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Depth and Perspective

Only in our imagination do we live in more than two dimensions, and with its help we attempt to enliven the flatness of our image with depth. All of a sudden it may dawn on us how foolish we are, we faddists of the two-dimensional picture with our constant urge to achieve unobtainable depth.

-M.C. Escher, 1947[1]:

Explorations

Begin learning about depth and perspective with:

Begin learning about impossible figures with:

Depth

A sort of miracle occurs when you view artwork: the flat image is transformed into a solid scene in your mind, without conscious effort. Throughout his life, Escher created prints that disturb this unconscious process, and force you, the viewer, to come to grips with the technique of art and the workings of your eyes. Escher does this choosing perspective so exaggerated that it can't escape notice, and by loading his prints with depth cues so contradictory that the three dimensional scene is entirely impossible.

We begin by asking the question:

What is it that gives a flat picture depth?

Before continuing, try to answer the question yourself with the Depth Exploration.

Detail of Tutankhamun's painted chest, c.1323 BC

One of the strongest visual cues that gives depth to a scene is overlap. If one object in a scene overlaps or obscures another, then the partially hidden object must be further from the viewer. Overlapping is the primary depth cue in ancient egyptian paintings. Egyptian artists followed a strict set of stylistic guidelines, and sometimes cultural rules overrode the natural representation. As an example, no object was allowed to obscure the face of the king, so that in the image of Tutankhamun the bowstring and arrow must be "behind" his head.

Men Feeding Oryxes, from the tomb of Khnumhotep, c. 2400 BC

In the painting Men Feeding Oryxes, there are a complicated mix of overlaps intertwining the standing man and oryx. The most interesting depth cue in this painting is that the standing man and oryx are slightly higher than the seated pair. It is a historically early instance of another fundamental way to show depth - objects further away from the viewer are higher, or at least standing on a higher ground line.

The Laestrygonians. Greek wall painting, c 100BC.

There are many other techniques for indicating depth in a flat artwork. For example, distant objects can be shown smaller, brighter objects appear closer to the viewer, shading can give the illusion of contour and shape, and objects in the distance can be shown with less detail. Ancient Greek art used most of these techniques and achieved a naturalistic three dimensional scene, such as the image of the Laestrygonians. Interestingly, the techniques of Greek art were lost (or possibly rejected) for centuries and not revisited until the Renaissance.

Christ before Pilate from the Rossano Gospels, 6th century.

In illuminated manuscripts throughout the dark ages, overlap and height are the primary depth indicators. In the illustration from the Rossano Gospels, people in the crowd surrounding Pontus Pilate are higher on the page the further back they are in the scene. Pilate's table is also shown with its top rising up and to the right.

During the Renaissance, artists such as Brunelleschi developed the highly realistic technique of linear perspective. The three images below trace the evolution of perspective. In Duccio's Annunciation of the Virgin's Death, lines which recede from the viewer in the scene are shown slanted in the painting. As these lines recede, those to the left of the viewer's eye run towards the right, lines to the right slant to the left, lines below the eye slope upwards, and lines above the eye slant down. However, these slopes are applied inconsistently throughout the image.

One hundred and fifty years later, Piero della Francesca's The Flagellation has a carefully calculated perspective. Lines that recede from the viewer all converge to a point just to the right of Christ, although some small inconsistencies remain. In Da Vinci's The Last Supper, all receding lines converge precisely to a point in the center of the picture, effectively focusing the viewer's eye on Christ.

Linear perspective develops
Annunciation of the Virgin's Death, Duccio, 1308 Flagellation, Piero della Francesca, 1455-1460 The Last Supper, Leonardo da Vinci, 1495-1498

Linear Perspective

Linear perspective was invented in the early 1400s by Filippo Brunelleschi of Florence, who painted the outlines of buildings onto a mirror. According to his biographer, Brunelleschi set up a demonstration of his painting of the Baptistry of St. John in the doorway of a facing building. He had the viewer look through a small hole on the back of the painting, facing the Baptistry. He would then set up a mirror, facing the viewer, which reflected his painting. When he removed the mirror, the veiwer saw the real Baptistry and could see that the painting and real building appeared nearly identical.

The fundamental idea of linear perspective is to treat the painted picture as a window, and trace sight lines from the viewers eye through the window and onto the scene. The problem of accurate drawing then becomes an exercise in geometry, and was well understood by the end of the 15th century.

The image above shows the mathematical assumptions of linear perspective. The red square, slanted away from the viewer, is positioned on the picture "window" by locating its corners at the points where the blue sight lines cross the picture plane.

In a perspective drawing, every collection of parallel lines in the three dimensional scene become lines converging to a point in the drawing:

Vanishing point
A set of parallel lines converges to a single point in the drawing, called the vanishing point.

Different sets of parallel lines converge to different vanishing points. Lines in the scene which are parallel to the ground (i.e. horizontal) converge to vanishing points on a horizontal line in the drawing, called the horizon line:

Horizon line
All vanishing points of horizontal lines lie on the horizon line.

The horizon line defines "eye level", in the sense that things above the horizon will appear to be above the viewer and things below the horizon will appear to be below the viewer.

In a computer generated scene, there can be many different vanishing points. However, traditional drawing and painting techniques tend to fall into one of three categories: one-point perspective, two-point perspective, and three-point perspective.

One-point perspective

One-Point Perspective.

One vanishing point is typically used for roads, railroad tracks, or buildings viewed so that the front is directly facing the viewer. Any objects that are made up of lines either directly parallel with the viewer's line of sight (like railroad tracks) or directly perpendicular (the railroad slats) can be represented with one-point perspective. Da Vinci's The Last Supper, shown above, is an excellent example.

One-point perspective exists when the scene is composed entirely of linear elements that intersect only at right angles. If one axis is parallel with the picture plane, then all elements are either parallel to the painting plate (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the painting plate are drawn as parallel lines. All elements that are perpendicular to the painting plate converge at a single point (a vanishing point) on the horizon.

Two-point perspective

Two-Point Perspective.

Two-point perspective can be used to draw the same objects as one-point perspective, rotated: looking at the corner of a house, or looking at two forked roads shrink into the distance, for example. One point represents one set of parallel lines, the other point represents the other. Looking at a house from the corner, one wall would recede towards one vanishing point, the other wall would recede towards the opposite vanishing point.

To draw a cube in two-point perspective, follow these steps:

1. Set down a horizon line (shown red).
2. Put two points on the horizon line, which will be your left and right vanishing points (these are the endpoints of the horizon line in the illustration)
3. Draw the front vertical edge of the cube (shown blue), which you can place anywhere between the vanishing points and at any height. If the front edge is entirely below the horizon line, you will be able to "see" the top of the cube. If the front edge is entirely above the horzion line, you will be able to "see" the bottom of the cube. If the front edge crosses the horizon, neither the top nor bottom are "visible".
4. Lightly sketch lines from the top and bottom of the vertical edge to the left and right vanishing points, four lines total (shown black).
5. Draw vertical lines to define the left and right visible edges of the cube.
6. If necessary, draw the top or bottom of the cube by extending lines from the back top or bottom corners to the left and right vanishing points.

You can place other cubes in the scene turned at different angles by choosing different vanishing points on the same horizon line.

Three-point perspective

Three-Point Perspective

Three-point perspective is usually used for buildings seen from above (or below). In addition to the two vanishing points from before, one for each wall, there is now one for how those walls recede into the ground. This third vanishing point will be below the ground, and is known as a nadir:

The vanishing point for vertical lines in the down direction.

Looking up at a tall building is another common example of the third vanishing point. This time the third vanishing point is high in space, and is known as a zenith:

Zenith
The vanishing point for vertical lines in the up direction.

Escher's work frequently involves three-point perspective, with extreme convergence of vertical lines to zeniths or nadirs. As an example, consider Tower of Babel, shown below. The lines which are horizontal in the scene converge to a pair of vanishing points on the horizon. Because the horizon line is so high above the drawing, the scene appears to be viewed from high above. The vertical lines converge downward to a nadir, reinforcing the impression of great height.

See other examples of Escher's use of perspective in Perspective Exploration.

Vanishing Points

M.C. Escher, Cubic Space Division, 1952.

Escher undoubtedly had a complete mastery of linear perspective. In many of his works he used straightforward linear perspective but pushed it to extremes, such as the bird's eye view in Tower of Babel or the technical complexity of Cubic Space Division and La Mezquita, Córdoba. Even his early landscapes are rarely shown from a traditional viewpoint. For example, in Castrovalva, the viewer is positioned on the precarious edge of a narrow cliff path, and so seems almost to be floating in space.

M.C. Escher, Gallery, 1946.

Many of Escher's works take the theory of linear perspective even further. A vanishing point is simply a point in the picture where parallel lines converge, and so there is no mathematical distinction between a zenith, nadir, and point on the horizon. Usually it is quite easy to distinguish the three from the context of the image, but not in Escher's work. In Gallery, a simple scene in one-point perspective, the central vanishing point is used in three different ways. Focus on the left and right walls of the gallery, and you see a fairly traditional one-point perspective hallway, with the horizon line emphasized by the planet surface and night sky in the backgrount. Focus on the top quadrant, and suddenly you are looking down into an abyss, the center point serving as nadir. Finally, the lower quadrant shows the bottom of an infinitely tall wall, with the center vanishing point now a zenith.

M.C. Escher, House of Stairs, 1951.

The use and re-use of vanishing points is at its most disconcerting in House of Stairs. Again, single vanishing points are used as zenith, nadir, and horizon for various parts of the picture. Unlike Gallery, however, the different parts blend together seamlessly via a technique known as the telegraph effect.

The telegraph effect refers to a series of telegraph poles extending far to the left and right of the viewer, while staying parallel to the picture plane. The mathematics of linear perspective dictates that these poles remain the same height in the scene:

The reality of human vision allows for convergence at both endpoints, with the parallel lines bending towards each other as they recede from the viewer:

Follow the curves in House of Stairs to see how Escher transitions each vanishing point between horizon, zenith and nadir. He uses a similar trick to accomplish the double vision of Up and Down.

Flatness

Pierre-Auguste Renoir The Luncheon of the Boating Party, 1881. An example of impressionism.

The accurate representation of a three dimensional scene began with linear perspective in the Renaissance. For the next centuries, the way to make a visual record was through art, and so realistic portraits, landscapes, and still lifes dominated the art world. The rise of photography in the 1800's allowed impressionist artists to move towards a more abstract representational style of painting. Impressionist painters were less interested in a precise representation of what they saw, and more interested in conveying a mood.

By the early 1900's, realism and the linear perspective that went with it were passé. Photography could keep the visual records, and artists had nothing to add to the technique of perspective - it was mastered. In the 20th century, many artists rejected the notion that art should represent reality, and Escher was among them. His extreme perspectives, fantastical worlds, and impossible figures are all examples of this. Escher's goal is to force his viewers to acknowledge that art is not reality. He wrote, about his print Balcony:

M.C. Escher, Balcony, 1945

We sometimes forget that we are by no means dealing here with a city... but with a sheet of paper. You are not willing to accept this? The dream vision is dearer to you? Then I will rudely destroy it by making a bulge in the sheet. -M.C. Escher[1]

The conflict between the flatness of the print and the three dimensional mental image is most evident in a group of prints Escher made in the 1940's.

Trace through these prints with Flatness Exploration.

M.C. Escher Dragon, 1952

As an example, consider Escher's Dragon, which at first glance represents a dragon. Looking closer, the dragon's neck and tail pass through slots in his body. The slots are open because the dragon has crease lines running vertically up and down his body, which can only be explained if the dragon itself is flat, a printed image in which Escher cut two slits and made six folds before representing it in the print. Escher is toying with you. Your eyes see the solid dragon, with its thick body and long tail, an unconscious process that takes depth cues and produces a three dimensional model. But your mind cannot reconcile the flat folds and thin slots, and so after some thought the solidity of the dragon is in doubt.

Illusions

Even more forceful than art that questions the viewer's reconstruction of a third dimension is art which attacks that process. Ambiguous and impossible figures are designed specifically to break the automatic process by which people construct three dimensional scenes from two dimensional images.

Ambiguous Figures

 Roman mosaic. Real Alcazar, Cordoba. Bench. Real Alcazar, Sevilla
Ambiguous Figure
An ambiguous figure is a figure which has two (or more) reasonable interpretations as a three dimensional object.

The Necker cube is an ambiguous figure, named after the Swiss crystallographer Louis Necker, who first published it in 1832. Are we looking down on the cube, or are we looking up at the bottom of the cube? Both interpretations are possible in the sketch as given.

Another well known ambiguous figure is the Thiery figure, so named in 1895 after the psychologist A. Thiery:

These figures are ambiguous because they lack depth cues to help the viewer decide what parts are in front and what parts are in back. In fact, every two dimensional image must be ambiguous, but usually there are enough depth cues and real world context to force a single three dimensional interpretation.

Escher's Convex and Concave is a tour de force of ambiguity. The edges and shadings of the various buildings in the print do not give enough information to distinguish convex objects from concave. The positions of humans, animals, and other objects resolve the issue but lead to an uncomfortable clash of interpretations in the center of the image. The same idea, that convex and concave can easily interchange, forms the basis for this animation by Charlie Deck:

Impossible Figures

Impossible Figure
An impossible figure is a two dimensional image of an apparently three dimensional object which cannot possibly exist.

An impossible figure usually contains a deliberate blending of foreground and background. Something that appears to be in the front in one part of the figure is forced to be in the back in another part. Escher enjoyed playing with these ideas and used them to create several famous prints.

The Penrose triangle, sometimes called a tribar, is a standard example of an impossible figure. If we look carefully at the image we see that there is a horizontal bar (at the bottom) with on the left a bar pointing to the back, while on the right we have a bar pointing up and slightly towards us. In our real world the receding bar and the bar pointing straight up can of course never meet. Yet in the drawing they are drawn as though they connect.

The print Waterfall is an example of an Escher print using the tribar. The water flows across three connected tribars and creates a situation where water flows upstream.

Impossible cube.

Another well known impossible figure is the impossible cube. The impossible cube is related to the Necker cube, because some overlaps are consistent with one interpretation of the Necker cube, and some overlaps are consistent with the other interpretation. The second vertical bar from the left represents part of the front of the cube when viewed at the top, but it represents part of the back of the cube when viewed at the bottom.

Escher's Cube With Magic Ribbons is a play on this idea. In this print the cube is actually well-defined when it comes to its placement in space, but the ribbons are drawn so that their placement is ambiguous. The bumps on the ribbons are the magic - follow them around and it is impossible to decide whether they protrude outward or cup inward.

Belvedere is an excellent example of this exchange of front and back. The top half of the print is unambiguous as is the bottom half, but they are joined togehter in an impossible fashion. If we take a close loook at the colums we see that they connect to the back of the building at the top and the front of the building at the bottom or vice versa. It's rather amusing to see the man sitting on the bench in front of the building holding an impossible cube.

Impossible staircase.

A third type of impossible figure is the impossible staircase, also designed by Roger Penrose. Escher used this figure in his print Ascending and Descending, where monks walk eternally up or down a loop of stairs.

Impossible Exploration and Impossible Exploration II are designed to have you look at some of these impossible figures. The official M.C. Escher website has several videos that explore Escher's Impossible prints, these videos are incorporated in the Impossible Figures and Escher Exploration.

Notes

1. Phoenix, Jaargang 2, Juni 1947, #4 Graphic Artists of the Netherlands Speak of Their Work