Constructing Cubic Curves with Involutions
Lorenz Halbeisen, Norbert Hungerb\"uhler
TL;DR
This paper revisits Schroeter's ruler construction for cubic curves, providing a simplified proof using elliptic curve theory and extending the method to construct tangents and additional points.
Contribution
It offers a new, simplified proof of Schroeter's construction and introduces methods to construct tangents and points on cubic curves using involutions.
Findings
Simplified proof of Schroeter's ruler construction
New ruler method for constructing tangents on cubic curves
Extension to constructing additional points on cubic curves
Abstract
In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles' Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter's construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions.
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