If you look along a straight road, the parallel sides of the road appear meet at a point in the distance. This point is called the vanishing point and has been used to add realism to art since the 1400's in Florence, Italy (via city.net, no longer active).
Suppose you want to draw a railroad track that vanishes into the distance. The rays from the points a given distance from the eye along the lines of the tracks are projected to the eye. The angle formed by these rays decreases with increasing distance from the eye. The picture below shows an overhead view of an observer (camera or eye) looking down the the track.
The next picture shows a side view. The observer's eye or camera is above the ground.
Draw these pictures on graph paper and try to figure out where the points would fall on the plane of the drawing. Can you draw the railroad track?
To draw in perspective, draw a horizon line and draw a vanishing point anywhere on the horizon. Lines which are parallel in real life are drawn to intersect at the vanishing point.
Distant figures appear smaller but have the same shape and proportions as they would close up. In geometry, we would say that the figures are similar.
The picture below shows a long hallway with a window in the left wall. The window is a trapezoid. Can you use your knowledge of geometry to draw another window further down the hallway? An entire row of windows? To start with the simplest problem, assume the window tops are all at the same height in the hallway and assume the window bottoms are all at the same level in the hallway.
Albrecht Durer used perspective in his engraving St. Jerome dans sa Cellule. (A smaller image can be found at the Australian National University.) Look for lines in this picture that would be parallel in real life. Can you determine where the vanishing point is? Sometimes artists use more than one vanishing point, but that's another story.
Durer also worked in mathematics, particularly geometry. How many mathematical objects can you find in his engraving Melencolia I. While you're here, you may wish to visit the WebMuseum.
You can also trace lines to a vanishing point in this photograph of the Portico of Attalos in the Agora. The photo is part of the Ancient City of Athens exhibit, created by Kevin T. Glowacki and Nancy L. Klein.
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