Perspective | Research & Encyclopedia Articles

Perspective

Mathematical perspective is the realistic representation of a three dimensional object on a two dimensional surface. In a perspective drawing, vertical lines in an image are drawn as vertical lines on the paper or canvas. Parallel horizontal lines, however, are drawn as oblique lines that converge to a single point, known as the vanishing point. A projection is a mapping of a geometrical figure onto a plane according to certain rules. The history of perspective and projection is an interesting one whose origins lie not in the field of mathematics, but in art.

Medieval paintings are noted for their lack of visual depth. Artists from those times focused on themes, using symbolism within a painting to convey a message. They did not attempt to create visual realism. The third dimension--depth--was not recreated. As a result, figures in medieval paintings looked flat. As the middle ages gave way to the Renaissance, artists became concerned with portraying the natural world as the human eye sees it. Toward that end, artists worked to develop techniques for accurately portraying the three dimensional world on a two dimensional canvas. The attempts of early Renaissance artists to master the third dimension led directly to the development of mathematical rules of perspective.

One of the pioneers of mathematical perspective was Leone Battista Alberti. His della Pittura, published in 1435, is the first known written account of mathematical perspective. Alberti described the following method for achieving the proper perspective in a painting. He placed an upright glass panel between the scene to be painted and his eye. Holding his eye in a fixed position, he imagined rays of light running from his eye to each point in the scene itself. Alberti referred to this set of rays as a projection. The intersection of this projection with the glass panel produced a set of points, which Alberti called a section. By painting that section on the glass panel, or on a canvas, Alberti was able to create a painting of a scene that gave the same impression on the human eye as the scene itself. This system worked well for creating realistic, three dimensional paintings of actual scenes. It did not work for recreating imagined scenes, however, as no glass panel could be set up between the eye and the imagination. Alberti recognized this problem, and thus recognized the need for a mathematical system of perspective that could accurately and consistently convey realism in painting.

Alberti made great strides in developing such a system. He applied his findings to the specific task of recreating checkerboard tiling patterns that existed in the ground plane. He was able to create a painting of a checkerboard floor that gave the proper appearance of depth. This was an important milestone in the development of mathematical perspective. However, Alberti's technique had one major limitation--it only worked for figures that existed in the ground plane. Though he could create a painting of proper depth of a tiled floor, he could not do the same for non-horizontal figures, such as a person or a building. Despite the limitations of Alberti's system, it introduced two concepts crucial to the development of an accurate system of mathematical perspective--vanishing points and horizon lines.

The Italian artist Piero della Francesca expanded Alberti's technique and perfected the science of perspective. In his work De prospectiva pingendi, published in 1478, Francesca outlined his system for creating three dimensional images on a two dimensional surface. The basic tenets of his system are as follows:

• A straight line in perspective remains straight.
• Parallel lines either remain parallel or converge to a single point. This point is known as the vanishing point.
• Vanishing points exist on a line known as the horizon line.

These rules, developed and refined by artists over five hundred years ago, are the same rules used by artists today to create proper perspective in paintings.

It may seem strange that what, in hindsight, appears to be a math problem was tackled by so many artists. However, many people in Renaissance times pursued more than one discipline. Artists were not just artists. Artists of the day were also the architects, engineers, and scientists of the day. As such, it was important for them to be able to create drawings and paintings with the proper appearance of depth. Scientists during the Renaissance times were also artists. To those scientists, art was a means of understanding the world around them. Being able to accurately portray the natural world on canvas was not just an artistic problem, but a scientific one as well. Thus, the pursuit of an accurate system of perspective was a mathematical one. None other than Leonardo da Vinci, the Renaissance genius whose painting The Last Supper is considered one of the finest examples of perspective painting, regarded painting as a science, because it revealed the reality in nature.

Girard Desargues, a seventeenth century French mathematician, extended the idea of mathematical perspective to an entirely new branch of mathematics. Desargues recognized that different perspective drawings can be created from the same scene by viewing the scene from different angles. Desargues asked a fundamental question: What do different sections of the same scene have in common? This question formed the basis for a new field of mathematics known as projective geometry. Projective geometry is the study of those properties of plane figures that are unchanged when a given set of points is projected onto a second plane. Though Desargues pioneered this field, his work went largely unnoticed for several centuries. French mathematician Jean Victor Poncelet helped revive interest in the field of projective geometry when he published the first known textbook on the topic in 1822.

One of the key differences between Euclidean geometry and projective geometry is their treatment of parallel lines. In Euclidean geometry, parallel lines never meet. In projective geometry, parallel lines meet at infinity. This difference is reconciled by the fact that Euclidean geometry is considered a special case of projective geometry.

Mathematical developments often arise out of attempts to solve particular problems. This is certainly the case with perspective and projection. A problem faced by Renaissance artists was ultimately solved through mathematical means. Not only did this result in the development of a mathematical system of perspective, it led directly to the creation of an entirely new field of mathematics--projective geometry. This, in turn, has been extended to many other fields, including cartography and aerial photography.