Ellipse 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 circle with respect to a point within the circle is an ellipse. The ratio of the distance from the center to the radius of the circle is the eccentricity of the ellipse. Obviously it must be greater than zero and less than one. The point is one focus of the ellipse. If you line the ellipse with mirrors and shine a light at one focus, the light will all be reflected to the focus.

As the eccentricity gets very close to 1, the radius of the circle becomes infinitely large compared to the distance from the focus to the circle. In the limiting case, the circle approaches a straight line and the construction approaches the case of a parabola.

The negative pedal approach is a quick and dirty way to sketch an ellipse or parabola if you know the focus and the major axis (for an ellipse) or the distance from the focus to the vertex (for a parabola). Use the corner of an index card to draw the tangent lines.

In the figure below, the circle is in black, the foci are blue, radii from the focus to the circle are magenta and the perpendiculars to the radii are in red.

Eccentricity; Must be between 0 and 1:
Angle between radii (degrees):

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Created 29 November 2010, Last Update 11 February 2012

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