Polar Plots of Conic Sections

Conic sections (the circle, ellipse, parabola and hyperbola) can all be represented by an equation in polar coordinates:
,
where is the semilatus rectum and is the eccentricity of the curve. For the circle, ellipse, parabola and hyperbola, the eccentricity has the values , , and , respectively.
One focus of such a conic is at the origin. After selecting values of and , you can use the "polar angle" slider to trace out the polar curve by sweeping from 0 to . For , the plot generates both branches of a hyperbola.

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A more general form for the polar equation is
,
where one focus is at the left-hand side of the conic for the minus sign (which we have used) but on the right-hand side of the conic for the plus sign. The alternative form
rotates the conics by 90°, with the semimajor axis oriented vertically.
The following illustration shows the derivation of the formula for the case of a parabola (), with the focus at and the directrix (the vertical line) on the left.
At every point on the curve, the defining relation is
or .
The semilatus rectum is equal to when . We obtain thereby
.
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