Step By Step Calculus » 14.2 - Sketching Polar Curves

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Synopsis
For drawing a curve from a polar equation, one can either
  • evaluate values of rr for several values of \theta\theta, then plot the (r,\theta)(r,\theta) pairs and connect them by a smooth line to form the polar curve,
  • or, draw the curve in the Cartesian co-ordinates of rr and \theta\theta, and then retrace them in polar co-ordinates.
In either case, the following symmetry test can be helpful.
  • Symmetry about the polar axis exists if a polar equation is even in \theta\theta.
  • Symmetry about the vertical line \theta=\pi/2\theta=\pi/2 exists if a polar equation remains the same on the change of \theta\theta by \pi-\theta\pi-\theta.
  • Symmetry about the pole exists (i.e., rotating the curve by \pi\pi radian doesn’t change the curve), if a polar equation remains the same on the change of rr by -r-r (i.e. even in rr), or \theta\theta by \pi+\theta\pi+\theta.
Note that passing a symmetry test is a sufficient but not necessary condition for a polar equation to exhibit that type of symmetry.