Friday, October 4, 2013

Reflective Property of the Parabola

All rays emanating from the focus of a parabola reflect off the parabola along parallel lines (each perpendicular to the parabola's directrix, not shown). 

Note how each ray "lines up" with the other rays after it has reflected off the parabola. The reason for this is related to the fact that rays emanating from one focus of an ellipse arrive at the other focus at the same time, illustrated in this post

A parabola is just a special ellipse. A parabola does not have just one focus; rather, it has two, but the other is located "at infinity." A future post will illustrate how an ellipse can be stretched to form a parabola. Therefore all rays emanating from "the" focus of a parabola travel toward the other focus (at infinity) and will arrive there at the same time, just as happens in the ellipse. 

Of course, this can sound like pure silliness - after all, what does it mean to "arrive at infinity at the same time"? One can rigorously defend this statement using limits, which might make for an interesting animated gif ...


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