Shadows revealed: mathematics and vision | Tangente

"The sun never sees a shadow," said Leonardo da Vinci, showing that the concept of shadow was already important in his time. Yet it had acquired significance long before him, having been used to measure inaccessible heights: ancient surveyors had understood that seeing an object's shadow already gives some idea of its dimensions. During the Renaissance, shadows became a tool for theorists of perspective, who knew that seeing an object's shadow already means recognizing it to some extent—and therefore knowing how to draw it with the illusion of reality, or, as we would say today, "in 3D."

Several centuries later, computer scientists used self-shadowing to convey objects' three-dimensional form on screen: who could forget the famous "Utah teapot," created by Martin Newell in 1975 and now a genuine cult object in the world of computer graphics? The concept of shadow therefore has a profound influence on how we perceive the space around us.

A famous computer graphics model: the Utah teapot.

Shadows in sunlight -------------------

Using shadows cast by the sun to measure inaccessible distances is an ancient practice, dating back to around the 6^th century BCE. Several sources report that Thales not only travelled to Egypt but also measured the height of the pyramids "by observing the length of their shadow at the moment when our own shadow is equal to our height." The idea? Because the sun's rays are parallel, we can use triangles—what we will later call similar triangles. The method? According to Plutarch, Thales showed, "by planting a stick at the end of the pyramid's shadow and forming two triangles through the action of the sun's rays, that the pyramid bore the same ratio to the stick as the pyramid's shadow did to the stick's shadow." Carrying out the calculation gives:

Since the sun's rays are regarded as parallel, a shadow is precisely the image of an object under parallel projection onto the ground—in other words, its drawing in parallel perspective. Knowing a shadow means already knowing how to move from three-dimensional space to the plane, and thus how to draw in perspective. The shadow of a line segment is always a line segment, and midpoints are preserved. The construction then follows naturally.

Sunlight streams through the window. The window with panes ABCD projects an illuminated area A'B'C'D' onto the adjacent wall, while its glazing bars cast shadows.

Shadows by torchlight ---------------------

The situation is different for a shadow said to be cast by torchlight, because here all the light rays originate from a single point, the torch. But knowing how to draw the shadow cast under these conditions means knowing how to draw an image by central projection. Shadows by torchlight thus correspond exactly to central perspective, so well illustrated by Dürer (1471–1528) with his device known as the Dürer window: a sighting hole fixes the position of the operator's eye, with a window placed between that eye and the object to be depicted. Each point of the object is represented by the intersection of the window's plane with the light ray joining that point to the eye, just as the shadow of a point is the intersection of the ray emitted by the torch and passing through that point with the plane of the screen—the wall, the ground or any other flat surface—onto which the shadow is cast.

The techniques of shadow drawing, which "flatten" a given three-dimensional object into two dimensions, are therefore also those of perspective drawing: it always pays to know how to draw (lines) faster than your shadow…

The shadow square -----------------

The shadow square is a surveying instrument, often engraved on the back of astrolabes, that came into use in the 9^th century. Its first mention in the literature appears in al-Khwarizmi's writings. It really is a square, and its name was inspired by the analogy with measuring heights from the shadow of a stick.

How is the shadow square used? In this illustration, an observer can use it to determine the height of a tower. Each side is divided into twelve equal parts. The observer stands a known distance from the foot of the tower and sights its top with the shadow square. This reproduces a right triangle whose proportions are the same as those of the triangle formed by the observer's position on the ground, the tower and the line of sight. As long as the angle of sight is less than 45°, the vertical scale (versa) is used; if it is greater, the horizontal scale (recta) is used. In the image, the angle of sight is less than 45°. Standing 120 paces from the foot of the tower, the observer reads an umbra versa of 10 (out of 12). This means that the tower's height is 10/12^ths of the distance of 120 paces, so the tower is 100 paces high.

References: Les angles (Angles). Bibliothèque Tangente 53, 2015.

Speaking of shadows -------------------

The word "shadow," which plays a central role in how we see space, also features prominently in the way we speak, and many expressions in our language include it. How many can you count in this short text?

"In looking for this holiday rental—in other words, a shady spot for the summer—you have, without a shadow of a doubt, given up the substance for the shadow and given up your comfortable apartment for a dubious seaside studio with a shaded terrace. The only shadow over the picture is that the owner, who clearly operated in the shadow of criminals, has just been put away… So his schemes have been exposed and brought out of the shadows—but what about you? And the deposit you paid under the table? Trying to recover it would be like chasing your own shadow. And the crooked landlord's irritable agent, that prickly character—you certainly do not like him casting a shadow over you! I can just see you flitting like a shadow beneath his windows, your head full of thoughts of revenge. You are already a mere shadow of your former self; there is no point putting on your own shadow play, when you were already afraid of your own shadow… So go home and… stay in the shade!"