At first blush, it is astonishing to find out that people must learn to see—must learn to interpret the visual sensations that are processed by the brain. We tend to think that what we see is simply an image of things the way they are, rather than being a creation of our nervous systems. And it is astonishing that what we have already learned becomes—itself—a part of the process of “seeing.” Such habits of processing may even cause us to make errors when we are faced with unfamiliar circumstances. One obvious example of this is the case of depth perception, which lies at the heart of the artists’ use of perspective. While it is true that we have binocular (two-eyed) cues to depth (we would call this parallax if we were talking about photography or optical range-finding), most of them are actually monocular cues that involve the use of only one eye. In fact, monocular cues can be manipulated to over-ride binocular cues, something that happens regularly in the road-side attractions, the “mystery houses” that are claimed to “exist in a space-time warp” or other such nonsense.
The most obvious example of a monocular cue is probably overlap. We can tell that one house is closer to us than another because it is “in front of” the other, which simply means that one image overlaps the other. This is a monocular cue, and it isn’t dependent on the size of the objects at all. A small house overlapping a large one is interpreted as being closer, as is a large house overlapping a smaller one. We will come back to this in a moment.
The difference in size of familiar objects is also an important cue. We expect an eighteen-wheeler to be larger than an automobile, and very much larger than a baby. If we were to see a picture taken with the camera placed on the surface of the street, a baby near to it, a car so far away that it appeared smaller than the baby, and a semi very, very far away, we would still know exactly what we were looking at. We say, “of course babies are smaller than semis,” but this is so obvious that it is easy to forget that it is something we learned. (There are other depth cues at work here, as we will see in a moment.)
- Hudson studied these two phenomena in Africa with the indigenous people as his subjects. He used a drawing of a hunting scene. On the left side was a man holding a raised spear and on the right side was an antelope. Both were standing in front of (overlapping) low hills. Between them, on a taller mountain behind (being overlapped by) the low hills, was an elephant, drawn small. The elephant’s elevated position, however, put it directly to the right of the spear point. Looking at this drawing, we would see a man about to spear an antelope in the foreground, and notice that there is an elephant on a mountain in the background. When the African subjects were asked what the man is doing they said that he was about to spear the elephant. This is a cultural difference in perception between the indigenous Africans and Westerners, which means that it is learned.
Now, let’s return to the baby example: one of the depth cues at work here is a texture gradient (also called a density gradient). It is related to the “size difference” mentioned above, but we aren’t directly concerned with familiar objects; instead, we have the expectation that the pattern of a “textured surface” such as the road will be “coarser” when seen close-up and “finer” when seen at a distance. This applies to the rocks on a gravel road, the sand on a beach, the grass on a lawn, etc. What we see in the picture is a textured surface which we know to be receding into the distance—and on this is a baby, a car, etc.
Most Western artists represent these things in their drawings and paintings without even thinking about it. Atmospheric perspective or aerial perspective is another matter, but it still doesn’t present many problems. Distant mountains appear more bluish than those close up. This happens because the blue light coming from the sun is “scattered” in every direction by molecules of gas in the atmosphere (mostly nitrogen and oxygen), and it is then reflected to our eyes by tiny droplets of water and particles of dust. As a result, blue light enters our eyes from every direction we look. (That is, of course, why the sky is blue. When you look up, you are looking into a deep blanket of scattering/reflecting atmosphere.) There is also much more scattering/reflecting atmosphere between us and distant mountains than there is between us and things close-up, so when we look at the mountains, more blue light is entering our eyes. (In fact, the further away a mountain is, the more closely it matches the color of the sky itself. When you see a very distant snow-capped mountain, the snow-cap–which remains white because it is highly reflective and reflects light of all colors well–may actually seem to be floating in the air.) Only experience teaches us that this change in the apparent color of things is a depth cue.
(There is another monocular depth cue, one that is rarely used, and that is focus. When we focus on things close up, the background is blurry, and vice versa. This effect is usually approximated in different ways by artists, who may use stronger colors, heavier lines, or well-defined edges to draw attention to the center of interest. Actual fuzziness in the image is not generally something to strive for.)
Linear perspective is the bugbear of all of the depth cues, but whenever we look around an urban landscape it is there to be seen. It is not “there to be seen” if we live in a forest or on a plain, so we don’t expect to find it developed in non-urban cultures, and this explains the experiments W. Hudson was writing about. As a matter of fact, it took thousands of years before it appeared in urban cultures, and it was not developed everywhere. We can see an astonishing history of “learning to see” played out for us in paintings from the Hellenistic period to the Renaissance. Before looking at that, however, we need to talk about the concept of “projection.” Various projection systems have been used naively for 2500 years to create drawings that give an indication of depth. Perspective, itself, is a type of projection.
An oblique projection of a cube can be made by drawing a two-inch square on a sheet of paper, and then drawing parallel lines that extend up and to-the-right from each of the top two corners and also from the bottom-right corner. (These “projected” lines could also be drawn downward from the bottom two corners and the upper-right corner if we want an upward view of the cube instead of a downward view.) The angle at which we choose to draw these is not very important, but thirty degrees from the horizontal works well. Then, we simply measure one inch along each of these “projections” and locate a point. To finish the drawing, we connect the three points we have made with two straight lines, one paralleling the top of the square, and one paralleling the right side. I suggested marking the “projections” at one inch instead of two inches because of foreshortening, another aspect of perspective. If you hold a pencil pointing away from you at eye-level, you just see only the circle of the end; if you hold it cross-wise to you at arm’s length, you see a rod; and if you tilt it so that it is angled away from you—and ignore what you know to be the case—it looks as if it is shorter than the “rod” you were just looking at. This is foreshortening. (If you have never taken a drafting or technical drawing class, you should try drawing an oblique projection of a cube following the instructions given above. A careful freehand drawing is good enough to make the point.)
A naïve form of oblique projection was shown in Greek vase paintings as early as 350 BCE and it became a common feature of Greek and Roman wall paintings. And there are two simplifications in such a drawing that keep it from being “perspective.” One is the fact that the “square” is drawn as if you were looking at it straight-on, although this doesn’t, by itself, keep it from being part of a perspective drawing. The second is that the “projections” are all parallel. Now, look at the wood piled for a funeral pyre (?) portrayed on an amphora dated 350-300 BCE. It shows the execution of Polyxena, the daughter of the King of Troy, which is copied in a simplified fashion in Fig. 1. (All of the figures are placed at the end of the essay.) The sides recede, and this is an enormous change from earlier representations.
Of course, we don’t always see buildings by looking straight-on at one wall, so it isn’t surprising that another approach was developed, a sort of axonometric projection. You can make an axonometric projection by drawing a parallelogram (a slightly diamond-shaped “square” with vertical sides on a piece of paper, and then projecting parallel lines from it in the same way that you did for the oblique projection. Figure 2 is a simplified drawing of one of Giotto’s St. Francis paintings from Assisi (1290s?). It shows St. Francis being mourned by St. Clare. If you look at all of the receding lines in the top of the building (roof, windows, eaves, etc.), you will see that they are created by axonometric projection, but beginning at about the top of the door-frame Giotto confronts the problem that this poses for him: he can’t go on to show the bottom of the building as if one is looking up at it. Consequently, the bottom portion of the painting is inconsistent, and this can be most easily seen if you compare the “horizontals” at the tops of the windows with the “horizontals” of the architraves above the lowest columns. The greater part of this inconsistency is concealed by placing the bier at the base of the building, which is a time-honored trick in which crowds, bushes or any non-geometrical forms conceal the geometrical ones. (In China, axonometric projection continued to be used instead of perspective, probably because it suited both the character of the art and the philosophical notions surrounding it.)
Religious scenes dominated Western art from late Roman times on, and until 1300 they tended to be decorative and highly stylized. Since people didn’t paint cityscapes for their own sake, perspective doesn’t appear until the early fifteenth century. When it does appear, it is in a form that is in analogous to oblique projection in one simplifying, regard— the subject is seen straight-on. Figure 3 is a simplified drawing of Masaccio’s wall painting, The Holy Trinity, the Virgin, and St. John (c. 1427), one of the first, fully worked-out examples—but a simplified (and inconsistent) example appears in Duccio’s Maestà altarpiece as early as 1311. A painting like this is called “one-point” perspective because the “projection lines” that make it possible to draw the architectural details of the walls and ceiling converge on a single “vanishing point,” which in this case is located at about the foot of the cross.
And now, it was as if artists had been given a new toy, and a virtual orgy of ill-conceived and obtrusively mechanical perspective painting ensued. Nonetheless, many artists continued to show little knowledge of it—or perhaps little interest in it—for a surprisingly long time. (By now it should be clear that convergence on a point is the essence of perspective, and such convergence is another depth cue in the baby example. Because of it, we see a receding road, and this knowledge tells us that the car and the semi are progressively further away.)
If we rotate the architectural shapes so that we are no longer looking straight-on at a flat wall, we have two-point perspective, and this, of course, is an analog of axonometric projection. As its name tells us, we need a second “vanishing point” to make such a drawing . . . or do we? It took very a long time for unambiguous two-point perspective to appear, and artists persisted in locating architectural features so that the viewer looks down a long river or street towards one vanishing point, though the buildings are drawn as if we are seeing the end walls almost straight on. There may be very little indication that the are not just rectangles—and crowds and shrubbery are often used to cover the base of a building, which may leave the viewer in doubt as to whether any convergence towards a second vanishing point actually takes place. It is amazing how long this practice persisted (you see it in Monet’s “Snow Effect at Vétheuil,” “Quai du Louvre,” and many other paintings). It is an amusing “museum game” to look for it: the more we know, the more interesting museums become, and it is fascinating to see how much can be done with such a simple technique.
It is important to recognize that “two-point perspective” is the name for the technique used to represent a single rectangular solid: it does not specify the total number of vanishing points in a more complicated image. If you were to scatter many large cardboard boxes on a stage—heaping them randomly, many tipped part-way over at various angles—and draw them, they would not be arranged in a rectangular grid, and each one would be represented separately and have its own pair of vanishing points. (This used to be a common art-school exercise—long, long ago. It was also a very good one.) “Three-point perspective,” as you might guess, would involve a vanishing point on which the “verticals” converge, but this is hardly ever a problem for practicing artists, even cityscape painters.
We don’t need to know a lot of complicated rules to make perspective drawings of buildings that are actually in front of us: we only need to observe well and draw accurately. (Nonetheless, such rules are necessary to make architectural renderings of buildings that don’t yet exist because we must construct them from scratch and have nothing to copy.) The principal reason for non-artists to know something about perspective is to sharpen their perceptions (just as in the “museum game”), and if we are artists we need to have a basic understanding of perspective that will allow us to solve problems when they do appear. Unfortunately, much of what you read about perspective is misleading if not incorrect. For example, you may be told that the horizon is always at eye-level and that the “vanishing points” are located on the horizon. This is clearly not true if you are looking up at a tall building or down at a brick on the ground. The real problem is that such bogus rules make it difficult to recognize the basic principle of perspective.
Let’s make a little experiment and define some terms before going on. Draw parallel one-inch lines, one inch apart from each other in a column on the page. If we mark the mid-point of the bottom line (where we will imagine your eye to be) and draw pairs of lines from this point to the ends of each of the other lines, we can see that the angle formed by the first pair of lines is fairly large and that the angles rapidly get narrower and narrower. If we extend the column of lines we have made to any great length, the angled lines we have been drawing will be so close together as to almost be a single line, and at a very great distance we could treat them as a single line for practical purposes. We will call this the sight-line, and we will call the point to which it extends the sight-point. This is actually never reached, since it is infinitely far away (and by definition parallel lines never intersect), but just as the two lines become inseparable for practical purposes, we can say that their end-point is located as far away as you can see. We will call the first point, which is the position of the observing eye (which is what makes these a “sight”-line and a “sight”-point) by its traditional name, the station-point. Now we can state a rule:
Lines parallel to your sight-line (the direction you are looking) always appear to converge on your sight point (the “infinitely” distant point you are looking at), no matter where it is located.
Let’s take the most clichéd example: you are standing on a straight railroad track in the desert that extends as far as you can see (this is like a gigantic “column” of lines such as the one we made in the preceding paragraph.) Looking ahead, you see that the rails converge on a point on the horizon. It is located on the horizon only because our bodies are constructed in such a way as to allow us to walk on the surface of the earth, and our eyes are placed in our heads in such a way as to give us the most useful view as we do so: we look outwards, parallel to the surface we walk on. The railroad tracks parallel our sight-line so they converge on our sight point, and that would be true if we were three feet tall or sixteen feet tall. (I feel almost stupid in saying this, but that is all there is to it.) If you lie on your back in the middle of the street looking straight-up, the corner-edges of the sky-scrapers (which parallel your sight-line) converge on your sight-point in the sky. If you are looking down at the same intersection from a helicopter, the corner-edges of the sky-scrapers will converge on your sight point at the middle of the intersection (but “infinitely” deep in the ground). If you are on a long, steep slope looking down it parallel to a road and a fence, the edges of the road and the rails of the fence will converge on your sight-point (which will be buried infinitely deep in the valley floor). Now, we have a corollary:
You can imaginatively put yourself anywhere in any scene—real or painted—and from your new “imagined” station-point you can locate new sight-points with their associated families of parallel lines.
Let’s go back to example of the road sloping down in front of us. If, at the bottom of the slope, it begins to climb up the hill on the other side of the valley, it is clear that we will need to locate a second sight-point in order to draw it as it appears from our present position. In our imaginations we can transport ourselves down to the valley floor and look up parallel to the climbing road. We will have a family of lines parallel to our sight-line, and they will again mark out the edges of the road and rails of the fence. Unless the road is “infinitely” long, it will not continue all the way to our sight-point (which may well be in the sky). Instead, it will go over the top of the hill, or around a turn. Now, and this important, if we return to the place we were standing when we looked down the slope, we can ask, “where is the new sight-point located?” If the road continues straight on in front of us, climbing the distant hill, it is clear that it must be directly above the first sight-point which we located at the bottom of the slope.
That’s it. You can now locate the perspective lines in any scene. But to help you see how this works, let’s call in our helper, a flying eyeball named Igor (pronounced “eye-gore”—with apologies to Marty Feldman and Mel Brooks). Let’s suppose that you are on a low sand dune looking down at a curved bay. There is a wharf extending out into the water at a moderate angle with respect to you. There is a bait-shack sitting squarely on the wharf with the end wall facing you (at a moderate angle). It has a door in the end and a window on the side. On the beach there is shed that is sometimes used by people who rent equipment. It is beside the access road, and sits at an angle to the wharf. It has a window and a door which open out onto the beach. A number of people are walking and playing on the beach, and some of them have blankets and beach umbrellas.
We send in Igor. He flies down and sights along the railing at the edge of the wharf. His sight-line locates a sight-point on the horizon, and makes a permanent luminous trace that you can see from your vantage point on the dunes. Immediately the whole family of parallel lines extending to this sight-point becomes illuminated as well. These trace the sides of the wharf, the eaves of the bait shack, the crest-line of the roof, the top and bottom of the window in the side of the shack, and the bottom edge of the shack as it sits on the boards of the wharf. Then Igor flies to the end of the wharf and sights along the railing that blocks the end. The family of lines parallel to his sight-line becomes illuminated, and they trace out the boards across the wharf, the bottom edge of the end of the shack, and the top and bottom of the door in the shack. All converge on his sight-point. Then, he flies to the side of the equipment-rental shed and sights along it. Immediately, the family of lines paralleling his sight-line becomes illuminated, reaching out to the horizon, converging on his sight-point. These trace out the eaves of the shed, the crest-line of the roof, the bottom edge of the building as it touches the parking lot, and the tops and bottoms of the window and door in its side. Then . . . Igor continues his work leaving a pattern of lines that we can see emerging as we sit on the dunes, drinking a soda.
Why have all of the sight-points (vanishing points) been located on the horizon? The answer is easy: Igor has been looking out parallel to the surface of the Earth (which is flat from our pygmy point of view). Now, let’s suppose that we ask him to sight up the nearest edge of the sloping shed roof. We would see a luminous line reach up to his sight-point in the sky, but how far out does this line reach? Where do we locate the point itself? This is like the example of the road going down the slope and up the hill on the other side, but to make this a little clearer let’s ask Igor to sight horizontally along the corners of the eaves that project out on both sides of the building. We have already located his sight-point, and it is on the horizon. Now we’ll have him slowly look up. His sight-point traces a line projected straight up from the horizon. Finally, he will be sighting directly along the edge of the sloping roof, and his sight point is at the end of the line he has been tracing, directly above the other sight-point which is on the horizon. Immediately the family of lines parallel to his sight-line is illuminated, converging on his sight-point, and that takes care of the other end of the roof as well. (Of course, Igor knew that all along, and his sight-point in the sky would naturally have been located just above the one on the horizon.) The same thing is true of the roof of the bait-shack on the wharf.
Whenever you look at a painting in perspective, you can imagine Igor lighting up the families of parallel lines to help you see just what the artist is doing. Of course, if you are the artist, you begin by making certain choices. For example you draw the horizon line (if it is within your picture plane); choose where you want to put a building; and mark in the lines that represent the nearest edges of its foundation—but having done that, your knowledge of how Igor works produces the necessary sight-points (vanishing points).
The development of various systems of projection we have looked at really has been a matter of learning to see what is before us—as opposed to seeing what we “know” to be true. It is rather like the fable of the elephant and the blind men. The one who felt the tail thought that an elephant was very like a rope. The one who felt a leg thought that an elephant was very like a tree. Just as children may draw both ends of a house because they know that they are both there, we all have a natural inclination to draw the tops and bottoms of walls parallel because we “know” that they “really” are parallel. For this reason, oblique and axonometric projections developed before perspective. In fact, once a certain way of drawing becomes the accepted tradition in a culture, other ways of drawing probably look strange. I’m sure that a perspective rendering of a house would look wrong to those who have always drawn houses in oblique projection and have never seen perspective: “That isn’t right at all—look, the walls aren’t square. How could the roof fit on!”
Notes If you are actually going to make perspective drawings you need to take note of a few common-sense reminders—just to help you see some things that aren’t obvious at first glance. Reminder 1: Drawing is a way of representing on a flat surface something that is really only part of an image, an image that could be drawn on the inside of a large transparent globe within which you are standing. Since this partial image must be flattened out to put it on the paper or canvas there will be distortion as we move out to the margins (think of a “fish-eye” view), but this is only a problem if we make the partial image too large. If you look straight at the subject of your drawing and imagine a cone constructed around your sight-line, that cone should never encompass more than about sixty degrees. Nothing outside of the cone should be drawn unless you are willing to put up with distortion. Corot said that you should begin the foreground of a landscape at least fifty feet away, and that is just another way of saying the same thing. If you are looking outward while painting a landscape, you cannot see the things at your feet. The problem is much more serious when buildings are involved. Don’t turn your head and include what you see there. Don’t tilt your head up and include what you see there. If you want to include a building, or a tree, or a rock that is outside of your cone-of-vision, drag it inside and correct the perspective. The problem of staying within your cone-of-vision is really why you need a “viewfinder,” a piece of cardboard with a window the shape of your canvas or drawing pad cut in it. You hold it out in front of you so that you look through the opening. You don’t really need one to pick a good view of a scene, but you do need one to remind yourself of the limits you must respect, at least until it becomes second nature. (Figure out how big to make it and how far away you need to hold it, and then punch a hole in one end and wear it around your neck on a loop of string. Just push it out until the string becomes taut to make it work for you. Some books will ignorantly tell you that perspective distorts a scene because in real life we look to the right and left, up and down to take it in. Of course we do. But if that is the criterion for what “real” vision is, all drawings distort the scene because they all fix it in place. (You might try looking out a window with one eye closed and the other kept in a set position—you will probably need some mechanical help for this, like resting your chin on the back of a chair. Then draw the image you see on the window-glass with a grease-pencil. It will be in perspective!) Other books will tell you that the camera distorts a scene, but if you are using the normal lens (which should probably be more like 50mm for a 35 mm camera, not 35mm), it shows you the same scene you would draw in perspective. Wide-angle lenses give you a “fish-eye” view, just like the one you would make if you were to make the drawing on the inside wall of a sphere. Telephoto lenses are said to “collapse” perspective (we are looking at such a small part of the scene before us that there is little to no parallax.) Reminder 2: Since the subject of our painting will usually be a part of the view we see as we look outwards, we draw all verticals as verticals. This holds true for ordinary landscapes and cityscapes, but it is not true if you are actually looking up or down. Then, you will need a third sight-point for the edges of the buildings, the sides of the towers, the sides of the trunks of gigantic trees, etc. to converge on. And if you are really looking up, you won’t have a horizon at all. Reminder 3: When you draw a picture looking slightly up or slightly down, the horizon line must be placed differently on the page. To the degree that you are looking up, the horizon will be low on your paper, and to the degree that you are looking down, the horizon will be high on the paper. Thus, If you are standing on a small bluff painting a beach scene, the horizon will be high. And if you are painting fields with a farm building from a distance and featuring the sky, you will have a low horizon. Since the degree you are looking up (or down in the case of the beach scene) is slight—and constant—you can place the sight-points on the horizon and ignore any slight convergence of verticals (and this is particularly true if the center of interest—the farm house in the second example—is far away). Reminder 4: The corner of a building formed by the front and side-walls as you see them is always drawn as an angle larger than ninety degrees, and this is true both for its appearance in the sky and its “footprint” on the sidewalk. Think about it for a moment. For such an angle to appear to be as narrow as ninety degrees you would have to have your eye merged with the building corner and be looking either straight up or straight down. You will sometimes see this principle violated by people who should know better, and you will even see incorrect examples in books on drawing. Reminder 5: The “mid-point” of a square or rectangle is always where the diagonals cross. Of course this is true. Imagine rotating a rectangle (one which has its diagonals marked) in space. The point at which the diagonals intersect doesn't move around on the square, we just see it differently. You can put this knowledge to good use. Suppose you have drawn the rails of a railroad track and have put in the nearest two ties. How do you space the others? This is actually very simple. First draw diagonals in the rectangle formed by the "ends" of the first two ties and the rails. (I am calling the points at which the ties intersect the rails their "ends.") Then project a line from the point at which they intersect (the mid-point) to the sight point upon which the rails converge. This line will cut across all of the midpoints, including that of the second tie. Then, draw a diagonal from one "end" of the first tie through this midpoint. (You are constructing a rectangle made up of two smaller rectangles.) This intersects the rail at one "end" of the third tie. Now, using one end of each successive tie as the end of a diagonal you can locate the other ties as far as you want to go. If you are looking straight down the tracks, you can draw the ties in as horizontal lines. If, as is more likely, the railroad tracks are running at an angle to you, you will want to remember to draw them as projections from the sight-point on which that family of parallel lines converges. The same process will space the posts of a fence or a line of telephone poles. (If this doesn’t seem clear to you on reading it, draw it to see what it means.) Reminder 6: A circle becomes an ellipse in perspective. To verify this for yourself, hold a plate before you and slowly tip it until it is edge-on. You can see that it becomes a narrower and narrower ellipse as you tip it. Now place the plate on a sheet of paper and draw a circle around it, using it as a template. Then, construct a square around the circle you have drawn and divide the square into four quarters. Tip this slowly and see how the square and the circle look as they are tipped. This shows you how a circle enclosed in a square looks in perspective. Finally, draw a perspective square and locate its mid-point with crossed diagonals (as in Reminder 5). Draw lines to the sight-points through this mid-point to divide the square into quarters. Now you can easily draw the needed ellipse in the square. (If you must be exact, you can use an ellipse template or one of the geometrical procedures for drawing ellipses, but that shouldn’t be necessary unless you are making an architectural or engineering rendering.) Reminder 7: People are like posts. Suppose you have people scattered on a beach, all more or less the same height. You can imagine every pair of them to be posts of a fence. Draw one where you want it (her/him) to be and then mark the spot where you want to put the base (feet) of the next one. Draw a sight-line connecting the feet, and that will give you the sight-point that you need to draw the sight-line connecting their tops (heads). Do the same thing for all of the others. If there is a child that comes up only waist-high, use a waist instead of the top of a head to draw the needed sight line. Perspective shouldn’t be thought of as being the inevitably “right” way to represent any image. To take such a viewpoint is ignore the role that color, mass, and other elements of form can play in organizing a painting. Cezanne painted many great still-lifes, works that are completely satisfying although they ignore linear perspective to a surprising degree. Nonetheless you should remember that perspective does faithfully represent the image before you, and you should be knowledgeable and purposeful whenever you choose to disregard it. Illusion in Nature and Art. R.L. Gregory and E.H. Gombrich, eds. New York: Charles Scribner’s Sons, 1973. This is a fascinating book, though not very systematic. It looks at almost every kind of illusion from animal mimicry to optical illusions. The material on Hudson’s study comes from Jan B. Deregowski’s chapter, “Illusion and Culture.