History and Significance for understanding vision

Poggendorf's illusion was discovered in Johabb C. Poggendorf, an editor of a Physics and Chemistry journal, in 1860. The illusion is based upon Zoellner's Illusion, another illusion that causes the viewer to misinterpret the angles of lines. In the illusion to the left, all the longer lines are parallel to each other, however, they appear askew. When this illusion was described to Poggendorf, he realized that people  "have a very difficult time being able to visually determine the differences in length between vertical and horizontal lines at varying angles to one another. ⁽¹⁾"

There are 2 main theories used to explain for Poggendorf's Illusion: Angular Displacement, and Depth Processing Theory.
 

When the bar covers the set of lines, it creates 2 acute angles and 2 obtuse angles as labeled. It is theorized that our visual system may be over estimating the degree of the acute angles and underestimating the degree of the obtuse angles. This would cause the diagonal lines to appear to bend away from each other and therefor not connect. (The orange line connects to the green line in the image to the right.)
The visual system is capable of this angle overestimation as proved by Hubel and Wiesel. They found that they could explain angle overestimation in a similar manner that lateral inhibition is explained. The visual system contains many different cells that respond very well to orientations. There are different cells that respond to different orientations. When presented with acute angles, some cells will respond vigorously while others will remain at their normal firing rate and yet others will fire at a below normal rate (inhibition). When the visual system is presented with an obtuse angle, those cells that responded to the acute angle will remain at their steady firing rate, while some of the cells that did not previously respond, will fire more vigorously, and other cells (those that are more attuned to other angles) will fire at a rate below normal. 
Placing an obtuse angle next to an acute angle can lead to lateral inhibition, where the various cells that are giving responses interact with each other and cancel out each other's responses. If one side's response and inhibition is stronger than the other, this can lead to an angle appearing larger or smaller than it really is. ⁽²⁾
However, angle displacement does not fully explain Poggendorf's Illusion, for if it did, removing the angles should diminish the effect of the illusion, and as you can see by the image on the right, it does not.
 
 

As you can see by the figure on the left, the disjointed line effect is greatly diminished when the receding lines are behind it. The receding lines add the illusion of depth and give the viewer the impression that the image is a standing block with a line placed at an angle behind it. The effect is even further diminished when the block is placed in the same plane as the receding lines. However this last image is not the same as Poggendorf's Illusion and so cannot be used to fully explain the effect of the illusion. ⁽³⁾