A graphic for exploring Watt’s curve:
The entire curve is given in polar coordinates by
\[ r^2 = b^2 - \left[ a \sin\theta \pm \sqrt{ c^2 - a^2 \cos^2\theta } \right]^2 \]Since b and c are both squared they can be assumed to have positive values. The parameter a is effectively positive as well, since altering its sign merely interchanges parts of the curve.
Unfortunately a straightforward attempt to display this curve as a polar plot goes awry, because the expression as given becomes complex over significant portions of the angular domain. It is simpler to expand the definition to remove the square root
\[ r^2 ( r^2 - a^2 - b^2 + c^2 )^2 + 4 a^2 y^2 ( r^2 - b^2 ) = 0 \]and then visualize the curve as an isoline using marching squares.
Complete code for this example: