Depth and Perspective

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Relevant examples from Escher's work:


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 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 basic depth cue - that objects further away from the viewer will be 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
Duccio-annc-virg-death-1308.jpg D-francesca-flagellation-1458.jpg Da-vinci-last-supper-1498.jpg
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.

Drawing Square in Perspective 2.gif

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 becomes a collection of lines in the drawing which converge to a single point, called a 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. 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.
  2. Put two points on the horizon line, which will be your left and right vanishing points.
  3. Draw the front vertical edge of the cube, 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.
  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. 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.

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.

Impossible Figures

Most impossible figures stem from a deliberate blending of foreground and background. Escher enjoyed playing with these ideas and used them to create several famous prints.

File:Penrose triangle.png

The penrose triangle, sometimes called a tribar, a 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 situaution where the water flows upstream.

File:Necker cube and impossible cube.PNG

In the above image we see two related but slightly different cubes. On the left we see the Necker cube and on the right we see the impossible cube. In the image of the Necker Cube the orientation of the cube was deliberately left ambiguous. Are we looking down on the cube, or are we looking up at the bottom of the cube? Both interpretations are ppossible in the sketch as given. In the impossible cube we see a deliberate change of perspective within the image. 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.

The 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.

Belvedere is an excellent example of this exchange of front and back. The top third of the print is unambiguous as is the bottom third. The interesting part of the print is the middle third. 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.png

A third type of impossible figure is the impossible staircase. Escher used this figure in his print Ascending and Descending.

Impossible Exploration


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. This is most evident in a group of prints he made in the 1940's which emphasize the conflict between the three dimensional world of reality and the two dimensional world of art.

Trace through these prints with Flatness Exploration.

M.C. Escher Dragon, 1952

Escher's Dragon, 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.


Depth and Perspective Exercises

Related Sites


Impossible Figures