Degrees
As noted above, the inverse of a curve of degree n has degree at most 2n. The degree is exactly 2n unless the original curve passes through the point of inversion or it's circular, meaning that it contains the circular points, (1, ±i, 0), when considered as a curve in the complex projective plane.
Specifically, if C is p-circular of degree n, and if the center of inversion is a singularity of order q on C, then the inverse curve will be an (n − p − q)-circular curve of degree 2n − 2p − q and the center of inversion is a singularity of order n − 2p on the inverse curve. Here q = 0 if the curve does not contain the center of inversion and q = 1 if the center of inversion is a nonsingular point on it; similarly the circular points, (1, ±i, 0), are singularities of order p on C. The value k can be eliminated from these relations to show that the set of p-circular curves of degree p + k, where p may vary but k is a fixed positive integer, is invariant under inversion.
Read more about this topic: Inverse Curve
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