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Eccentricity of a Conic

Animation of graphs of a conic with a varying eccentricity 


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   $r(\theta)=\dfrac{1+e}{1-e\cos\theta}$   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   $\dfrac{\left(x-\frac{e}{1-e}\right)^2}{\left(\frac{1}{1-e}\right)^2}+ \dfrac{y^2}{\left(\sqrt{\frac{1+e}{1-e}}\right)^2}=1$.

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   $e=-1$.   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.

48 superimposed conics