equal lengths and meet at their common midpoint. Referring to the figure, this implies that s = t = r. F Figure 5.5. Looking down on the plane H and the top edge of the picture plane. M r r s t M r r s H H Figure 5.6. When working in twopoint perspective, art students often tape strips of paper to their drawings so they can spread the vanishing points far apart. This makes for a larger viewing circle, one that is more likely to be occupied by the casual viewer s eye. Theorem 5.1 explains a common trick used in art classes. Notice in Figure 5.4 that the farther apart the vanishing points and are, the bigger the viewing circle ; that is, the farther away the potential viewpoints will be from the picture. We know from Chapter 3 that when viewpoints are unusually close to pictures, viewers perceive distortions, because they won t suspect that the correct viewpoint is so close. To prevent this from happening in two-point perspective drawings, art teachers often have their students tape strips of paper to their drawing paper (as in Figure 5.6) so that one or both of the vanishing points can be located beyond the edges of the paper. An art teacher would say, We spread the vanishing points to avoid distortion. In view of Theorem 5.1, we could also say that We spread the vanishing points to enlarge the viewing circle. Is a small viewing circle really so bad? To convince yourself that it is, look at Figure 5.7. It s a drawing of some boxes in two-point perspective, with both vanishing points and in the drawing, making them very close together. The drawing has been set up so that the viewpoint is directly in front of the midpoint C of and. Imagine a semicircle coming out of the page with and as its endpoints. It should be clear that the viewing distance in this case is just the radius of the semicircle, which is the distance between C and (or ). Close one eye and hold the page so that your open eye is very close to C and directly in front of it. Gaze at C for a second, then let your eye roll down and look at the boxes. You ll see
Two-Point Perspective 73 that they are just ordinary boxes with square tops! Now move the page away from your eye and see how distorted they get. Figure 5.7. Vanishing points close together. Close one eye, put the other one very close to C (at a distance C), and gaze down at the boxes. From there you will see they are perfectly normal boxes with square tops!
74 Chapter 5 xample 1 (uncropped photograph). At this stage we know that the viewpoint must be on a semicircle, but where on the semicircle? The answer requires more information about the picture. A simple case is when we know that a picture such as Figure 5.1 is a standard, uncropped photograph. When a standard photograph is not cropped, the viewpoint must lie on a line orthogonal to, and through the center of, the photograph. Let s assume this is the case in Figure 5.8. xplain why the construction in Figure 5.8 correctly determines the viewing distance T U and the viewing target T (the point in the photo that should be directly in front of the viewer s eye). Figure 5.8. Solution for an uncropped photograph. The viewing target is T and the viewing distance is T U. Why does the construction work? T U Notice that in order to solve the problem, we don t need to know to know the proportions of the building; that is, whether it s a cube or an elongated box of some kind. In fact, it s possible to use the solution to figure out the proportions look for this same photograph in the exercises!
Two-Point Perspective 75 xample 2 (horizontal square). When we re not sure that a twopoint perspective picture is an uncropped photograph, other information about the picture or the subject can be helpful. In Figure 5.9 someone has started a perspective drawing of train tracks. Assume that the rectangles between railroad ties are actually squares. Find the viewing target and the viewing distance. (Or if you like, look at Figure 5.10 and explain why the construction works.) When you ve finished this example, try looking at Figure 5.9 (with one eye) from the correct viewpoint. You may find that you need to use your right eye, so that your nose doesn t block the view! assume this is actually a square Figure 5.9.
76 Chapter 5 T W 1 W 2 assume this is actually a square U The solution. The viewing target is T and the viewing distance is TU. Why? Figure 5.10. Solution of xample 2.