Area-Invariant Pedal-Like Curves Derived from the Ellipse
This work addresses a theoretical geometry problem for mathematicians, offering incremental insights by extending known properties from Steiner's 1825 result to new cases.
The paper tackles the problem of identifying area-invariant pedal-like curves derived from an ellipse, proving that six such curves maintain constant area when the pedal point lies on either a concentric circle or the ellipse boundary, with explicit expressions provided for all invariant areas.
We study six pedal-like curves associated with the ellipse which are area-invariant for pedal points lying on one of two shapes: (i) a circle concentric with the ellipse, or (ii) the ellipse boundary itself. Case (i) is a corollary to properties of the Curvature Centroid (Krümmungs-Schwerpunkt) of a curve, proved by Steiner in 1825. For case (ii) we prove area invariance algebraically. Explicit expressions for all invariant areas are also provided.