Parabola as a Negative Pedal

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Pick a curve (call it C) and an arbitrary point, which we'll call P. Draw a line from the point to the curve, and where they intersect, draw a line perpendicular to the first line. If you do this for all possible lines between the point P and the curve C, the perpendiculars will outline a new curve which is called the negative pedal of C with respect to P.

The negative pedal of a straight line with respect to a point not on the line is a parabola. The line is called the directrix of the parabola, and the point is the focus of the parabola.If you line the parabola with mirrors and shine a light into the parabola from a long distance, the light will all be reflected to the focus. This property makes the parabola immensely useful as a shape for telescope mirrors and communications antennas.

In the figure below, the directrix is in black, the focus is blue, radii from the focus to the directrix are green and the perpendiculars to the radii are in red.

Focus-Directrix Distance: Angles Between Radii (Degrees):

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Created 18 January 2008, Last Update 11 February 2012

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