7.1: Projective Geometry

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Before drawing a 3-dimensional object on a flat surface, such as a sketch book or a canvas, one needs to learn some principles of perspective. This skill, based on projective geometry and developed during the Renaissance, allows the artist to draw things more realistically. All of us have seen, for instance, pictures of railroad tracks and straight highways where the two lines, though parallel in space, appear to converge to a point, referred to as the principal vanishing point.

Perhaps you have seen the famous art piece, The Last Supper, by Leonardo da Vinci (1452—1519). In that picture, the parallel lines on the ceiling are drawn in such a way that they, if extended, would converge in the middle of the picture, directly behind the head of Christ.

Figure \(\PageIndex{1}\): The Last Supper, by Leonardo da Vinci is in the public domain

During the 15th Century, many more artists started to use perspective, practicing it by sketching polyhedral and buildings, while exploring the mathematical basis and rules for such a technique. It was no surprise that da Vinci was also an accomplished mathematician, scientist, and engineer. Interested readers can look up other painters who contributed to the geometric theory of perspective, such as Filippo Brunelleschi (1377—1446), Leon Battista Alberti (1404—1472), Paolo Uccello (1397—1475), Albrecht Dürer (1471—1528), and Wenzel Jamnitzer (1508—1585).

An old sketch on faded paper. It shows a triangle with 10 lines extending from the vanishing point at the top to wide bottom side of the triangle. On the left of the triangle are a series of letters. A through K. A line extends diagonally from each letter to the bottom of the triangle. — Figure \(\PageIndex{2}\): Leon B. Alberti’s study on perspective is in the public domain.

In particular, Alberti is famous for the concept of “visual pyramid,” which gives us the foundation for perspective, as seen in this picture:

Figure \(\PageIndex{3}\): Visual pyramid (CC-BY-SA image by Martin Kraus)

Without going into all the details developed over the centuries, we now state some basic principles of perspective.

Four sets of railroads and powerlines hanging above them. The tracks extend so far in the distance that the tracks appear to converge in the background of the picture. — Figure \(\PageIndex{4}\): Image by Moajjem Hossain is licensed by CC-BY-SA-4.0

clipboard_e500695f74cd32d5ce6bac6554ce20de0.pngThis image shows a vanishing point in the center of the page. It is shown as a horizontal line. In the foreground are street lamps and a bench. It could be a train station or a bus stop. The lines of the 'street' in front of the bench are wide in the foreground, and they narrow to the point where they meet the vanishing point. — Figure \(\PageIndex{5}\): Image by mellowchu

This image shows a 3 dimensional cube in the foreground with one corner close to the viewer. In the background of the image is a horizontal line. From each end of this line, a red line extends to each corner of the cube. Where these red lines converge on the horizontal line, it is labeled 'vanishing point'. — Figure \(\PageIndex{6}\): Image by Wikimedia Commons is licensed by CC-3.0

Principles of Perspective

As the first two figures show, all lines perpendicular to the canvas (the plane on which the objectives are drawn) are parallel in space and will converge at a point, called the “principal vanishing point.”
All lines parallel to the canvas, such as railroad crossties (wooden pieces below the railroad track) and horizontal edges of the floor tiles, will not converge on the canvas but stay parallel and horizontal on the canvas.
All other lines parallel in space will converge at their vanishing points, as shown on the third picture, and all of the vanishing points are on the horizontal line (which also contains the principal vanishing point).

Knowing these three principles, or rules, of perspective will help you draw objectives in 3-dimensional space in a much more realistic way.

Reference

References (16)

Contributors and Attributions

Saburo Matsumoto
CC-BY-4.0