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.
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).
In particular, Alberti is famous for the concept of “visual pyramid,” which gives us the foundation for perspective, as seen in this picture:
Without going into all the details developed over the centuries, we now state some basic principles of perspective.
Principles of Perspective
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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.”
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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.
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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.
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Contributors and Attributions
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Saburo Matsumoto
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