Inverse curve

Similarly, the inverse of the curve defined parametrically by the equations with respect to the unit circle is given parametrically as This implies that the circular inverse of a rational curve is also rational.

Results for other circles of inversion can be found by translation and magnification of the original curve.

For a line passing through the origin, the polar equation is θ = θ0 where θ0 is fixed.

To summarize and generalize this and the previous section: The equation of a parabola is, up to similarity, translating so that the vertex is at the origin and rotating so that the axis is horizontal, x = y2.

In polar coordinates this becomes The inverse curve then has equation which is the cissoid of Diocles.

The polar equation of a conic section with one focus at the origin is, up to similarity where e is the eccentricity.

The general equation of an ellipse or hyperbola is Translating this so that the origin is one of the vertices gives and rearranging gives or, changing constants, Note that parabola above now fits into this scheme by putting c = 0 and d = 1.

This family includes, in addition to the cissoid of Diocles listed above, the trisectrix of Maclaurin (d = −⁠c/3⁠) and the right strophoid (d = −c).

Examples include the circle, cardioid, oval of Cassini, strophoid, and trisectrix of Maclaurin.