Picturing Space: Projection and Perspective

Space is the breath of art.
— Frank Lloyd Wright

There are two kinds of space in advanced forms of mimetic painting: the flat, two-dimensional space of the canvas on which the painter places lines and colors, and the illusion of three-dimensional space, called pictorial space, which is suggested by those shapes, colors and lines.

In the earliest times, artists were concerned above all with the representation of objects, singularly or in groups. Untrained artists, as well as painters of ancient Egypt, Crete, India, Islam and pre-renaissance Europe, depicted objects and surroundings independently of one another. To do this they were viewed from a direction that presented their most characteristic features, rendering them as they are known to be, rather than as they are seen by the eye. Only when painters became increasing concerned with the representation of naturalistic appearances of the broader world did they begin to consciously ponder not only objects and their individual or symbolic qualities, but the spatial relationships among them.

While the medium of the painter is to some degree complimentary to the appearance of natural objects—both present different degrees of color, brightness and texture—paint shares nothing with space, since space is not a substance but the lack of it. If single objects are accurately represented by the artist, they do produce the effect of the space that immediately surrounds them, in other words, the space that objects displace (e.g., the space inside a drinking glass or the space between two fingers of a hand), but the great arena of space which extends beyond them is in no way expressed. Only linear perspective can achieve this objective precisely and in a mathematically predictable manner.

Knowing the laws of linear perspective allows the artist not only to represent the shapes and positions of real objects correctly foreshortened in space, but to create such images from mathematical measurements. To do this, the shapes of objects are distorted, but according to consistent geometrical laws. Before linear perspective was codified by Leon Battista Alberti in 1435 (De Pictura), artists used various types of projection methods, sometimes referred to as "perspectives."

Methods of Expressing Spatial Dimensions on a Two-Dimensional Surface

Oblique Projection
Oblique projection, sometimes referred to as "parallel projection" or "parallel perspective," is a simple type of graphical projection used to produce two-dimensional images of three-dimensional objects. The objects are not in true perspective, because they do not correspond to any view of an object that can be obtained in practice. Nonetheless, oblique projection yields convincing and useful images that allow the viewer to perceive the semblance of spatial depth.

In oblique projection the horizontal edges of the opposing sides of an object remain parallel to each other. They produce no vanishing points, and no horizon line, so no part of the object gets smaller with distance as when projected with perspective. Oblique projection is "less faithful to appearance, but more faithful to fact; it shows things nearly as they are known to the mind. Since oblique projection removes the vanishing point, all things are represented as being the same distance away and the eye of the spectator is everywhere at once."^{Jan B. Deręgowsk and Denis M. Parker, "Convergent and Divergent Perspective," Perception, vol. 21, issue 4, 1992, 441–447.} It would appear that the Chinese have never had any scientific interest in perspective or its rules and typically adopted the parallel projection when representing buildings or geometrically regular objects (fig. 1). Such objects are viewed obliquely and their orthogonal sides recede along two main angles.

Isometric Projection
Isometric projection is similar to oblique projection in that it allows to represent three-dimensional objects in two dimensions. Like oblique projection, the horizontal edges of an object in isometric projections remain parallel to each other, but the X and Z axes are inclined to the horizontal plane at the angle of 30 degrees. Because all edges of an isometric object are inclined at the same angle, they are equally foreshortened, which allows the measurement of each side of an object using the same scale making it particularity useful for technical and engineering drawing.

Perspective

Divergent Perspective
Divergent (or "inverted," "reverse," or "Byzantine") perspective, was used by artists of the Russian Orthodox tradition, as well as by Byzantine, Chinese and Japanese artists, and many pre-Renaissance cultures (fig. 2). In divergent perspective, receding parallel lines appear to move away, or diverge, from one another. This has the visual effect that objects farther away from the viewing plane are drawn as larger, and closer objects are drawn as smaller, in contrast to the more linear perspective for which closer objects appear larger.

The use of divergent perspective was regarded as arbitrary or even as erroneous in spite of the fact that entire schools of art exist in which this kind of perspective was regularly used. ^{Ian P. Howard, Robert S. Allison, "Drawing with Divergent Perspective, Ancient and Modern," Perception, 2011, vol. 40, 1017–1033.} Curiously, when asked to draw a simple cube, about 50% of a group of university students unfamiliar with perspective used divergent perspective.^{Dominique Raynaud, Studies on Binocular Vision: Optics, Vision and Perspective from the Thirteenth Centuries, New York: Springer, 2016, 15.}

Convergent Perspective
Convergence is a type of distortion caused by the medium of human sight, although it is largely overridden by the perceptual mechanisms, such as distance constancy. In fact, convergence is rarely noticed except in highly specific conditions, which rarely occurred until man began to modify nature and create constructions which produce many long parallel lines.

Giotto (1267–1337) may have been the first post-classical artist to intuit that lines which run parallel to the line of sight appear to eventually intersect at a single point, whereas physically, they do not. Both Duccio (c. 1255/1260–c. 1318/1319) (fig. 3) and Giotto employed convergence, but the projected lines of different objects in the same picture (ceiling rafters, floor tiles or furniture) converging at different locations, not at a single vanishing point as is the case in linear perspective.

Linear Perspective
Linear perspective is often referred to as "geometric perspective" or more simply, "perspective." Perspective is a measurable relationship between object, pictorial plane and observer. The two most characteristic features of perspective are that objects are smaller as their distance from the observer increases; and that they are subject to foreshortening, meaning that an object's dimensions along the line of sight are shorter than its dimensions across the line of sight.

Linear perspective creates a picture on a flat surface in such a manner that the various objects represented in it appear to have the same sizes, shapes and positions, relatively to one other, that the actual objects as located in actual space would have if viewed from a single immobile point of view in a single direction. With the aid of perspective geometry the relative sizes of different objects can be assessed at a distance for the first time.^{James Burke, The Day the Universe Changed, London: Little, Brown & Company, 1995, 76.} Distant objects could be reproduced with fidelity, or created to exact specifications in any position in space and then manipulated mathematically.

Because perspective reproduces the way our eyes see objects, it creates distortions that oblique projection does not. Although a perspective projection of objects is considered more realistic than an oblique projection, oblique projections are useful in technical applications, since the parallelism of an object's lines and faces is preserved. Perspective itself rejects the property of parallelism for the principle of a gradual reduction in size, reproducing as closely as possible the conditions of natural vision."^{Science Art and Technology, What is Perspective?" (website no long available).}

From a painter's point of view, linear perspective may be thought of as a practical means of establishing a measurable relationship between the shapes of objects as definitely located in space when observed from a single point and direction of view (fig. 5). It is essentially the same as tracing some scene on a window while keeping one eye closed and the other eye absolutely stationery.

Unlike aerial perspective, which utilizes gradations of tone, color and detail to evoke the sensation of distance, linear perspective makes use of visible or implied lines associated with the objects represented. Although particularly useful for the portrayal of man-made geometric objects, such as buildings and furniture, linear perspective may be applied to organic objects as well, particularly when they are the same height (fig. 4 & 6). Through linear perspective the viewer becomes aware of a tactile and spatial sensations through the sense of sight, which sees objects in perspective.

Perspective—perspectiva in Latin means "see through"—enables not a view of the world of objects in space, but rather, a view of the world as it would appear seen through a theoretical window with an immobile eye, and drawn on the window pane.

Specifically, a perspective projection is an intersection of a plane, called the picture plane in painting, with a virtual pyramid or cone of sight (fig. 6 & 7). As Leonardo da Vinci wrote, perspective is "nothing else than seeing a place behind a plane of glass, quite transparent, on the surface of which the objects behind that glass are to be drawn. These can be traced in pyramids to the point in the eye, and these pyramids are intersected on the glass plane."

Assuming the a photographic camera has a simple single lens, and the viewing screen is flat, then the perspective geometry of the image will be exactly the same as that of a conventional perspective drawing. Paradoxically, perspective, which is indispensable for creating highly realist images, does so through by the distortion, often extreme, of tactile reality.

There are basically three types of linear perspective: one-point, two-point and three-point perspective. Vermeer employed both one- and two-point perspective in his interior scenes, sometimes simultaneously in the same composition. He did not use three-point perspective.

One-point perspective is typically used to draw the exteriors or interiors of buildings whose facades or background walls are directly facing the viewer (fig. 8). One-point perspective is highly adapted for representing objects within such environments that are positioned parallel or perpendicular to the viewer's line of sight and whose principal structural edges are at right angles to each other. All lines that are parallel to the picture plane, called transversals, are drawn as parallel lines on the surface of the image. All lines that run parallel to the viewer's line of sight, referred to as orthogonals, converge at the so-called vanishing point. The vanishing point falls on the horizon line.

"In brief, one-point perspective begins with a horizon line, which defines the farthest distance of the background and a central vanishing point. To this vanishing point, orthogonals may be drawn from the bottom of the picture plane, which defines the foreground of the space. The orthogonals, vanishing point, and horizon line establish the space in which the artist may arrange figures, objects, or architecture such that they appear to exist in three dimensions. Once these basic elements have been set in place, the artist may add further elements to create a more complicated, yet more realistic, space. For example, to represent a square-tiled floor, the artist chooses another point on the horizon line, called the distance point, and draws a line through the orthogonals to a point at the bottom of the picture plane. The points at which this line bisects the orthogonals establish the points at which horizontal lines, called transversals, may be placed. These lines represent the perspectively correct regression of the square tiles into space. These elements of linear perspective link the science of three-dimensional geometry with the art of illusionistic representation."^{Bruce MacEvoy, "Two Point Perspective," 2015. https://www.handprint.com/HP/WCL/perspect3.html.}

Instead, a two-point perspective drawing contains two vanishing points. Two-point perspective is used to draw the same type of objects that can be represented with one-point perspective, but rotated, for example, looking obliquely at the corner of a house (fig. 9) or the inside of a room. For objects viewed at an angle, the parallel lines of left-hand side of an object recede towards a vanishing point to the left of the central vanishing point (of the one-point perspective system) while those of the right-hand side converge at a vanishing point located to the right. As in one-point perspective, vertical lines that are parallel to the picture plane remain vertical. Distance points increasingly move away from the object as the observer recedes from the object; and vice versa, the points converge as an observer approaches it.

Two-point perspective greatly expands the range of pictorial effects and emotional associations of a perspective image, creating a palpable contrast between the viewer and the object, highlighting the accidental or arbitrary viewpoint of the viewer, or the distinctive physical presence of the object. ^{Richard Talbot, "Speculations on the Origins of Linear Perspective Including Analyses of Masaccio's Trinity and Piero's Flagellation," Nexus Network Journal, vol. 5, no. 1, 2003.}

When both one-and two-point perspectives are combined in the same drawing, all lines converge at vanishing points located on the horizon line. Objects with curves can also be rendered with one linear perspective but the process is much more complicated.

The principles of one-point perspective were eventually used to develop the shape of shadows cast by solid objects (fig. 10).

Three-point perspective is often 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 the vertical lines of the walls recede. For an object seen from above, this third vanishing point is below the ground. For an object seen from below, as when the viewer looks up at a tall building, the third vanishing point is high in space.

Three-point perspective exists when the perspective is a view of a Cartesian scene where the picture plane is not parallel to any of the scene's three axes. Each of the three vanishing points corresponds with one of the three axes of the scene. One, two and three-point perspectives appear to embody different forms of calculated perspective, and are generated by different methods. Mathematically, however, all three are identical; the difference is merely in the relative orientation of the rectilinear scene to the viewer.

The Role of Perspective Manuals in Practice

During the Renaissance and Baroque periods, the study of perspective was a crucial aspect of artistic training and practice. Perspective manuals played a significant role in teaching artists the principles of linear perspective, which was a fundamental technique for creating the illusion of depth and three-dimensionality on a two-dimensional surface. For many artists, the mastery of perspective was seen as a mark of their technical proficiency and understanding of the natural world. As artists strived to depict reality in increasingly accurate ways, the knowledge and application of perspective were key components of achieving that goal.

These manuals were not just theoretical texts; they were practical guides that provided artists with step-by-step instructions on how to construct accurate and convincing spatial representations. The most ambitious artists of the time were deeply interested in exploring the mathematical and optical principles that underpinned perspective, as it allowed them to achieve a higher level of realism and naturalism in their works.

Perspective manuals covered a range of topics, including one-point, two-point, and even three-point perspective, as well as techniques for depicting complex architectural structures, foreshortening and other visual effects that could enhance the illusion of depth. These manuals often included diagrams, illustrations, and geometric constructions to help artists understand how to apply these principles to their own artwork.