The technique of linear perspective allows artists to simulate or construct the appearance of three dimensional space on a two dimensional surface. It is one of the major innovations of European art, with an extraordinary impact on western visual culture from the 15th to the 19th centuries.

Linear perspective is the artist's slide rule, an antique technology that has been replaced by film or digital photography, opaque projectors, and computer assisted design and animation software. It lives on primarily in art historical monographs, architectural visualizations and the digital animation of intergalactic space.

So why bother with it? Knowledge of perspective greatly enhances your perception and understanding of light and space, it attunes you to spatial recession as the power line of visual design, it is a powerful guide to drawing in all situations, and it is a fascinating case study of the ways that a painting is shaped by purely conceptual considerations. It is also indispensible to understand the design problems that inspired and guided artists of the past.

Although the Greeks and Romans could draw foreshortened limbs or architectural edges, the construction methods of linear perspective were first developed and used by 15th century Florentine artists with strong mathematical skills: the visionary architect Filippo Brunelleschi (1377-1446) and the humanist scholar Leon Battista Alberti (1404-1472). They systematized the Tuscan workshop methods of pseudoperspective or near perspective which are already apparent in frescos by Giotto and Duccio. (It's interesting that technically advanced northern Italians were working out the geometry of linear perspective at about the same time that optically aware artists in the Low Countries were adopting projection devices to create more realistic pictures.) In 1435 Alberti wrote the Latin manuscript De Pictura (On Painting), which he revised and translated into Italian as Della Pittura in 1436. This was the first book to describe perspective methods and the visual design of paintings, and within a century artists such as Piero della Francesca (c.1470), Leonardo da Vinci (c. 1490), Albrecht Dürer (1525) in Germany and architect Jean Pélerin (1505) in France wrote more ambitious perspective studies. During the 16th and 17th centuries linear perspective diffused across Europe and was honed through the analysis of specific representational problems: illusionistic frescos, foreshortened human figures, architectural interiors, and geometrical solids (often illustrated as intarsa or inlaid wood designs). In the 17th century the methods became thoroughly grounded in projective geometry, which was however too complex for artists to use. No matter: artists meanwhile had continued using perspective in a variety of innovative ways, often by combining LP drawings of architectural spaces with informal free perspective drawings of figures and objects -- a balancing act that Raphael brought to perfection. Perspective became one of the core studies in art academies, which taught a tedious and formulaic form of perspective to its extinction. Near its demise, artists such as J.M.W. Turner made interesting explorations in optical color mixing and aerial perspective, preparing the break from perspective traditions in the 19th century.

In the traditional account, Brunelleschi either discovered or verified linear perspective by literally painting views of buildings onto a mirror or windowpane. This "draw on glass" procedure was probably not Brunelleschi's method, but it soon became a standard way to convince drawing students that perspective really works -- though the landscape of the problem was often reduced to a table top.

one more convert to perspective
from Charles Hayter, Introduction to Perspective (1813)

Want to try it? Punch a small hole in an index card, staple or tape the card to one end of a wood yardstick, then tape the other end of the yardstick to the back of a chair, so that the hole in the card is at your eye level as you straddle the chair. Put the chair in front of a large window, sit, and look through the hole in the card with one eye. Use an erasable marker pen or grease pencil to draw on the glass the buildings or objects in view, then hold a white paper behind the glass to see your finished design. As you draw, you may notice that this procedure feels confining. That's because it is -- illustrating how awkward perspective drawing can become if it's pushed too far.

a renaissance perspective machine
from Albrecht Dürer, Underweysung der Messung mit Zirckel und Richtscheyt ... [Instruction How to Measure with Compass and Straight Edge ...] (1525)

The peephole or "peep show" tracing method does not make clear that linear perspective is really created by straight lines passing through a vertical plane. For that, the perspective "machine" by the German artist Albrecht Dürer (1471-1528) is a better example. By looping a weighted string through an eye ring in the wall, one artist could hold the opposite end along the contours of a lute; a second artist could measure the height and horizontal position of the string where it passed through a rectangular frame, then transfer this point to a paper or canvas (shown as a hinged panel). By repeating the measurements dozens or hundreds of times, a "connect the dots" perspective view of the three dimensional lute could be constructed on a two dimensional surface. Through similar empirical methods, Renaissance artists worked out the geometric fundamentals of perspective technique.

Dürer's machine demonstrates that linear perspective explicitly defines a physical point of view -- it shows the location, vantage and orientation of the viewer as precisely as it shows the physical form of the objects in view. The eye ring represents the eye of the viewer or center of projection; the stretched string represents the straight beams of light or visual rays that converge on the eye from all objects in the field of view; the hinged surface is the artist's canvas or image plane. Everything comes down to that point of view defined by the wall ring or hole in index card: the point of view, not objects in space, is the fundamental perspective theme.

LP is fundamentally a geometrical method, not a mathematical one. This simply means you draw, you don't calculate. All you need are the standard construction tools from high school geometry -- a straight edge (ruler), pencil and a compass (or a long strip of cardboard and some push pins). Unfortunately, these tools are awkward to use when the perspective view is extreme or the drawing is very large. So I explain numerical calculations that can help you through those problems, and provide a simple vanishing point calculator. But these are not necessary to apply LP methods in your drawing.

There is a sprawling and stale literature on linear perspective, but two points deserve mention. As James Elkins and Martin Kemp point out, the historical uses of perspective construction have always been opportunistic and evolving. Many seemingly "obedient" perspective paintings reveal, on closer examination, "fuzzy" or multiple vanishing points, objects that are "in perspective" and objects that are not. Most artists had to make "corrections" to vexing projection distortions, often in pursuit of a more pleasing finished composition: linear perspective often creates more design problems than it solves. In that sense, LP is not any better or worse at making "realistic" images of objects than the various types of parallel perspectives used by architects and engineers -- elevation, section and plan, or military, cavalier, isometric and diametric projections. In all these, similar distortions are accepted and ignored. The moral is that perspective must be used gently: it is a very cool drawing tool, not the ultimate code of drawing correctness.

The best approach is to rely on LP to lay out basic proportions and shapes, to clarify difficult drawing problems, and to guide your intuitions about forms and textures in space. Freehand perspective and trust in your visual judgment should not be pushed aside, but nourished and exercised by the insights linear perspective can provide. As Vasari wrote of Michelangelo, "he held his compasses, that is to say his judgment, in his eyes and not his hands."