page 1: perspective in the world
This page tries to clarify these paradoxes in terms of perspective geometry and the structure of vision. My aim is to present the assumptions that allow linear perspective to simplify human seeing and provide a powerful method for artistic design. It is possible to carry the geometric methods of linear perspective too far, especially in hackneyed architectural settings. To get a feel for what perspective is really about, consider that it is visible everywhere and in everything — even when architectural edges and corners are nowhere to be seen.
Anything that helps us see the relative distance of objects in space is called a distance cue. Fundamentally, all distance cues are made possible by the geometrical regularity of three dimensional space, and it is this regularity that linear perspective attempts to describe. First, vision takes advantage of the fixed characteristics of our two eyes to make sense of what we see. The most powerful distance cue, binocular parallax, is the disparity between the images created by the two eyes that arises because they are located about 5cm to 7cm apart. This causes near objects to shift back and forth against a distant background as we close first one eye and then the other. The mind uses this parallax to infer the distance of objects in the field of view: the larger the left to right shift, the closer the object. We also use motion parallax, which occurs when we move our head, stoop or turn, walk or run through the environment. Parallax is a very powerful and accurate distance cue, and it is effective across an enormous range of distances — binocular parallax from the tip of our nose out to about 20 meters, and motion parallax (depending on the speed of movement) out to several kilometers. Parallax cues depend so heavily on the fixed attributes of space and the location of our eyes that it takes infants only about four months to learn how to use parallax to guide reaching and grasping. Other cues related to eye position, such as lens focusing (accommodation) and crossing the eyes to see close objects (convergence), are comparatively weak — they are only useful within a few feet. However, in the two dimensional, fixed surface of a painting, all the cues from parallax, convergence and accommodation disappear. So the artist must rely on other distance cues to create the illusion of three dimensional space. Some cues appear in the optical properties of monocular (single eye) retinal images. In three dimensional space, objects close to us appear larger than those far away, so retinal image size is an important distance cue, especially for objects we recognize. Objects at our feet or just overhead appear much lower or higher in relation to the horizon than objects far away, so the vertical position of objects in our visual field — compared to each other or to the horizon — also serves as a distance cue in natural environments. All these distance cues seem related to our view of detached objects. However, equally powerful depth cues arise in the visual appearance of surfaces, especially the textures and colors of the natural world. distance cues in changing textures In the foreground, within our physical range of motion, we usually distinguish separate objects, in part by using the occlusion of one object outline by another. The simple rule is, whatever covers is closer, and this rule applies across any distance (even when the sun sets behind a far mountain). distance cues in overlapping forms However, the main distance cue in the Adams photo is the change in visual textures across space. The foreground rocks appear large and extremely rough; with distance they grow smoother, the spacing between them becomes smaller, and the rocky surface appears flatter, less irregular. Beyond the rocks, the mountains and clouds have irregular outlines but appear smoother than the rocky plain. And beyond everything is the sky — the only perfectly textureless "surface" in nature. If the object or surface is far enough away, it is "behind" a considerable distance of atmosphere, which can obscure the object with suspended particles of dust, smoke or molecules of water vapor. The cumulative effect of these obscuring particles creates aerial perspective in large objects visible from a great distance, especially mountains, buildings and desert or ocean horizons. Depending on the time of day and strength of light, aerial perspective can make distant objects appear less distinct, less saturated and darker or lighter in value. Smoke or dust shifts the hue of distant objects warmer (toward red, yellow or yellowish white), while water vapor shifts the hue toward blue. We have to use the recognizable continuity of an object's outline, or its "completeness of form," to see occlusion, which is more difficult if objects are far away or very small, dimly illuminated, or unfamiliar to us. Look again at the Adams picture, and you'll see that one rock clearly covers another at the bottom of the image, but in the middle distance these overlaps become harder to see. Instead, everything merges into the average spacing or spatial frequency of the rocks — that is, the rocks do not separate themselves from the texture as distinct forms. Wherever objects become too small or complex to show occlusion clearly, texture takes over. This transition from form to texture means that visual experience is a combination of objects filled in by visual textures. Increasing distance in space transforms the appearance of objects into structurally or visually related textures. And at extreme distances, texture itself dissolves into pure color. So we have the following sequence that applies to large vs. small or near vs. far visual elements: pattern —> texture —> color perspective transforms pattern into texture and color Occlusion works because we can compare the outlines we see with our idea of the objects we look at: anything partly covered is a "broken" or "altered" form of itself. So our knowledge and expectations of the world are essential to create effective distance cues. However, the boundary between what we "see" with our eyes and what we "know" with our memory and mind is not at all clear. In fact, we can create the illusion of a recognizable form entirely through the visual completion induced by forms around it. Finally, these transitions from occluding objects to patterns to textures to colors as distance increases do not happen in the same way for all objects — unlike the effect of aerial perspective or fog, which causes all forms to fade equally from view. Increasing distance creates characteristic visual transitions in different objects, especially in natural forms where there is a distinctive structure at different scales of view. Trees are the classical example, much studied by 19th century artists, because different species of trees express a different branching pattern that is recognizable from twigs up to large branches; the tree's branching pattern, in turn, determines the tree's overall form and the clumped appearance of the trees in copses or forests. the unique sequence of patterns created by perspective changes in oak trees Linear perspective is space drawn as the geometrical idea of itself. But we do not see the idea of space: we see a world of light, colors, textures, objects and opportunities for action. As we explore the artistic uses of perspective, we will repeatedly grapple with the fact that our visual experience of the world is much richer and more complex than our idea of the geometrical space in which it appears.



First, the foundation of linear perspective, is that all rays of light travel in a straight line (1 in the figure at right). These are sometimes called visual rays if they represent light to the eye and projection rays if they represent lines of sight in a perspective construction.
When light is reflected by the naturally dull and rough surfaces of the physical world, it is reflected or scattered in all directions. This means light is always abundantly radiating in all directions on all sides: this is how we are able to see the world in every direction, from any position. How can this tangle of rays create a three dimensional image? The second fact is that light creates vision at a single point (2 at right). We only perceive light that reaches this viewpoint. The location of the seeing eye is the viewpoint or center of projection. The only visual rays that matter to our view of the world are those that converge on that point; all other light is effectively invisible to the viewer. This untangles the scattering weave. The eye is really a small sphere, so we have to simplify the facts of sight somewhat. Confusingly, this depends on what we mean by looking the world: • Light enters the eye through the cornea and lens, which have width and depth and therefore do not define a point: but the light bending effects of a lens can be summarized as a single point (called the nodal point). If we use only one eye looking only in one direction, this nodal point is located slightly behind the center of the lens (because light has already been refracted by the cornea). • If we use only one eye but are allowed to look in different directions, the spherical shape of the eye as it moves becomes more important, so the viewpoint shifts to the center of the eye. • If we use both eyes, then the "viewpoint" is really the average of two retinal images, so it is arbitrarily located near the center of the head. We can't see light through the back of our head, and the lens and cornea of our eyes focus most clearly along the eye's optical axis, which is roughly in the center of our visual field. This is the third perspective fact: we see clearly light coming from one general direction (3 at right). This specific direction of vision is our direction of view (also called the central ray or principal visual ray). The limits to our vision around the direction of view define a visual cone or visual pyramid, which is centered on the direction of view with the viewpoint as its point or apex. Any visual ray intersecting the viewpoint from within this visual cone defines a line of sight with one end at the viewpoint and the other end on whatever point we see in our field of view. The human visual field actually has a very complex structure — crisp central vision and fuzzy peripheral vision — but linear perspective assumes everything inside the visual cone is visible with equal precision. This is a specific example of how linear perspective does not represent what we actually see with our eyes, but rather what we know about optics and the geometry of the physical world. Finally, because all visible light rays pass through the viewpoint, all light rays are viewed head on, not from one side. This means all visual rays (lines of sight) appear to the eye as points (4 at right), in the same way that a small flashlight appears as a ray if seen from one side, but as a point of light if turned directly into the eye. The visual field can be thought of as a vast map of points, like a sky rich with stars, each point or star created by the visual ray arriving at the viewpoint from a specific location in space. This description of light rays as straight lines, arriving from objects in space to a viewpoint with a specific direction of view, allows us to use a geometrical method for describing the visible world on a two dimensional surface from a single point in space. This is called a central projection. Geometry in turn gives us the procedures necessary to construct these central projections using the simplest tools: a pencil, a straight edge and a compass.

the four perspective facts: 
My explanation proceeds step by step, so that you can see the logic behind the procedure. The actual evolution of linear perspective was more haphazard; the section on early Renaissance methods describes how the first perspective constructions were made.
The Visual Cone. We start with human vision as described in the four perspective facts: the visual cone (or visual pyramid) extending and expanding in front of the viewer along a specific direction of view. These create what we would call the visual field in other contexts. All visible visual rays (those that enter the eye) are gathered in a single place, a single view of the world, experienced uniquely by a single viewer: no one else can have exactly the same view at the same time. the visual cone The visual cone comprises a flood of visual rays from all parts of the visible world. Because these all meet at the same point, any two visual rays define a visual angle between them, which is the apparent distance between two points in the visual field. The widest visual angle that can be contained within the visual cone was determined by medieval optics to be 90°. Later perspective practice adopted this 90° limit as convenient, for reasons explained in the next section. We can determine the spatial width of the visual cone at any distance by taking a circular section through the cone, perpendicular to the direction of view. This defines a 90° circle of view; the radius of this circle is equal to its distance from the viewpoint. As mentioned before, the visual cone simplifies the visual field in several important ways. In linear perspective, everything inside the 90° circle of view can be described with equal and unambiguous clarity. In visual experience, this clarity is only experienced in foveal vision, at the center of view; motion and binocular parallax are omitted. Linear perspective is actually a cognitive, not perceptual, rendering of the visual world: perspective lets us draw what we know, not what we actually see. The Ground Plane. This visual cone is not floating in space, but originates in a human being, and the characteristics of the human body and posture put restrictions on the physical surroundings for vision. the ground plane The ground plane can be very far below the viewer, for example if the viewer is standing on a cliff or mountain or at the top of a tower or bridge. In all cases, it represents the largest or dominant level surface below the viewer. As it extends outward in all directions, the ground plane cuts the visual field in half, blocking the range of vision downward. This naturally orients the direction of view straight ahead, so that the direction of view is parallel to the ground plane. Because the surface is flat and level, we can also define the viewer's orientation in space: standing upright, perpendicular to the ground plane, balanced against the downward pull of gravity. The Metric Grid. The ground plane is also our reference for location in space, and therefore the distance from the viewer to any objects in space. To specify these concepts of location and distance, we can define a measurement system on the ground plane. covering the ground plane with a metric grid This metric grid is one of the foundation elements of perspective. Early Renaissance artists actually painted in this measurement grid as a pavement of square tiles, often in a strongly contrasted checkerboard pattern. Limiting Visual Rays. The visual cone is filled with an infinite number of visual rays, arriving to the viewer from every visible object and surface in space. To study how perspective works, we can limit our attention only to visual rays from corners in the metric grid, and ignore the rest. We assume (correctly) that any methods we develop from these few visual rays will apply to any other visual rays in the visual cone. visual rays from rows of points to the viewpoint The visual rays from these points allow us to answer two fundamental questions about recession in space. What happens visually to a series of objects standing in a straight line parallel to the direction of view (as defined by the line dx and the matching line on the opposite side)? And what happens to objects arranged in equally spaced rows perpendicular to the direction of view (represented by the transverse lines ending at each labeled point)? The answers to these questions are presented in the next section. Perspective Geometry. By limiting the perspective view to a handful of visual rays from the corners of the metric grid, we have started to simplify or abstract the viewing situation, reducing it to its geometric essentials. Let's complete that process. perspective geometry The viewpoint is at the tip or apex of the visual cone, and the origin of the direction of view. The direction of view is already parallel to the ground plane. So we can define a median line extending from the station point and parallel to the direction of view, which divides the ground plane into symmetrical left and right halves. The distance between the viewpoint and station point is the viewing height, which depends on the viewer's size and his location in relation to the ground plane (sitting, standing on the ground, or standing at the top of a tower). We can also specify a viewing distance between the viewpoint (or station point) and any image or object in front of it. All the right angles within the metric grid and between the grid, median line and viewpoint put distance measurements and the visual angles of visual rays within reach of basic trigonometry. (Linear perspective works just as well if there are no right angles in the setup, but it is geometrically somewhat harder to explain and was not theoretically resolved until the early 18th century.) At this point linear perspective becomes a precise measurement system. In fact, as linear perspective developed during the Renaissance it was closely associated with methods of surveying, mapmaking, navigation and astronomical observation; procedures for measuring representational drawings and for measuring the physical world were often explained in the same book. Making the viewing situation geometrically abstract imparts a similar abstraction to the identity of the viewer of the image. Paintings that create an identity or presence for the viewer as an individual recognized by persons in the painting, as in Velázquez's Las Meninas, are rare in the perspective tradition, especially in academic or history paintings. More often the perspective viewpoint implies a timeless or universal witness, an abstract vantage that can be filled equally well by any adult viewer. The Image Plane. Next we insert the image plane through the visual cone. This represents the distance and angle to view of the painting surface or support — the paper, canvas, board, wall or tapestry on which the perspective drawing will be made — so the image plane is perfectly flat. As before, to keep the geometry simple, and to mimic the vertical viewing position of a painting or fresco, we orient the image plane so that it is perpendicular both to the direction of view and to the ground plane. the image plane in the basic perspective setup The image plane is conventionally divided by two extensions of the human posture toward the world. The horizon line corresponds to the visual limit of the ground plane, and divides the image plane horizontally; the median line corresponds to the median line on the ground and is perpendicular to the horizon line. The two lines meet at the principal point, which locates the direction of view as it passes through the image plane perpendicular to it. The principal point defines the two most important measurements on the image plane. The distance between the principal point and the viewpoint is the viewing distance to the image plane; the distance between the principal point and the ground line is the viewing height. (These distances are the same as the distance from the station point to the ground line, and from the station point to the viewpoint.) Two important details: the horizon line is not necessarily defined by the visible horizon on the surface of the earth. As we'll see later, it actually depends on the orientation of the viewer's head. (There would be a horizon line in outer space.) Similarly, the image plane does not have to be perpendicular to the ground plane, or even to the direction of view, but defining it that way makes it easier to work out a perspective problem. The Perspective Image. The image plane is commonly described as a window opening onto the world. This means all visual rays pass through the image plane on their way to the viewpoint. The final step is to identify the intersection of each visual ray with the visual plane. The intersection of a line and plane is a point, so all visual rays become points on the image plane. images of points on the image plane In the figure, point a' on the image plane is the intersection or image of the point a on the ground plane; point b' is the image of point b, c' is the image of c ... and so on for all the points on those two parallel lines of points we decided to study. We have defined a way to map three dimensional space onto a two dimensional surface, literally by plotting visual rays point by point. Which means we can more or less paint anything we see — that is, anything we can outline as points on the image plane. In practice, perspective constructions are made by plotting only significant points, especially vanishing points and corners or edges that can be connected by straight lines or freehand curved lines. Perspective builds in steps: from corner points, to straight lines connecting the points, to geometrical forms composed of straight lines, and finally to patterns, curves or forms that are inserted freehand, using points or lines as drawing guides. The guiding rule: it is most efficient and convenient to build the image from the fewest possible points and lines. Viewing the Image. Once the artist has used these principles to transfer the three dimensional world onto a two dimensional canvas, and from the significant points and lines developed a completed perspective painting, a second situation arises that must be governed by perspective principles: the cultural encounter between the perspective image and a human viewer. If we whittle this encounter down to its essentials, we are left with the vertical (wall hanging) orientation of a faceless museum or gallery painting, and the ghostly center of projection, perpendicular to a line from the center of the painting, that is the viewpoint implied by the perspective facts in the image. the visitor and the artwork Because the viewers of paintings and frescos rarely choose (or are able) to stand at exactly the viewpoint implied by a painting's geometry, perspective typically works as a system of symbolic representation rather than visual illusion. That is, if we happen to see a Malevich square from one side (foreshortened), we do not mistake it for a painting of a rectangle; we unconsciously compensate for our skewed view. Again, linear perspective is about what we intellectually know, not what we optically see. Even when a painting is viewed from exactly the center of projection, a perfect perspective drawing creates apparent perspective distortions that are intrusive and often objectionable in a painting viewed from many angles. Artists spent many centuries attempting to understand, minimize, or use for expressive purposes. As a result, the eventual cultural encounter can intrude on perspective problems even before the painting is started.
 
the perspective framework in elevation (above) and plan (right) If we also specify the locations of the viewpoint and the image plane in the elevation and plan, then we can draw visual rays from objects in the elevation and plan to the viewpoint, measure where they intersect the image plane, then transfer these horizontal and vertical measurements to the painting format. The figures show this done twice: horizontal green lines for measurements done on the elevation (which gives the distance of the points above or below the horizon line), and vertical green lines for measurements from the plan (which gives the distance of the points to the left or right of the median line).
If we make and measure these schematic drawings carefully, then connect the dots to construct the perspective image of our metric grid on the image plane, we discover that the receding rows of points appear as converging lines of image points, as shown below.
a perspective image of the metric grid on the ground plane
First, a quick orientation to the key components. The image plane is shown as a square that roughly fills the visual field; it is bordered along the bottom edge by the ground line, which is the intersection of the image plane with the ground plane (floor, pavement). The image plane also intersects the visual cone to create a 90° circle of view (or any other size circle of view we want to define), centered on the principal point or intersection of the direction of view with the image plane. The direction of view is perpendicular to the image plane; the principal point is at the same height as the viewpoint. The distance on the image plane from the principal point to the ground line is the viewing height or the distance of the viewpoint above the ground plane. (In this and previous figures, because the viewing distance from the viewpoint to the image plane is slightly greater than the viewing height, the 90° circle of view extends below the ground line.) The horizon line and median line intersect the principal point at right angles to each other. By convention the horizon line is parallel with the bottom of the image rectangle (the ground line), and the median line is perpendicular to the ground line and parallel with the sides of a rectangular image format. However, both also imply our posture in relation to the image — standing with squared shoulders and erect spine. Now let's examine the perspective image of the metric grid. First of all, we find that it still consists of straight lines (in red). Connecting pairs of metric points parallel to the direction of view has created the image orthogonals (the mathematical term for "perpendicular," which reminds us that they are perpendicular to the image plane). Because we know the lines of the metric grid on the ground plane are in fact parallel and equally spaced, we conclude that the orthogonals define a constant interval of width. We immediately discover that lines parallel to the direction of view appear to converge at the principal point, which is therefore their vanishing point (vp) — a term coined by Brook Taylor in 1715. Because the vanishing point is in the center of the image, this kind of perspective is called central perspective, discussed on the next page. Connecting pairs of metric points parallel to the image plane has created the image transversals, which are parallel to the ground line and horizon line. Because we know these are also equally spaced lines in the metric grid, we conclude that they define equal intervals of perspective distance from the ground line toward the horizon. We can hardly appreciate today the extraordinary sense of discovery that early Renaissance artists must have experienced as these simple drawings first took shape under their hands. We sense their delight in the loving care they invested in solving perspective problems, and their awe in the austere clarity with which the solutions were transformed into paintings. Basic principles were recognized early. Parallel lines actually converge to a point, the equally spaced transversals crowd together with distance, and the spacing between transversals narrows more quickly with distance than the spacing between orthogonals (the squares of the metric grid become narrow rectangles). That is, the artists had unlocked the fundamental regularity of foreshortening, which compresses distances or dimensions more strongly as they become more parallel to the direction of view and which was always the most awkward aspect of Roman and medieval painting. This narrowing transversal spacing also explained the perspective gradient of ground plane textures, such as urban pavements or ocean waves, as they recede in space. Indeed, the early woodcuts of artists studying perspective show them in situations where foreshortening is obviously the major focus of the drawing. Several decades later, these early artists also realized that that the diagonals of the squares created by the orthogonals and transversals must also be parallel lines (like the parallel diagonals of a chessboard) and therefore must also converge to two points on the horizon line on either side the principal point: the diagonal vanishing points (or dvp's). The dvp's were first described by Jean Pélerin in 1505, who showed how they could be used to find the depth (transversals) of a square grid from the widths (orthogonals) measured along the ground line. (The dvp's are sometimes called distance points, for two reasons: (1) the dvp's are used to measure recession or distance from the viewpoint in central perspective, and (2) the distance from a dvp to the principal point is exactly equal to the viewing distance from the viewpoint to the principal point. This means they can be used to reconstruct the center of projection implicit in a perspective painting.) The Circle of View Framework. Because the dvp's powerfully tie together the principal point, the viewpoint, the viewing distance to the image plane and perspective recession within the image plane, we can greatly simplify the perspective framework if we standardize everything around the viewing distance. This brings us to the modern version of the basic perspective setup, the circle of view framework. This is a very flexible two dimensional system that lets us control the viewing geometry of any perspective problem or any perspective point of view. the circle of view framework The specific measurements depend on the stature or vantage of the viewer. For an average size adult whose viewing height is about 1.6 meters (63 inches), the circle of view at the image plane will be about 3.2 meters (10.5 feet) wide. (The viewing height is always measured from the viewer's eye level.) The 90° circle of view is essential to work out the perspective geometry, but to minimize perspective distortions the actual drawing fits into a much smaller circle of view, such as the 60° or 40° circles shown in the diagram. For example, the watercolor full sheet (22"x30") would appear as shown in the diagram — nicely contained within a 30° circle of view. The massive emperor sheet (40"x60") is slightly larger than a 50° circle of view, unless we assume a viewing distance farther than 63 inces away. The 90° circle of view can also be applied to different viewing distances. Most book illustrations are viewed from a reading distance of about 20"; many art books are printed in an 8.5"x11" page format. Full page reproductions would define a 24° circle of view across the page width. Once you firmly understand that the 90° circle of view framework explicitly links together the principal point, viewing distance, viewing height, ground line and all diagonal vanishing points, you will be able to apply this simple system to solve any perspective problem.
The terms viewpoint, line of sight, image plane and principal point have been defined above. 1. A line of sight appears as a point on the image plane. If we look straight down any line or visual ray, we only see a point in the image. Thus, the direction of view only appears as the principal point, and the origin of any visual ray appears as a point. 2. Any straight line that is not a line of sight projects a straight line on the image plane. A stretched piece of string always looks straight in a perspective drawing, no matter which way the string is viewed. 3. Any two points on a straight line, projected onto the image plane, define that line on the image plane. We can create the image of a line, if we can project any two points from the line onto the image plane. Any other points on the actual line in space will also lie on the image of the line on the image plane. 4. The image of a straight line ends in two points: its intersection with the image plane and its vanishing point. The only exceptions to this rule are lines parallel to the image plane (they never intersect the image plane, and they do not converge to a vanishing point), and lines of sight, for which the intersection and vanishing point are the same. This rule, which the English perspective theorist Brook Taylor called "the principal foundation of all the practice of perspective," has important consequences that we will explore in the next page. 5. The vanishing point for any line is the parallel line of sight. If our direction of view is exactly parallel to any line, then we are looking directly at the vanishing point for that line. (Given a fixed viewpoint, there is only one line of sight parallel to any line in space.) In our perspective setup, the elevation and plan show that the direction of view is parallel to the gridline of points abc, so those two lines never actually meet in the real world. Even so, the visual angle between the direction of view and any point on the gridline becomes smaller as the point moves farther away from the viewer — the visual ray from point d is much closer to the direction of view than the ray from point a. As points get farther and farther away, the visual difference between the direction of view and a very distant point becomes so small that it visually disappears, and the gridline of points merges with the principal point, as we see in the two red lines connecting the points on the image plane. (This fundamental principle of recession was first proved by Guidobaldo del Monte, in 1600.)
6. All parallel lines converge to the same (single) vanishing point. If any two lines are parallel to a third line, then they are parallel to each other. So a single line of sight defines the vanishing point for all lines parallel to it.


7. Lines parallel to the direction of view appear to converge at the principal point. This is only a specific case of rule 6, but it is very useful. We saw with the lines drawn on the metric grid that the orthogonals define a constant width or length across the receding transversals in the image. The principal point, and its associated orthogonal lines, define the controlling measure of depth in any perspective image.
This is a fundamental fact of everyday vision and of perspective constructions alike. Straight railroad tracks on level ground (right) are the most striking example. (Here a camera lens, rather than the eye, creates the perspective viewpoint.) Sunlight provides another case — the sun is so far away that its rays of light are essentially parallel at the earth's surface, and seem to converge when broken into shafts by clouds. We know the tracks and sunbeams are parallel, yet the illusion of convergence is very powerful. The basic rules developed for lines can also be applied to planes, which lets us define the large surfaces that may contain many different lines. By knowing the location and orientation of a plane, we also partially define the location and orientation of any lines it contains. 8. A plane that contains a line of sight appears as a line on the image plane. This matches rule 1 for lines: the vanishing line and intersection of the plane are the same, and the plane itself disappears, like a playing card viewed edge on. 9. Any two lines in a plane, projected onto the image plane, define the plane on the image plane. This is the matching principle to rule 3 for lines. 10. The image of a plane ends in two lines: its intersection with the image plane and its vanishing line. Thus, the ground plane defines the ground line (its intersection) and the horizon line (its vanishing line). This is the matching principle to rule 4 for lines, and similarly the only exceptions are planes parallel to the image plane and planes that contain a line of sight. Note that the intersection and vanishing line of a plane will always be parallel to each other on the image plane. 11. The vanishing line for any plane is the parallel plane containing a line of sight, or a line connecting the vanishing points for any two lines parallel to the plane. This matches rule 5 for lines. 12. All parallel planes converge to the same (single) vanishing line. This matches rule 6 for lines. In the standard perspective setup, the horizon line is the vanishing line for the ground plane and all planes parallel to it, such as ceilings and cloud layers. 13. The vanishing line of a plane contains the vanishing points for all lines in the plane and all lines parallel to the plane. This is a powerful rule, because it makes the vanishing line of a plane the location for an unlimited number of vanishing points for any and all lines that lie on the plane and all lines parallel to them. Thus, the horizon line, which is the vanishing line for the ground plane, contains the principal point, which is the vanishing point for all orthogonal gridlines actually on the ground plane and for the parallel direction of view, and it contains the diagonal vanishing points as well. 14. The vanishing lines for all planes parallel to the direction of view intersect the principal point. This matches rule 7 for lines. All lines on the ground plane, or parallel to the ground plane in any direction at any distance above or below it, must converge to a vanishing point located somewhere on the horizon line. All planes parallel to the ground plane any distance above or below it must also converge to the vanishing line for the ground plane, which is the horizon line. In the vertical dimension, the median line is the vanishing line for all vertical planes parallel to the direction of view, all lines they contain, and all lines parallel to them. Both vanishing lines pass through the principal point on the image plane.
Our leverage on these problems comes from triangular proportions, the principle at the heart of all perspective images. And it's very simple. If you have two triangles with the same three inner angles, then all sides of the smaller triangle will be in a constant proportion to the larger. (The proof appears in Euclid's Elements, Book 6, Proposition 2; and the the optical implications were developed in Euclid's Optics, written c.300 CE.)

parallel railroad tracks converge parallel sunbeams converge 

Let's start with distance. Given the perspective setup and the fixed proportions of the circle of view framework, a point located beyond the image plane will have a specific location on the image plane. But where?
distance and constant triangular proportions
Because the median line X is parallel to the direction of view, the line Y defines the same angle 1 at the viewpoint and at A. Because the line Z is parallel to the image plane, the line Y defines the same angle 2 at the viewpoint and at point a. By subtraction the remaining angle in the two triangles is the same (it is a 90° or right angle), so the three angles within xyz are equal to the matching angles in XYZ. This means there is also a constant ratio or proportionality between the lengths of all matching sides of the two triangles, as follows: x/X = y/Y = z/Z = constant ratio. So if we know the measurements of any two matching sides of the large and small triangles, we can divide them to find the constant ratio, then apply this ratio to determine the other sides of the triangle. Thanks to the circle of view framework, we already know x and Z (the viewing distance is the same as the viewing height, for example 1.6 meters), and we can specify the object distance X (either arbitrarily or by measuring it), which means we can compute the length z — the distance from the principal point to point a — as: (1) distance z from principal point to point a = Z*(x/X). Or, if we don't know the actual distance of an object shown in a perspective drawing, but we can measure the distance of its image a from the principal point, and can specify the viewing height and viewing distance used to construct the perspective view, then the actual object distance is: (2) distance of object X = x*(Z/z). That's fine for distance, but what about size? How big should the object appear in the drawing? Here again the triangular proportions give us the solution. object size and constant triangular proportions (3) ab/AB = x/X. So if we know any three of these measures, we can easily find the fourth. These triangular proportions allow us to coordinate our drawing size to match the actual size and distance of objects, given the expected distance from the viewer to the drawing. We only need to specify three of the four measurements — drawing size, viewing distance, object size or object distance — to determine the fourth: (4) AB/X = ab/x, or
Because the drawing size is largely under our control, and the object size is not under our control, the artist must choose an apparent viewing distance to the object that creates an appropriate image size, given the chosen format (dimensions of the drawing or painting) and viewing distance. The format (and viewing distance to the image) set limits on the apparent viewpoint we can take in relation to the object: the smaller the format, the farther away the object must appear. As a rough guide, consider the drawing size of three illustrative objects: a contemporary single story home (20' to roofline) to represent architecture, a man of average height (5'9") to represent figures, and a basketball (9.5") to represent still life objects, as they would be drawn in a painting viewed from 1.6 meters (63").
Stepping back gives us more options. If an object is about 1.75 meters high (the adult male), and you want to draw it as it appears from a distance of 100 meters, you find that the 1.6 meter viewing distance is about 1.6% (1.6/100) of the distance to the figure, which makes the appropriate drawing size 1.6% of 1.75 meters, or 2.8cm (about 1.1"). This would be suitable for a book illustration. The table is too inflexible for practical use, but can provide a rough check for design questions and when using formulas (3) or (4). Many design situations impose specific viewing distances — for example, book illustrations and architectural drawings are normally viewed from a distance of no more than 2 feet. But if the apparent size of objects is correctly scaled to fit within the usual picture formats, the image will produce a characteristic "presence" that cannot be reproduced in any other format. Botticelli's actual Birth of Venus as it hangs in the Uffizi (Florence) creates an impact utterly different from any book illustration. Rightly used, linear perspective can play with or against the characteristic sense of intimacy or grandeur that we associate with different genres and format sizes. It's useful to develop experience with the apparent size of objects (that is, widths in the visual field) as a function of the object's size and distance. It's handy to explore these relationships with a common hula hoop, as the hoop serves as a standard unit of distance, as the object dimension, and as the circle of view.
These hoops are manufactured in diameters from 28" to 36" (70cm to 90cm). Just buy a hoop (or make one out of a length of black irrigation tubing), measure its diameter, then use the following table to convert the distance to a hoop perpendicular to your direction of view into degrees of a circle of view (apparent width as a visual angle).
The normal procedure is to pick up a piece of paper, scale the object to fit within it, and make the drawing. The point is not that this procedure is necessarily wrong, just that it is unconscious. O'Keeffe thought differently about the relationships between the viewer, the object, the format, and the gallery encounter, and produced some stunning botanical images. The format and the painting are only the mechanism to create a sense of presence and impact through the image. Think about the distance relationships implied by the viewing situation, and you'll gain control of that impact.
Image Plane Orientation. First, let's revisit the point mentioned earlier that the image plane is not necessarily perpendicular to the ground plane, but is always perpendicular to the direction of view. In a basic sense, the image plane represents a hidden or symbolic context for the art experience, because the image plane itself is arbitrary. In terms of projective geometry, we can just as easily and accurately record the optical facts of the world on an image plane that is not perpendicular to the direction of view (or to anything else); we can use a curved surface as effectively as a flat one, or use curved lines rather than straight lines to draw figures. The form and orientation of the image plane are primarily conventional or functional. We assume the image plane is perpendicular to the ground plane because we expect the finished image will be hung for viewing on a vertical gallery or museum wall. We assume the image plane is flat because stretched canvas and drawing paper are flat. We might think of these as the media conventions contained in the image plane. In addition, the image plane represents the working location of the artist making the painting — how the image plane was seen by the painter. We might call these the orientation conventions of the image plane, because they govern our conclusions as to "the right way to look at the painting" or "the right way to hang the painting." We can specify these in terms of the the orientation of the artist's head in relation to the image plane, as the following image makes clear. natural orientation of the image plane By convention the artist's head is aligned with the edges of the standard rectangular format — the direction of view is roughly through the center of the format and perpendicular to its surface, the pupil line is parallel to the top and bottom edges of the format, and the head midline is vertical through the center of the format. We assume the image plane is like a windshield or visor hung in front of the artist's face, keeping a fixed orientation to the position of his head. Finally, there is a third kind of structure folded into the image plane, which implies the posture convention of artist's viewpoint and direction of view in the real world. The convention here is simply that we say what the "artist's view" really was, based solely on the appearance of the world represented by the image. ("In this painting, the artist is looking down into Niagra Falls.") We expect, for example, that if the horizon line is parallel to the top and bottom of the image plane, then the artist's pupil line was parallel to the horizon, even though the artist may have been looking down while working. Through experience with landscape and architectural paintings we come to expect that the artist's head was erect and his direction of view was parallel to the ground plane when the image was recorded. And we assume (in some cases wrongly) that the gallery viewer can stand at the implied viewpoint in order to view the image. Paintings often gain visual drama when the posture conventions are violated — for example, when the direction of view is downward or upward in relation to the ground plane, or the major edges in the image are not parallel to either the horizon or median lines. But if the artist's head is tilted to one side, or looking upwards or downwards, the image plane will have a complex relationship to the planes and lines of the actual ground plane and actual horizon, as discussed under three point perspective.
effect of changing only the direction of view
The only difference between the two views is in the direction of view, and therefore in the orientation of the image plane — the pinhole was kept in exactly the same location. The image at left shows a direction of view perpendicular to the face of the arch; the horizontals appear parallel to each other and to the horizon line. When the direction of view is shifted 25° to the left, the horizontals now appear to converge, and only those at the horizon are parallel to the horizon. That is, simply by changing the direction of view, we've transformed a central perspective view into a two point perspective view. Once again, linear perspective is primarily about a specific location and direction of view in space. All these are folded into the assumptions that govern our use of the image plane as artists or viewers. Changing the direction of view has as much impact on the image as changing the location of the viewpoint. Both need to be considered carefully when composing the image to be painted. Horizon Line and Viewpoint. An important and useful fact of landscape perspective is that all objects exactly as high as the viewpoint are intersected by the horizon line. This is true regardless of how far above level ground the viewpoint may be, and even when the direction of view is not parallel to the ground plane.
horizon line and viewpoint in landscape perspective
All the perspective relationships between other figures or objects in the image and the horizon line (orange) change with the changes in viewing height. When the viewer is sitting, the horizon line appears to cross the waist of standing figures around him. When he is standing on level ground, the horizon line is at the eye level of all standing figures as tall as he — no matter how near or far they are from the viewer. When the viewpoint is from a raised platform, all figures on the ground below appear below the horizon line. (Note also the changing location of the horizon line against the central pillar.)
This fact arises from rule 12: all parallel planes converge to the same vanishing line. In this case, the first plane is the ground plane, whose vanishing line is the horizon. The viewing height, extended in all directions, creates a second plane parallel to the ground plane, like the surface of a large lake flooding the ground plane in all directions at a constant depth. This surface will also converge to the horizon line. Regardless of the direction of view, all objects lower than this plane will be "under water" and therefore below the horizon line. All objects above it will be "above water" and above the horizon line.


Many visual illusions of size depend on the position of the object relative to the visible or presumed horizon line, even when other perspective cues are removed. The famous and delightful Ames room (right), contrived by Aldebert Ames Jr. in the 1940's, is a large trapezoidal enclosure that appears perfectly square when viewed through a peephole. Figures appear to grow or shrink in opposite corners of the room because the "short" corner is substantially lower and farther away, reducing both the size of the figure and her relative position to the "horizon" defined by the windows and floor.
Any object appearing above a horizon line is higher (taller) than the viewing height, and any object appearing below the horizon line is lower (shorter). For example, in the photo of train tracks above, the horizon line is level with the bottom edge of the red passenger car, just above the wheels. This is much lower than the standing height of a man, so we know that the photographer was crouching or sitting (or the camera was on a low tripod) when the picture was taken.
And there's the catch. Even if the perspective drawing accurately represents a specific viewpoint, we typically don't look at the perspective drawing from the "correct" viewpoint and direction of view. The drawing may be done at a scale that conveniently fits the space available within the picture format, but creates a viewpoint that is too close to or too far from the picture surface; the painting or fresco may be positioned too far above the floor; or the painting may be viewed from different distances or angles as it is hung in a room or gallery; and, of course, we always look at it with two eyes. What happens if we look at a perspective drawing from a different location? The following diagram illustrates the crux of the problem. perspective geometry and viewing distance But this is a large painting, so we decide to step back a few feet (to 90") and look at it again. Now the drawing vanishing points no longer correspond to our visual vanishing points as defined by our 90° circle of view. As a result, the edges and angles of the building seem to place the vanishing points too close together, and the building appears exaggerated in perspective proportions — the front angle of the building seems more like 70° than 90°. There is a second problem caused by oblique (sideways) angles of projection onto the image plane. This is most glaring with spherical or rounded objects that show a near circular curvature from all sides (such as spheres or the heads of human figures) or with any repeated form (such as columns, arches or floor tiles) arrayed parallel to the picture plane — as often happens in central perspective. In a perspective projection these can appear highly deformed for the same reason that a flashlight beam appears circular if directed straight at a wall but elliptical if directed obliquely to one side. As an illustration, the figure below shows a correct perspective projection of identical spheres resting on top of identical rounded columns standing in rows parallel to the picture plane.
perspective distortions in rounded forms
These distortions have distinctive features worth memorizing: • Horizontal thickening. The spheres and columns far from the direction of view on either side appear thicker than those at the center of view. • Vertical stretching. The spheres far above or below the direction of view appear vertically elongated. • Diagonal emphasis. The distortions appear more extreme along diagonal (oblique) directions, where the effects of horizontal thickening and vertical stretching combine. • Diagonal tilting. Horizontal surfaces, such as the orange flat tops of the columns, appear tilted along the image diagonals rather than forward toward the viewer. • Peripheral crowding. Equal intervals (such as the spaces between columns) appear smaller near the periphery than at the center of view; eventually the spaces between the columns disappear and the columns seem to overlap. The common diagnosis for all these perspective distortions is that the width of the drawing is too large a proportion of the 90° circle of view. (This is more often expressed as, "the vanishing points are too close together.") For example, in the office building painting, the width of the paper is equal to the viewing distance (radius of the circle of view), so the format spans a 53° circle of view. As a result, when the drawing is viewed from different distances or angles, we see proportionally large discrepancies between the drawing's and the viewer's diagonal vanishing points. The solution is to move the vanishing points much farther apart: that is, to place all parts of the principal forms within a more restricted circle of view — equivalent to drawing the objects as they would appear from a viewpoint much farther away. (Once again, we're back to the importance of apparent distance and size to the visual effect of an image.) As a result, the viewing distance to the image is a small proportion of the apparent distance to the principal form, and the drawing can be acceptably viewed from a wider range of viewing distances. The peripheral distortions in rounded forms and crowding of sequential forms are cropped out of the image entirely, and moving toward or away from the drawing represents a change in viewing distance that is small in comparison to the radius of the circle of view. The practical limit on the field of view is conventionally set at a 60° circle of view — a suggestion made by Piero della Francesca in c.1470 and repeated often since then. In fact, depending on the geometry of the principal form and the location of the vanishing points, a 40° circle of view or less is much more typical. Leonardo da Vinci devoted many pages in his notebooks (c.1490) to the analysis of perspective distortions, and he especially disliked the exaggerated apparent size of the perspective grid as it reached the ground line of the image plane (for example, as in the ground squares of this image). He recommended painting an object as it appears from a distance of 3 to 10 times its actual dimensions (e.g., a standing figure 1.75 meters tall should be viewed from 5 to 18 meters). This is equivalent to placing the figure within a 19° to 6° circle of view. In fact, modern vision research has found that most people say an object "fills their field of view" once it occupies approximately a 20° circle of view. Some famous problems are simply cases of incorrect analysis. For example, artists from Leonardo down to Flocon & Barre have been vexed by the paradox that physical edges parallel to each other (such as the top and bottom of a straight wall or the sides of a cylindrical column), when viewed perpendicular to the direction of view, appear taper toward the opposite edges of the circle of view; yet they are drawn (in central persepctive) as parallel straight lines. In fact, the perspective rendering of edges does not bow or arc toward the extremes of view because the same triangular proportions that foreshorten the parallel edges of the wall or column also foreshorten the parallel lines in the canvas image. The increased oblique distance from the viewpoint to the object is proportionally matched by the increased oblique distance from the viewpoint to the parallel canvas surface: so no "correction" is necessary in the drawing. Such perspective distortions, and the restricted circle of view solution for them, were well known to artists from the beginning of perspective practice. However, these artists also realized that many distortions they could identify from geometrical reasoning were not equally objectionable to a viewer. Apparent distortions in rectangular forms are more objectionable than distortions in curved forms; distortions in the horizontal direction are more obvious than distortions in the vertical direction (in part because the format is usually wider than it is high); distortions in unfamiliar objects are more acceptable than distortions in familiar objects; distortions in the apparent location of vanishing points are more acceptable than distortions in the outline of forms; distortions in "rigorous" perspective drawings are more objectionable than those in "free" drawings; and so on. As a result, if artists were working with a large fresco or canvas format, or wanted a panoramic effect, they often adopted radical solutions guided by the context of the painting: they would "correct" or disguise perspective distortions wherever they appeared objectionable. This was almost always done for figures, rounded forms, the spacing between columns of a facade, and so on. Often several kinds of "corrections" were used at the same time. raphael's philosophy (1511)

an ames room 

A fine example is Raphael's large fresco Philosophy (commonly known as The School of Athens) which fills an almost 30 foot wide section of Vatican wall. This huge format clearly imposes a panoramic context on the image design, which Raphael utilized in novel ways. He framed the perspective construction within a relatively restricted 40° circle of view, which crops extreme distortions from the image — although as a result the correct perspective viewing point is not even in the same room! The perspective effects are limited to the strongly cropped floor tiles and the enormous central passageway, whose vanishing point is hidden by the approaching figures. The rest of the picture space is filled by walls parallel to the picture plane, displaying a pair of square columns on each side. These are cropped at the top and hidden at the bottom by standing figures, eliminating the repeated sideways intervals or diagonal corners that would accent perspective distortions. The semicircular front arch of the barrel vault is also cropped, because it would otherwise appear to be elongated vertically. The floor tiles on either side of the foreground are hidden by groups of figures. The foreground stairs help to separate the figures vertically and interrupt the perspective continuity of the tile floor. Most important, all figures are drawn as if centered on the direction of view — that is, with no perspective distortion. This is easiest to see in the two astronomers shown holding celestial globes (at right). Both figures are located at the righthand edge of the fresco, beyond the 30° circle of view. Rather than draw the spheres in correct but elliptical perspective projections, Raphael simply drew them perfectly round. Thus, the architecture enclosing the figures is presented in a carefully edited and arranged perspective projection, while each of the figures is drawn in its own, "head on" perspective space. Yet this hodgepodge of perspectives is perfectly judged. Even though the fresco is normally viewed from a viewpoint too close to the image plane and several feet below the center of projection, in context the overall perspective space appears harmonious and convincing.
As painters developed dozens of similar tricks to exclude, hide or counteract perspective distortions, thus minimizing the effect of viewing a painting from "incorrect" locations, they discovered that perspective distortions could be used for expressive effect. The most extreme examples are anamorphic images — especially popular in the 16th and 17th centuries — which appear as unrecognizable smears or blurs unless viewed from an extreme angle or with a corrective mirror. These strange paintings suggest how far artists were willing to extend the geometrical implications of perspective in search of visual illusions and expressive effects. In general, rendering a single three dimensional form within a circle of view greater than 40° (that is, as the form would appear to the naked eye from a close distance) has four important effects on its visual impact: • principal forms become more dynamic — buildings or figures seem to loom, surge and expand • perspective space is enhanced — the convergence among vanishing lines is more emphatic, creating a vertiginous depth of space • the front surfaces of the form dominate — the sides of the form may disappear from view, or appear smaller or highly foreshortened, and the side surface textures are viewed at a more grazing angle • vertical dimensions dominate — in particular, the extreme corners of the form may appear to jut or loom out of proportion with the rest of the figure. Renaissance and Baroque artists who experimented with these effects understood that perspective paintings are effective even when they are not viewed from the center of projection. This is sometimes called Zeeman's paradox, but the paradox is purely conceptual: it assumes we view a perspective representation as a retinal simulation, when in fact we view it as a two dimensional painting. In other words, perspective constructions create visual symbols, not visual illusions. The key is that paintings lack the depth of field cues created by binocular vision; we are always aware a painting is flat rather than deep. And that is how our mind interprets it, adjusting our understanding of the painting to compensate for our position. Of course, linear perspective can produce compelling illusions, but not easily — the image must be in exact perspective, the edges of the image must be hidden, and the image must be viewed with a single eye from the center of projection, in what is called a "peep show" or peephole arrangement. Binocular photography and an optical viewer that fixes our point of view can create a much more vivid illusion from flat stereoscopic images, but even slight changes in the point of view will distort or destroy the effect. As "peep show" setups were never a serious goal of fine painting, artists were free to ignore "exact" perspective projections and instead exploit perspective for its representational, expressive effects — mixing correct perspective buildings with "incorrect" perspective figures, obeying perspective recession but "bending" long foreground lines, and always adjusting the circle of view and center of projection to suit the motif, format, and installation of the work. The appearance of ironclad geometry and intricate drawing procedures disguises how much exploration, improvisation and creativity artists historically allowed themselves when using perspective methods. So we end with an important insight: geometry can justify and guide perspective constructions, but it can't determine the best design use of perspective or show us how perspective should be "corrected" or adapted for more effective results. Raphael's figures and celestial spheres do not need to be in correct perspective because they combine so well as icons on an elegantly designed surface. The rules of linear perspective only help us to create the symbols, not combine them into works of art. Continue to Page 2 Last revised 08.01.2005 • © 2005 Bruce MacEvoy 
correction of perspective distortions in Raphael's Philosophy (1511) 
