TobyBartels on Nostr: : But notice that \( (x^2 + y^2)^2 = x^2 - y^2 \) is not analogous to \( x^2 + y^2 = ...

: But notice that \( (x^2 + y^2)^2 = x^2 - y^2 \) is not analogous to \( x^2 + y^2 = 1 \) but rather to the rightward-shifted half-unit circle \( x^2 + y^2 = x \). Whereas on \( x^2 + y^2 = 1 \), the coordinates of a point are the sine and cosine of a related arclength; on \( x^2 + y^2 = x \), some related arclengths are the sine and cosine of the distance of a point from the origin. And on \( (x^2 + y^2)^2 = x^2 - y^2 \), the analogous arclengths are the lemniscate sine and lemniscate cosine of the distance of a point from the origin. See https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions#Relation_to_geometric_shapes for diagrams.