Eccentricity of a Conic This animation displays the effect of variations in the eccentricity of a conic section. We have fixed one vertex of the conic (the black point located at (-1,0)), and one of the foci (the fixed red point at (0,0)). The moving red point is the other focus.

These choices produce the polar equation for the conic, where the absolute value of the parameter e is the eccentricity. If converted to rectangular form, the equation of the conic is

To produce the display, we have allowed the parameter e to vary among all real numbers, both positive and negative. Recall that the conic with eccentricity larger than one is a hyperbola, and the conic with eccentricity less than 1 is an ellipse. A circle has eccentricity zero, and a parabola has eccentricity one.

The discontinuity that occurs when the hyperbola changes into the ellipse (with the momentary display of a horizontal line) is worth some discussion. Using the rectangular form of the equation, this horizontal line occurs when . The same value of e in the polar form produces a zero radius for all angles excepting 180 degrees, when the value is undefined. We have chosen to display the rectangular result.

For those of you who like a little "op art", we offer the 48 frames of the conic superimposed.