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The Mathematics of Perspective: An Introduction to the Cross Ratio

David A. Thomas

Department of Mathematics

University of Idaho

History

Projective geometry was developed by Renaissance artists and architects faced with a common problem: The geometric features of buildings and landscapes appear to change depending on one's point of view, or perspective. For instance, lines which are known to be parallel appear to converge in the distance, and the near side of a rectangular wall appears to be longer than the far side. Steps at the bottom of a staircase appear larger than steps at the top. Effects of this sort are an inevitable consequence of projecting a 3-dimensional object onto the human retina, a 2-dimensional surface. Figure 1 illustrates a few of these effects.

² Using the mouse key, move the drag point in Figure 1 first to the right then to the left. How does the view of the cube change?

² Using the mouse key, move Vanishing Point 1 and Vanishing Point 2. Describe the apparent changes in your view of the cube.

² Using the mouse key, move Point E. Describe the apparent changes in your view of the cube.

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Figure 1 Perspective View of a Cube [GSP File]

The first person to develop a systematic approach to perspective drawing was Leone Battista Alberti. Alberti was born in Genoa, Italy in 1404, second son of a wealthy merchant. Alberti acquired his early education at Gasparino Barzizza's Gymnasium in Padua, graduating in 1421. From there, he went to the University of Bologna, where he studied law but excelled in literature and geometry. After graduating in 1428, Alberti worked for several years as a secretary in the Papal Chancery in Rome, writing biographies of the saints in elegant Latin and traveling extensively in Europe. By 1432, he was living in Florence, working with famed artists Brunelleschi and Donatallo. He also collaborated with Toscanelli on the development of maps later used by Columbus on his first voyage in 1492.

Of his many interests, Alberti was most enthusiastic about the development of a geometrical basis for perspective drawing. In 1434 he wrote the book, Della Pittura, the first written exposition on how to add a realistic third dimension to paintings. In it he said, "Nothing pleases me so much as mathematical investigations and demonstrations, especially when I can turn them to some useful practice drawing from mathematics the principles of painting perspective and some amazing propositions on the moving of weights." Alberti's influence on the development of Renaissance painting was significant and long-lasting. Leonardo da Vinci is know to have taken passages directly from Della Pittura and incorporated them into his Trattato (a common and legal practice at the time). He also greatly extended Alberti's ideas and techniques. Alberti's ideas did more than advance the development of painting, however. They revitalized the study of geometry. By introducing a new geometry, students of science were suddenly able to view Euclidean geometry as a geometry, rather than as the geometry. Few thinkers have enabled such a profound change in perspective. Alberti died in Rome in 1472.

Alberti's method for creating a perspective view is illustrated in Figure 2. In this figure, the array of tiles seen in the Map View is converted into the array seen in the Perspective View. In the Perspective View, the eye is naturally drawn into the center of the image by the apparent convergence of parallel lines in the distance and by the realistic scaling of the tiles positioned different distances from the observer. Because there is only one vanishing point, the view is called 1-point perspective.

Figure 2 Alberti’s Method

Alberti knew that his method worked, in the sense that it produced realistic view of three dimensional objects. What he didn’t know was why it worked. This article presents the key mathematical concept behind Alberti’s method, the cross ratio.

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