A curve and its polar reciprocal with regard to the fixed conic have the same Halphen transformation.
Halphen's Transformation
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 346-347, 1959.Referenced on Wolfram|Alpha
Halphen's TransformationCite this as:
Weisstein, Eric W. "Halphen's Transformation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HalphensTransformation.html