February 20, 1981


Dear ***:

I thoroughly enjoyed your article in the July College, but on the topic of "curvilinear" perspective, herewith are some comments:

Figure 1 shows a duck hunter. He pivots about point O, which is at distance b from the plane the ducks are flying in. The ducks maintain a constant spacing, a. As the file of ducks stretches, on into the distance, angle theta (measured from b) at which he is aiming increases in smaller and small increments, approaching a limit of theta = pi/2. Theta is the angle whose tangent is na/b (counting n from b), so the limit of theta as n approaches infinity is pi/2. If we mark out the path the end of his shotgun traces, we get a spherical map which records the angle at which he was aiming, as equal angles at the center of a sphere trace out equal arcs. A quantity proportional to theta, namely arc length, can be read directly from the map.

If we take away his shotgun and supply him with a telescoping sight vane and a glass plane, PP, at distance d from O and parallel to the plane of ducks, he will give us another map, this time on a plane. The triangle formed by O and any two ducks is similar to the smaller triangle formed by O and their traces on the plane, as θ is common and PP is parallel to the duck plane. So, length y traced out on the plane will bear the same ratio to x, distance between ducks, as d bears to b. Therefore equal lengths in the duck plane have equal traces on the parallel picture plane, however far one wishes to extend these planes. The length traced on the plane is not proportional to angle theta. Given distance d and the point where it strikes the plane, we could measure y from that point and compute theta: theta = tan-1(y/d); but we cannot read any proportional quantity directly off the map. This map has the compensating property, though, that quantities in a plane parallel to the picture plane are drawn in their actual relations, proportionately reduced.

Figure 2 shows how to draw things on a sphere, after the manner of the duck hunter. (As with perspective, it makes no difference - except in size - whether the projection surface is placed behind or in front of the object.) As plan and elevation do not sufficiently identify where the projection lines hit the sphere, auxiliary planes are required, as I have shown. Suppose we wish to transfer this spherical picture to a plane. This is the same problem as making a map of the earth, and it is important to keep in mind that there is no completely satisfactory solution to it, as the sphere is not a developable surface and cannot be developed in a plane without distortion. [ See "The Ideal Map", p. 28 , Elementsof Map Projection, Charles H. Deetz & Oscar S. Adams, U. S. Dept. of Commerce, Special Publ. #68, G.P.O.] Usually those who suggest-this as a drawing system seem to intend using orthographic projection to transfer the drawing to a plane, but this is far from being the best way to do it. As my little drawing indicates, the very property of the sphere we were interested in, that equal angles subtend equal arcs, is negated by this method.

This is the drawing system discussed by Panofsky in his treatise on perspective (unfortunately never translated into English). To my knowledge Leonardo was the first to propose it. John White in the Birth and Rebirth of Pictorial Space also argued this thesis, while introducing numerous misconceptions as well. The reason why this drawing system is attractive is as follows:

Definition 4 of Euclid's Optics [Morris R. Cohen & I . E. Drabkin, A Source Book in Greek Science] is "That things seen under a larger angle (angle at the eye) appear larger, those under a smaller angle appear smaller, and those under equal angles appear equal." If this is true, then the spherical drawing we just did has properties which are precisely analogous to those of vision; that is, equal angles subtend equal areas (analogous to "appear equal"), larger angles a larger area, and a smaller angle a smaller area. So the argument in the first place for this drawing system is that it enables us to match the result of Euclidean optics by means of projection.

What did the Greeks mean by the visual angle? If they meant, that angle found by sighting, through an astrolabe or some such instrument, then the vertex of this angle is at the center of the eye (more or less - because the optical axis is off-center). The 'apparent size' with which this angle was correlated would have been found through introspection. This raises some questions, in that, although judgments about apparent size and shape can be made with some readiness for the central region, they are quite difficult in the peripheral region. Does Euclidean optics apply to the entire field of stationary vision, or only to moving central vision? A luminous sphere like a street light is a good test; it subtends at the eye equal angles vertically and horizontally and for central vision will always appear as a circle, but does it appear as an ellipse for any part of the field, as it might if equal angles do not always correspond to equal apparent sizes?

It is by no means obvious why the visual angle should be an important quantity for stationary vision, as the eye is not an angle-ineasuring instrument like an astrolabe. Since the crystalline lens has been displaced as the effective organ of vision, the visual angle cannot itself be a direct datum of sense. The only way it can be used for vision is if it can be somehow read or interpreted from the retinal image.

Figures 3 and 4 show some natural appearances considered both in terms of Euclidean optics and also rendered in perspective. The observer before the row of dryers in a laundromat cannot be equally distant from all of them. According to Euclid Optics Prop. V "Equal magnitudes situated at different distances from the eye appear unequal, and the nearer always appears larger...", and, indeed, we see that the visual angle comprising the height of the furthest one is smaller than that of the closest one. But if a picture plane is sited parallel to the dryers, each dryer's projected height will be the same no matter how far away they are from the observer. Likewise with the door seen from below. The last drawing on 3 is of a string held taut and moved up and down.

Figure 4 shows that three spheres, equidistant from the observer, will all subtend the same visual angle, and each will subtend the same angle vertically and horizontally. But in the perspective projection they cannot be drawn as equal in size, and only the one centered about the CVP can even be drawn with a circular outline. With the cube I have tried to show by these (admittedly primitive) graphical means that, if the observer can see more than one side of the cube, then he is closer to one edge and by Prop. V, this edge must appear larger. Yet in the perspective projection, if the picture plane is sited parallel to a side of the cube, both edges of that side will be drawn equal in size regardless of which is closer to the observer. (If you want to check all these appearances, use one eye.)

This is to show that no amount of attentive observation will lead one to produce a picture in linear perspective, as it requires one to draw things which are never seen. This brings up the question of perspective in antiquity.

You commend your Figure 9 for its adroit handling of linear perspective. Let us try to reconstruct what this is a picture of. We will have to make some conjectures, such as: the central temple/gazebo has a square floor plan (with cut-out edges) as at 5, the retreating sides set at 45 degrees with the picture plane. If we continue the top and bottom edges leftward, we find they converge to a point. If our conjecture about the building's shape and orientation is correct, this must be one of the two distance points. Continuing, we find the other DP and take the mid-point between them as the CVP (central vanishing point). (If the DP's occur within the picture the angle of vision exceeds 90 degrees, which is already alarming.) The distance between the CVP and the DP gives us the distance between the station point and the picture plane, enabling us to set up an elevation. As the figure with the sticks seems to be quite close to the picture plane, we can bring it forward a little and take its height off the plane, and transfer it to the elevation. Now, shall we place the temple close to the figure or further away? If close, we can give it a plausible size, but there isn't much room for the animals to roam. If further away, the figure could scarcely even clamber up onto it. What about the man and the goat? We can place the man by assuming he is as tall as the other figure. If we assume the goat is the usual goat size we must place him near the man; but isn't the man advancing on a bridge of some sort, so that the goat must be further back? But as we move him back, he inexorably grows larger and larger.

If we wish to determine if pictures like this are in perspective or not, the best way is to try to construct a plan and elevation which could yield the given picture as a central projection, as this method is without bias toward any particular construction points or methods. But trying to construct plan and elevation for these pictures only shows how much worse they are than they look.

Your Figure 10 is of interest also, as John White alleges that the upper orthogonals (invisible in the reproduction) converge to a single point. It is odd that if the artist knew the upper orthogonals should converge to a point, he did not know the lower ones had to converge to the same point. (Seems like insufficient generalization.) The lower part of the picture can at best (with a little charity) be said to embody a vanishing axis construction, which is itself a puzzle. How can we obtain this? We might begin by attempting to set up an auxiliary plane (Figure 6), and then rotate it back in line with the other. This would give a vanishing axis of sorts, but does not permit the further column to have its frontal plane parallel to the picture plane, as it does. The same objection would apply if we summed many such planes into a cylinder.

We can obtain it if we permit the observer to move, or if we are only able to specify the observer's position as, 'he is to the right of this column... he is to the right of that column as well' without being able to quantify it further. (If we are permitted a moving observer, we can indeed get White's "bird's eye view" - vertical oblique projection - from a bird. We can construct an orthographic drawing machine along similar principles.)

As to the supposed written evidence of a mathematical theory of linear perspective as found in Vitruvius' text, I assume you rely on White's argument so I will discuss that. These are the two passages: "In like manner, scenography is the sketching of the front and of the retreating sides and the correspondence of all the lines to the point of the compasses', and "...how it is necessary that, a fixed centre being established, the lines correspond by natural law to the sight of the eyes and the extension of the rays, so that from an uncertain object certain images may render the appearance of buildings in the paintings of the stages, and things which are drawn upon vertical and plane surfaces may seem in one case to be receding, and in another to be projecting." White professes to find described in these passages a little picture like Figure 7, which he further alleges is a perspective construction. In this little picture, the circle represents the intersection of the visual cone with a flat plane, the center is where a perpendicular from the vertex or point of sight intersects the plane; and the radii represent lines perpendicular to the plane which are projected onto the plane. First some remarks: even if perspective were taught by presenting little pictures showing the various patterns which may be created by perspective construction, which it is not, surely a word about the conditions of applicability of these little pictures would be in order. In this case, the lines which are to converge (in White's lexicon, a synonym for correspond) to the center of the circle are orthogonals, or perpendiculars to the picture plane, but Vitruvius nowhere mentions orthogonals. To read the text White's way, one must have all lines converge there, and White repeats this several times without noting anything remarkable about it, although this is a few more lines than we want.

Another thing to note is that, if this is the word on 'scenography', then everything must be in one-point perspective. But it wquld be a little difficult to develop a mathematical system of one-point perspective only, and I'll note in passing that the first perspective picture - Brunelleschi's - was in two-point perspective.

What evidence is there in the text for supposing that the "fixed centre" and "point of the compasses" are points on a plane? White's arguments on this topic are egregrious question-begging ('scenography' implies drawing...we draw on a plane...therefore this is a point on a plane). Vitruvius' only clear reference to a plane is to things which are drawn upon "vertical and plane surfaces", which seems a curiously tentative reference if he has been referring to this plane all along. Why not simply say "the aforesaid plane"?

What bothers me about this entire controversy is that it seems almost to share in the popular fallacy that the CVP is the be-all and end-all of perspective. In fact, the CVP is just a special case of the general law in perspective that all parallel lines not parallel to the picture plane converge to a point found by taking a mutually parallel line from the point of sight, which intersects the plane at the desired point. Instead of looking for particular construction points or methods, which might have more than one interpretation, we should I think focus our attention on the definition of perspective, given by Alberti thus: "a certain cross-section of a visual pyramid" (or cone). Vitruvius is certainly aware of the visual cone (in fact, I would read the "fixed centre" and the "point of the compasses" as references to the vertex of this cone), but he does not say anything implying section or projection.

This is not just an obvious extension of the idea of the visual cone or the only way it could be applied to drawing. (Indeed for people like the Romans who did not have transparent glass it might not have been altogether easy to consider the wall the artist was drawing on as a transparent plane sited between himself and the objects depicted.) In what way could the visual cone be employed in art if not to section it? Roman art begins from a base of oblique projection [See Drawing Systems, Fred Dubery & John Willats, Van Nostrand Reinhold, N. Y. 1972, an invaluable book] (and retreats there when pressed), but then a process of criticism and correction began to be carried out in light of the placement of the observer (possibly a typical observer), such as, the observer standing on the ground cannot see the roof of the temple. But, since the only measure in Euclidean optics is the visual angle, which does not have any obvious planar expression, this analysis in the absence of a projection plane could proceed only so far and cannot be quantified; as Piero della Francesca says, "If there were no plane, one could not understand how much the objects were foreshortened, and thus they could not be drawn". [p. 258, A Documentary History of Art, Vol. 1, ed. Elizabeth G. Holt, Doubleday Anchor.] Though I don't think it necessary to suppose Vitruvius meant anything particularly vague, in fact without a projection plane this analysis could proceed only in relative terms and not in metric or quantitative terms. Of course the results could not be criticized either, as the viewer would also lack means for making this analysis.

Even if this were a legitimate exposition of perspective, it would be most unusual to begin with the CVP, before even mentioning the picture plane, much less the observer! "A fixed centre being established" would more likely refer to the point of sight, which, in theory, comes first.

Though we can't find a definition or theory of perspective in Vitruvius' text, is what we find there a brief allusion to a simple but servicable perspective construction, the little circle? Let's see how serviceable it is:

From point of sight O, project the set of perpendiculars behind the plane onto projection plane PP (see Figure 7). This is not quite like White's picture, so let's produce one of these little 'radii' out to the circumference. What are we doing? (Remember, everything drawn on this plane has a meaning). What we are doing is continuing this orthogonal out in front of the picture plane until it hits the visual cone. One can very well draw things in front of the picture plane as well as behind it, though I don't think people often do both simultaneously. While not impossible, it would be confusing as the plane of the picture would tend to get lost. If we consider the picture plane as a pane of glass (which is most usual in art), continuing these lines is meaningless.

Now let us continue two of these lines back to the center. What we are doing in this case is extending the orthogonals toward infinity, and we can see that as we extend them the line on the plane which is the projection of the distance between them decreases in length. If the length of the two orthogonals is x, and the distance between them as projected on the picture plane is y, then the limit of y as approaches infinity is 0, in short this line vanishes to a point, which is thus a kind of 'ghost of a departed quantity'. Thus the analogy with the radii of a circle is not altogether illuminating, as no one has ever suggested that radii approach the center of a circle as if it were a kind of asymptote. (White is the only person I know of who even find it natural to say that the radii of a circle converge toward the center of the circle.) Most of the construction points and lines used in modern perspective are limits of this sort, and I have also shown in 7 the derivation of the horizon line. Assuming a flat ground plane stretching to infinity, the horizon line as seen on the projection plane is a consequence of the fact that you can't look up at the ground plane; but you can't strictly speaking look straight out at it either for any given distance, so this line is again a limit. These are not excessive refinements of perspective theory, as even in the most unpretentious little perspective handbooks one rarely finds vanishing points mentioned without a mention of infinity. It is the un-ancient character of this reasoning which leads me to doubt that the ancients would have gone down this road. Euclid's Optics does not provide a framework in which this argument can be made, as he conceives of visual rays as discontinuous. One cannot let them fail more and more closely together as the argument requires. (Oddly, however, they move (Prop. 1), though apparently not continuously.) Prop. III states that "For every object there is a distance at which it is no longer seen", because it falls through the cracks of the discontinuous rays. Thus we might say that, at some given distance of the railroad tracks (7), the tie between them fell through the cracks and was no longer seen, but what about a railroad tie of twice the width? Presumably it would no longer be seen at a different distance. (In fact, for the question of why a thing can not be seen at some particular distance, one might wish to suppose some discontinuity, in the retinal receptors or whatever, but perspective theory supposes no discontinuity in anything.)

Euclid shows that parallels converge, but does not trace this convergence to a limit. The point which is the limit of this convergence is a point on a projection plane and not a point 'out there' or an apparent point, so one has to employ projection to find it, but Euclid does not use projection to study appearances. One might argue that as the railroad tracks approach infinity, the visual angle of the ties approaches 0 degrees, but collapsing an angle does not give a point. I wonder therefore if vanishing points, which may on occasion have been used in antiquity, were ever given a mathematical derivation or thought of as other than irrational appearances. If they were given a mathematical treatment, it would have to change our conception of the capabilities and inclinations of ancient mathematics.

Since we have seen that the radii of this little circle are distinctly problematical, it remains to consider its center and circumference. White misconstrues Alberti's centric point as being necessarily the center of the rectangular base of the visual pyramid. Bizarrely, he quotes this passage for support: "First where I have to paint. I draw a quadrangle of right angles...then within this quadrangle, where it pleases me, I mark a point which occupies that place where the central (visual) ray strikes and for that I call it the central point." Does "where it pleases me" mean "in the center and nowhere else?" One should not be misled by the term "central ray" into supposing that this ray must be surrounded symmetrically by the other visual rays or that it must strike the picture plane at its center. Alberti employs the term "central" by analogy with the "centric line", [Leon Battista Alberti, On Painting... (sorry I can't be more informative, but when I xeroxed this I neglected to xerox the title page)] or diameter, of a circle, in this way: that, just as the centric line strikes off equal angles on the circumference of the circle, so the central ray makes equal angles with the picture plane, in other words is perpendicular to it. (The fact that the only circle in this analogy is in a plane perpendicular to the picture plane doesn't stop White from claiming it as his own). For a given picture plane, one may make the central ray strike where one pleases; in short, the visual pyramid need not be a right angle pyramid as its vertex is movable. One might understand this better by thinking that, for any given pane of glass or grid of threads, one may place ones'eye where one likes, and where a perpendicular from the eye strikes the plane is the CVP. And, in practice, the CVP is scarcely ever sited at the center of the picture, the resulting symmetry being thought undesirable. If we apply the same reasoning to the visual cone, we see either that the CVP need not be the center of the circle, or, if we insist that the visual cone be a right angle cone, then its section need not be a circle but might be an ellipse. (White describes the visual cone and visual pyramid as being generic terms differing only in the base being taken as typical, but for some of the ancients I think the visual cone may have been derived from the assumption of a maximum visual angle of less than 180 degrees, which upon rotation would produce a right angle cone. If the ancients meant by the visual angle, that angle measured by sighting through an astrolabe, then it seems correct that this angle is less than 180 degrees.)

Note that, if this little image is to be understood at all, one must realize that the visual cone is to be sectioned with a plane, or else it cannot be understood why its boundary is a circle rather than the picture rectangle. In short, one must understand the basic perspective geometry...but be able to win from this understanding nothing but an unusable little picture. One has to assume the ancients were a sort of precocious child with some unaccountable learning disability. Even White realizes that this little picture is not "to modern minds" (p. 256) an accurate perspective construction, but that doesn't stop him from saddling Vitruvius with it. But had the ancients thought up the basic perspective set up, there is no reason why they could not have developed accurate and usable construction methods. If nothing else, they could have used Brunelleschi's construction from plan and elevation, which is not so difficult or cumbersome as people often imagine. One can in this way make perfectly accurate perspective constructions without having any knowledge at all of particular construction points or methods. For reasons given above, I do not think they would have developed short-cut methods using vanishing points, but they might well have worked out some system of proportional diminution and convergence which could give equally accurate results. (It is only the modern habit of supposing all lines to be infinite unless specified otherwise that makes it seem inevitable to extend these lines to their limit.) There is no reason, however, to suppose that they did so, unless we are willing to speculate about pictures never seen and texts never unearthed. The closest thing in antiquity to perspective is one of Ptolemy's map constructions (if you like I can send a xerox) in which certain lengths are projected from a point onto a line, but cannot as a whole be said to be a central projection onto a plane (even if he had done what he set out to do, which he doesn't). It might have been a step on the way but has no precursors or repercussions I am aware of.

But to get back to White's little circle, we should note it has never been used or suggested as a perspective construction in the modern era, except by White. The reason he suggests it is because that is what a perspective picture looks like to him, until "overlaid by the ensuing spatial illusion". It is far less helpful to the artist than he thinks to point out what something ought to look like. Can it be used as a construction method? I think what White is really thinking of is a picture I show at 8 - an empty room. He seems to misconstrue this as the basic perspective architecture of the picture, but it is not, nor even really a construction method, but rather a picture of a particular thing - a room - which is used as an all-purpose stage setting. (Changing the outline from a rectangle to a circle turns it from a room into a tunnel and renders it useless.) The orthogonals in this case do indeed come out to the border of the picture, because that is where we set the walls; had we taken them elsewhere, they would not have; so, they are like radii only accidentally if ever. Besides, for the same reason that "snow-flake" is not an acceptable thumb-nail proof of Euclid Book IV Prop. 15, so this little circle is not an acceptable perspective construction - it doesn't matter what pattern things make!

I would suggest there are two flaws in White's procedure which make his analysis of ancient art less enlightening than it might otherwise be. One is his notion of pictorial space. The 'space in the picture' is a problematical expression, the picture being flat. Unless it simply refers to the real space which is represented therein, it can at best be an illusory appearance, something which seems to be but isn't. As with a 'fake rose', a complete inventory of the properties of this supposed space - that one object may occlude another, etc. - could only lead to the conclusion that such properties are not the properties of any space. Although terms like this have a legitmate use, they should not be used to imply that there might be a geometry of pictorial (or visual) space, any more than there could be a verterinary science of painted dogs. The supposition of such space is often needless. When I look at a map and say, "84 meets 86 at Hartford", there is no need to suppose I am making a statement about 'map space', a sort of halfway house between the 'real' 84 and 86 and the squiggly little lines on the map. Though the map may be unlike the real 84 and 86, I can read correct information about them from it. One should be wary of excessive interest in 'the space in the picture' - or the space behind the mirror for that matter - lest through thinking too much about this space, one should disappear into it.

Secondly, the only way White is able to distinguish between drawing systems is by (what he believes to be) the characteristic orientation of the drawn object. However, there is no drawing system so impoverished that it can handle only objects in one particular orientation. In medieval practice, each object the artist turned to represented a completely new problem which demanded its own unique solution, so we see tables and chairs drawn one way, people another, etc. This was all swept away by the Renaissance which set out one projection plane and one mode of projection to be applied to any and all objects. It doesn't help for the historian to be stuck in the medieval mode.

Here aresome consequences of something I've already mentioned: For a given point of sight O, with picture plane at distance d, and a given line h parallel to the PP whose distance from O is x, what will be the length of y, projection of h on the picture plane? Since h is parallel to PP, and theta is common, the larger triangle is similar to the smaller one, and therefore y = hd/x. As h and d are assumed to be constant, we can say y = k . lx, in other words, y - the projection of h on the picture plane - is inversely proportional to x, distance of h from the observer; so that, if h is twice as far away, its projected size will be half as large. But, by the same token that y1 is in fact twice y2 (see 8), it will not subtend a visual angle twice that subtended by y2, as the second equal segment is further from O than is the first. (See Euclid Optics Prop. IV.) In a case like this, Euclid would point out that the second segment is more distant than the first, but Alberti would point out that it is seen under a more oblique angle, thus more foreshortened. He states it thus: (p. 48) "The central ray (i.e. perpendicular) is that single one which alone strikes the quantity directly, and about which every angle is equal. This ray, the most active and the strongest of all the rays, acts so that no quantity ever appears greater than when struck by it." Since we measure distance from a point to a line by taking a perpendicular, it comes to the same thing in this case, but both ideas are helpful.) If we wish to find visual angle theta for any given distance x of the given line, we can find it thus: theta = tan-1 (h/x). So, it is not true (as is often stated) that if something is removed to 4X the distance it was, it will appear 1/4 as large; bearing in mind it is the visual angle which is important, (setting h = 1 unit and initial distance = 1 unit) the ratio will be apparent size at 4 units of distance/apparent size at 1 unit of distance = tan-1 ¼/tan-1 1 = .24497866/.78539816 = .31248012; in fact, this ratio is closer to 1/3. (Euclid does not strictly say that apparent size is directly proportional to the visual angle. It is difficult to make judgments about apparent size other than equal, greater, or less, but one can generally make a judgment of bisection. In this case it is not meaningful to suppose a constant of proportionality might be found.) A graph of the two functions (y = 1/x and y = tan-1 1/x) is given at 9. The first increases more rapidly, which is why in a movie it is so unsettling when an object approaches the camera. A practical check of this is to look straight out a ruler and place one finger at a distance twice that of the other finger. Does the first one, in appearance, bisect the other? In fact it looks somewhat larger, as we would expect. This is without prejudice to the fact that, if one placed another finger atop the further one, those two fingers would subtend a visual angle equal to that of the closer one; the fallacy lies in assuming that the additional finger in this case subtends a visual angle equal to the one to which it is added, which it does not, as the additional one is more distant from the point of sight.

This next one is a minor case, but is so adamantly mistated that I will slip it in. Figure 10 shows a foreshortened circle drawn in perspective. What we get appears to be an ellipse, and I provide a proof (of sorts) that, in this case at least, the major axis of the projected ellipse bisects the minor axis, as it should in a true ellipse. So, the two segments of the bisected minor axis are in fact equal, but just because they are equal, they cannot appear equal, as they are not equally distant from the observer. One can check this out with a cardboard circle.

Let us next consider the problem of quantities on a line with the eye, the supposed problem of seeing 'depth'. All lines emanating from the eye (such as the projection lines we have been using) will appear completely foreshortened to the observer, in other words will appear only as a point. Since these lines comprise all distances from objects to the eye, these distances cannot be seen directly. But it seems misleading to suggest that these lines comprise the third dimension, as we do not usually mean by the third dimension lines radiating in all directions from a point. At most one of these lines will coincide with any given axis of a Cartesian co-ordinate system (in fact, if one sites the origin of the co-ordinate system at the point of sight, more than one axis will coincide with one of these lines). Similarly, if one looks at a cube, at most one of its edges can be foreshortened to a point (11); the other parallel edges must be seen. It is true also in a drawing system employing parallel projection lines such as orthographic or oblique projection that lines coincident with the projection lines are projected as a point, and in this case one might well say that a dimension has dropped out. One cannot say this of central projection, whether on a sphere or on a plane; however, if on a plane, one can distinguish between quantities in a plane parallel to the picture plane and other quantities, which will have different characteristics on the projection, but one can't even do this with central projection on a sphere. Figure 11 also shows at what angle the "foreshortened" side of a square exceeds in length the "unforeshortened" side for both vision and perspective. (Piero della Francesca, A Documentary History of Art), gives an odd an interesting "proof" as to why the visual angle in a perspective picture cannot exceed 90 degrees, which points this out. This angular limitation is not an inherent limitation in the system, as anamorphic art demonstrates; the picture will still be illusionistic when viewed from the point of sight, nor is it in fact a limitation of vision.) Those who talk about the third dimension being absent in either perspective or optical projection have this in mind: a.) the picture is in fact two-dimensional; b.) therefore there is nothing at all problematical about reading the first two dimensions of space from it; c.) where, however, could the third dimension be hidden? But only a.) makes any sense. Returning to the distance of points from the eye, which is in fact lost in the projection, let us note first: the observer is not in fact a point in space, and other distances - such as distance from hand to object - which are not foreshorted completely - might be of greater practical interest; and also, any given line is completely foreshortened only for a projection point along that line; by taking a different viewpoint, whether by moving the eye (even rotation will do the trick as we'll see later) or by opening one's other eye, one can see any quantity that may have been missed. But, still supposing it a defect that quantities along the projection lines must be foreshortened to a point, how might we supply this defect? One nice way is to suppose an environment filled with objects of equal size, in which a judgment of "equal apparent size could be read off as a judgment of equal distance. The distance in this case is distance from a point, the point of sight, and so we will wind up with a system of spherical polar coordinates in which the position of any point can be specified by two angles and a distance. The texture gradient theory supposes such an environment, but as stated is marred by the substitution of linear perspective for the visual projection. Gibson justified this on the grounds that there is a point to point correspondence between a picture in linear perspective and the picture on the retina. But the properties we are interested in are the metric properties of the projection which need not be preserved by such correspondence. Figure 11 shows an object lying on a tile floor; but the three tiles it lies on do not for vision provide equal units with which to measure the object, as, being unequally distant from the eye, they do not subtend equal visual angles. So to measure with them would be using a shrinking and expanding yardstick.

In Renaissance perspective, proportions on planes parallel to the picture plane are represented undistorted. One can take proportional measurements off each such plane, marching on in serried rows to infinity. This was not the least of the attractive features of this system to Renaissance artists, who tended to identify beauty with harmonious proportion. For vision, though, the only surface on which proportions could be seen undistorted is a sphere concentric with the eye. Euclid Prop. IV shows that equal intervals on a straight line cannot be seen as equal, as a straight line cannot be everywhere equally distant from a point, and the same is true of a plane. And, this is the reason why perspective 'works' - provides an illusion - in spite of the fact that, strewn about the picture plane, are ellipsoid spheres and other things no one has ever seen. They are not seen by the observer at the point of sight either, because he sees this plane at unequal distances, itself foreshortened and diminished. In order to get the illusion he must not correct for these foreshortenings, but this is easier said than done, because of what is called shape constancy. If the observer can detect the foreshortening of the plane he will likely correct for it. The first perspective picture was intended to be viewed through a peephole, and this provides the ideal solution, both to keep the observer one-eyed at the station point, and also to keep him from detecting the shape of the plane. Most people have never viewed a perspective picture under these conditions and so do not know what is meant when these pictures are called illusionistic, but only call them so through courtesy. But the viewer at this peephole will not see either the harmonious proportions, because they are worked out with respect to the plane and not the point of sight, or the surface design, because the effect of the peephole is to make the surface invisible. Perspective has been used much less often in Western art than might be thought; about all that it has ever been used for is the architectural stage setting of the picture, where the undistorted frontal planes are considered veridical. Spherical or rounded objects (like human beings for instance) are almost never drawn in perspective. Most books on perspective advise ellipsoid spheres not be drawn, but instead suggest setting out auxiliary picture planes for them (a kind of surreptitious cubism). This pattern of use suggests that, where perspective has been used, it has at least in part been because, like other drawing systems, it is veridical in some ways that vision is not.

Since the discoveryof the retinal image a not entirely happy analogy has been drawn between that image and a perspective picture. The naive imagine that the retinal image is in perspective; the less naive but still erring think that it ought to be. Somehow the words 'correct' and even 'scientific' have attached themselves to linear perspective, so that one can often read of a picture (or the retinal image) that it is not in 'correct' perspective, and is left with a disquieting feeling that the picture just doesn't measure up - whether it's intended to or not. But there would be dreaful consequences if we did see in linear perspective, as follows:

Gregory (p. 165, Eye & Brain) gives as an illustration a picture by Canaletto which is "a fine example of perspective" and wonders has he painted "the geometrical perspective, as given in the image in his eye?" Let us summon the shade of Canaletto and suppose he answers 'yes'. We suppose him to have had a pin-hole box camera head, as at Figure 12, and depict him trying to draw a still life with oranges. (Recall that he draws by copying "the image in his eye") . Looking straight ahead, he produces the first drawing; turning to the left, he corrects it to 2; glancing to the right, he scratches that out and produces 3, but, looking straight ahead again, he sees he was right to begin with! Even though the point of sight and visual angles have not changed, the pictures are completely different. Frustrated, he determines to look straight at each orange, and doing so can draw them with circular outlines. When he does this, what he is doing approximates to spherical central projection. (He couldn't draw everything he sees this way on the flat paper without distortion.) But, for one glance, there is no way he can see each orange the same, even if they are identical and at equal distances from him. Despairing of his art, he lapses into insanity, with the sad results that we see.

In perspective the projected shape depends not only on the object's shape and distance from the observer, but on how obliquely its projection line hits the plane, so that for a given object and observer the projected shape depends on how the picture plane is oriented. In spherical projection, where every projection line is perpendicular to the projection surface, this cannot occur. A practical analogy is to take a flashlight and play its beam over a wall. The same beam produces a circular or elliptical trace depending on its angle of incidence, but if one were seated at the center of a planetarium, one could only get a circular trace. (If you wish to check these assertions, a mechanical way is by using a pin-hole camera (as Pirenne does), which always gives pictures in accurate central projection. One can make the hemispheric central projection by replacing the flat screen with a hemisphere centered about the pin-hole. There is also a mechanical way to make parallel line projections: take long, thin tubes (like drinking straws) and dip them in thinned black paint. Set them out in a bundle to project either perpendicularly (orthographic) or obliquely (oblique) onto the projection plane. One can get an image of a candle flame, for instance, which, while not acceptably in focus', nevertheless has the virtue of being in oblique or orthographic projection. (If one made the tubeslong and thin enough, one could I think make the image sharper - though dimmer.)

Perspective pictures are sometimes adduced as an argument that vision is deceptive, as if one would say 'we would see what we see just the same if we were looking at a flat picture.' It is as if these pictures were unaccountable productions of nature rather than art. If one considers how they are done, the more striking thought is 'we would see just what we see if we were looking at real solid objects which obey the conditions of Euclidean geometry', because this is what perspective is premised on. Most people, having in mind the impressionists tramping off with their paint boxes, conceive art to be a system which takes appearance as input and spits out another appearance as output. This is not what the Renaissance artists did, though - their compositions were all inventions. Brunelleschi's method takes reality as input and yields appearance as output (one can use the method to do the reverse also). This is true even of Durer's lute-drawing machine - the only appearance involved is the output, the picture of which one says, 'my, that looks just like a lute'. For the Renaissance artists, to imitate the appearances was a problem to be solved and not a self-evident description of their activity. I would guess that the most striking think about Brunelleschi's little demonstration to his contemporaries was that 'the appearances', those maligned things, could be neatly constructed by projective methods and so were as rational, mathematical, elegant and true as anything else. So, it is surprising to find Berkeley phrasing the question of the appearances just as if Brunelleschi had never lived, i.e. either appearance is simply identical with reality, or else there is no intelligible relation imaginable between them. It is as if a navigator, being given a map, were to say 'but the world is large and round! This piece of paper is small and flat! It is evidently useless', and, casting it aside, attempt to tell where he was in the world by scooping up seaweed and taking water temperature.

One might object that the perspective picture is not an appearance but only another thing in the world that presents the same appearance as the given original. But, employing the correction of the system suggested by Leonardo of using a curved projection surface, one can arrive in one leap from the real object to its appearance. As you know there is a prescribed series of ablutions and genuflections one must perform before naming the Name, Appearance, ineffable, inexpressible, and not to be compared with any profane thing. It is true that color sensations (the blueness of blue) are not describable or definable, but to postulate that 'patches of color' are the basic data of vision is not neurologically defensible. Certainly there is no merit in approaching the subject from its least tractable angle. The basic data of vision are shapes, which can be described geometrically or algebraically without traducing their nature. One might object, we do not see an (apparent) circle, we see an indefinable something which we have associated with a circle (known presumably to touch) by experience. But this is not true, because judgments of (apparent) distance can be made for vision, and one can define a circle for vision just as it is usually defined, as the locus of points at equal (apparent) distances from a given point. Because such judgments can be made, the visual field can be developed geometrically as a two-dimensional (not flat) surface), which has a necessary projective derivation from the three-dimensional world.

(It is also true, though of less importance, that in some respects the visual projection is identical with the object, if one considers only topological properties. The order of points on a line, for instance, is the same for vision as in reality. According to Piaget [The Child's Conception of Space, W. W. Norton & Co., Inc., N.Y.], this may be the only type of information infants and very young children can get from vision, although it is by no means the only information available.)

We brought up the question earlier of whether Euclidean optics - and consequently the projection system based on it, spherical central projection - is applicable to the entire visual field of stationary vision, or only to moving vision. Why it must be applicable to moving vision was suggested by our Canaletto: for a projective surface at right angles to a visual ray, which rotates about a point, as the projective surface becomes vanishingly small, the resulting composite picture will approach to a spherical projection from that point. (In the eye, however, the center of rotation is not a point on the optical axis, which complicates matters.) It would be helpful to suppose that the stationary visual field is the same kind of picture (although it can't be taken from the same point of sight), for this reason:

Helmholtz mentions that many people are unaware of the lack of definition in the peripheral field, because as the eye darts about, the fovea fills in detail and builds up over time a coherent and detailed visual field. But if a framework supplied by one sort of projective system were being filled in with details from another, one should expect discrepancies to arise as, for instance, a circle is substituted for an ellipse. There are in fact some discrepancies between stationary and moving vision arising from the non-coincidence of the two points of sight, but other than this, it is quite hard to track down any discrepancies.

Figure 13 shows a certain drawing system, in which equal angles do not subtend equal areas on the spherical projection surface, because the projection is taken from a point neither on the circumference nor the center. Although the object spheres are equidistant from the point of sight, they subtend unequal arcs on the equator. In fact, the problem is worse than this, because (according to Pirenne [M. H. Pirenne, Vision & the Eye, & Optics, Painting & Photography is the best authority on these topics]) this is an approximation offered only for the region about the optical axis. Away from the axis the rays are supposed to become more constricted, compounding this inequality. I have attached as Figure 14 a diagram, a simplified verson of which is the basis of Figure 13. Bizarrely, the projection point K,, is the center of curvature for much of the projection surface, giving us the desired picture over much of its extent. But this can only be coincidental. Helmholtz thought the eye was not truly spherical, but flattened at the back. I have not encountered this opinion expressed elsewhere; usually the eye is represented as being approximately spherical.

One might suggest that this picture could be corrected by means of a grid, to yield the desired information. But if the picture is not a central projection to begin with, and Pirenne says it is not, then it cannot be corrected to be one. There is thus no alternative but to deny that Euclidean optics is valid for the entire visual field of the stationary eye.

This perhaps is not as alarming as it might sound, as the construction which yields these disturbing results is obtained on a pure a priori basis. The various surfaces and curvatures of the eye are measured, little difficulties - like the fact that the lens is not homogeneous - are dealt with by ad hoc assumption, the formulas are plugged in, and the points fall on the retina where they may. What is not then done, that I am aware of, is that conclusions are worked out about the geometrical properties of the retinal projection, which are then compared with the real retinal projection. This would not be impossible to do - see what Pirenne does with eyes from albino rabbits in Optics, Painting & Photography. One could, I would think, take measurements from these eyes with compasses and determine if equal visual angles subtend equal retinal areas (visual angle has to be defined as we will define it shortly). If not, then what? - because, if one marked off a grid on these eyes and projected objects onto it, surely one would get a result which could be described somehow in projective terms. It is essential to do this because, just as with a map, what projection system is used determines what information is available and what not.

One is led to be somewhat sceptical of this construction, because there seems to be some lack of communication between art history and physiological optics. Pirenne, though aware of Panofsky's thesis, does not seem sensitive to the odd consequences of denying it. The entire problem does not seem to have been approached in the best direction. The eye is not an organ for detecting light; who was ever interested in light? Are we plants that seek nourishment in the sun? Light rays are only useful for vision to the end of obtaining a picture or map of what is out there, as their straight line propagation makes them usable as projection lines. (In fact, they are not altogether suitable as projection lines, being subject to reflection and refraction, which projection lines are not.)

In any case, supposing Panofsky's thesis not to have been refuted, here is how to draw the retinal image as a central spherical projection:

We cannot use light rays for projection lines. Light is propagated in a straight line only in a homogeneous medium, which the eye is not, and our projection lines must be straight. The lines we will use for projection are called 'visual rays". Leonardo shows in an ingenious experiment that these lines are straight:

"...place a light upon a table, and then retire a certain distance away, and you will see that all the shadows of the objects which are between the wall and the light remain stamped with the shadow of the form of the objects, and all the lines of their length converge to the point where the light is.
Afterwards bring your eye nearer to this light, using the blade of a knife for a screen so that the light may not hurt your eye, and you will see all the bodies opposite without their shadows, and the shadows which were in the partitions of the walls will be covered as regards the eye by the bodies which are set before them." [p. 991, The Notebooks of Leonardo da Vinci, ed. Edward MacCurdy, George Braziller, N. Y., 1939]

(This is reminiscent of Goethe's remark that the sun never sees a shadow. It is, in fact, why shading specifies the shapes of things; it gives another view, the view of the light source. The boundary between the lighted side and dark side of the object is its outline as seen by the light source.) Visual rays are straight, as are light rays outside the eye; this is why we can aim things by sight - like artillery barrages - with results so happily in accord with our expectations. But, in this experiment the eye can only be brought to coincide approximately with the light source. Do the visual rays in fact emanate from a point (making the visual projection a central projection), and if so, where is this point? First let us define visual rays thus: those lines on which two or more points will be seen as one point. (Only lines defined like this can be used for projection purposes.) We can locate such lines by punching pin-holes in 2 or 3 cards and placing them before a candle. When the three holes are aligned along a visual ray, the eye will see a light, otherwise not. Keeping the eye fixed, one can set up a number of these and ask, do the resulting pencils of light meet at a point? (One can shine a brighter light behind them and catch it on a screen.) This is trickier than it sounds, largely because the eye can see a pencil of light shone obliquely into it; instead of being visible and then invisible, the light changes from a bright circle when they are aligned to an oblique streak and then disappears. In the periphery, it is a little hard to tell the circle from the streak. Someone who is able to do this precisely might be able to do it to better purpose than I was able to.

A slightly more recondite way would be to make use of the fact that a given segment of a sphere uniquely determines the point of sight from which that sphere is seen. (As was often pointed out in the Renaissance, the closer the sphere is the less you see of it.) If there is one unique point of sight, orbiting the sphere in a circle about it, neither more nor less of it should be seen. If there is no such point, the retinal image cannot be a central projection.

Helmholtz and Pirenne assert that lines such as these (on which points at different distances will be seen as one) meet at the center of the entrance pupil of the eye (the natural pupil as it appears through the cornea). But this is not true, as one can determine approximately by lining up a pen at the eye so that one end covers the other. If one does this for the peripheral field (wiggle the pen) and then looks in a mirror or has a friend look, the pen will not be found to be pointing to the center of the seen pupil at all. The fact that Helmholtz and Pirenne make these lines meet at a point and then do not use them to construct the retinal image makes their construction of this image difficult to understand in graphical or projective terms, as these lines have to be used as projection lines. Their construction method is for objects simultaneously in focus - in other words, at the same distance from the eye.

The point we are looking for is a point in space, and we are not now interested if it means anything in the eye. Assuming there is such a point, where is it? Piero della Francesca says forthrightly (p. 262, op cit) "This eye, I said, is round and from the intersection of the two nerves that cross comes the power of vision at the center of the crystal fluid." (This view has some odd consequences in regard to how far one can extend the visual angle.) So, for Piero, there is not even any theoretical inaccuracy in linear perspective, as the vertex of the visual angle is the same for both stationary and moving vision. But, we can see that it is not, thus: align three of the pin-holes in front of a candle flame; by looking aside, one can send them out of alignment and cause the light to go out. Since rotation of the eye causes a translation in space of the point of sight, the point of sight we are looking for is not the center of the eye but somewhat in front of it. One can see this readily by placing the first joint of your thumb right before the eye; by moving your eye, you can cause different parts of the background to become visible or occluded.

I have not found it possible to throw one of these lines off by accommodating the eye to different distances, although it might seem plausible.

Suppose we have such a point to serve as the point of sight (see Figure 15). Now, in order to do the drawing, we are by no means obliged to simply continue these lines onto the retina, as we did in 13. This is not necessary under the given conditions of the problem: the retinal projection must be such that 3 points along a visual ray are projected as one point, but it is not necessary that this retinal point be in fact collinear with the three points. If equal angles at the point of sight are to subtend equal retinal areas, the projection point must be either the center of the spherical surface or a point on the circumference. By translating the point of sight to the center of the sphere and taking the projection from there, we can obtain the desired picture, one which, without correction or transformation, meets the conditions of Euclidean optics. In short, we may say, equal visual angles produce equal apparent sizes for stationary vision, because these angles subtend equal areas on the retina. It is not conceivable that light rays could follow this path, but it would be felicitous if they did end up at the same points on the retina, in other words that light from points A, B, C, D & F would reassemble itself at a, b, c, d, & f. Although the physiologists will not permit us to draw this picture, neither have they come out and said that we see spheres as ellipses, so the thesis can't be regarded as having been disposed of.

This started off as a brief comment on your article, but now I suppose it has eclipsed your article in length. In spite of this excessive length, hopefully you will find it of interest,

Fredericka G. Lohr


Is there such a thing as "curvilinear" perspective?: Text
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