|Ivars Peterson's MathTrek|
January 7, 2002
If you were a surveyor, you could measure angles and then use high-school trigonometry to calculate distances. That's great for drawing a map or establishing property lines, but it's more work that you want to do for just an on-the-fly estimate.
One cue that you might use is the angle of an object with the ground--in effect, where the object is in your field of view. If you have to look down toward your feet to see it, you can assume that it's close to you. If you have to peer toward the horizon to view it, it's far away. In effect, your distance estimate depends on the angle between your line of sight to the horizon, which is at eye level, and your line of sight to the object of interest. This quantity is known as the angular declination.
Researchers now have evidence confirming that people actually use the angular declination as one cue to decide how far away something is.
Teng Leng Ooi of the Southern College of Optometry in Memphis, Tennessee, and her coworkers set up an ingenious experiment in which volunteers wore goggles equipped with prisms and looked at illuminated objects against a dark background. Changing the direction in which light travels, the prisms increased the angular declination, making objects appear lower in the field of view than they actually were. Volunteers wearing the prism goggles consistently said objects were closer than they really were. When they tried to walk toward the targets after being blindfolded or threw beanbags at them, they nearly always missed.
Interestingly, when volunteers were allowed to adapt to the prism goggles beforehand by walking around in a lighted room while avoiding obstacles, they tended to judge distances correctly. Then, when they took off the prisms, they temporarily went the other way by overestimating distances and overshooting objects when they walked toward them.
“Our study provides direct support for the angular declination hypothesis, that the visual system can access the angular declination below the horizon to determine absolute distance,” the researchers concluded in the Nov. 8, 2001 Nature. “Our study reinforces this notion by relating the changes in the angular declination below the horizon to perceived eye level, and demonstrating that eye level serves as a reference for the visual system to compute the angular declination below the horizon.”
“This general topic is not only of intrinsic interest,” psychologist Jack M. Loomis of the University of California, Santa Barbara commented in the same issue of Nature. “It also helps us to understand how we move and act in three-dimensional space, and is central to developing realistic computer graphics, including virtual-reality displays.”
Copyright 2002 by Ivars Peterson
Beckman, M. 2001. A new angle on sight. Science Now (Nov. 9). Available at http://www.academicpress.com/inscight/11092001/grapha.htm .
Klarreich, E. 2001. New angle on vision. Nature Science Update (Nov. 8). Available at http://www.nature.com/nsu/011108/011108-11.html.
Loomis, J.M. 2001. Looking down is looking up. Nature 414(Nov. 8):155-156.
Ooi, T. L., B. Wu, and Z.J. He. 2001. Distance determined by the angular declination below the horizon. Nature 414 (Nov. 8):197-200.
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A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book Mathematical Treks: From Surreal Numbers to Magic Circles.