Polar Plots of Conic Sections
Conic sections (the circle, ellipse, parabola and hyperbola) can all be represented by an equation in polar coordinates:[more]
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.[less]
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
The semilatus rectum is equal to when . We obtain thereby