Tangent and Supporting Lines, Envelopes, and Dual Curves

TL;DR

This paper explores the mathematical relationships between differentiable curves, their tangent lines, dual spaces, and transformations, providing methods to find dual curves and analyze their properties.

Contribution

It introduces new techniques for transforming between original and dual spaces using various tangent line forms and explores geometric methods for finding dual curves.

Findings

Transformation between x,y-space and dual space depends on tangent line form

Methods to find dual curves from tangent lines are developed

Different tangent line representations lead to different dual spaces

Abstract

A differentiable curve y = y(x) is determined by its tangent lines and is said to be the envelope of its tangent lines. The coefficients of the curve's tangent lines form a curve in another space, called the dual space. There is a transformation between the original x,y-space and the dual space, such that points on the original curve and the curve of the coefficients of the tangent lines are transformed into each other.The dual space and the transformation depend upon the form that is used for the tangent lines. One choice is y = mx + b, so that the coordinates in the dual space are m and b, where the curve representing the tangent lines is b = b(m). Each point of the curve b = b(m) in the dual space corresponds to a tangent line to the curve y = y(x) in x,y-space. We present other choices of the form for the tangent lines and explore techniques for finding a curve from the tangent lines, transformations between the original space and a dual space and among dual spaces, the differential equation, whose solutions are exactly the equations of the equations of the tangent lines and the original curve. Geometric constructions and other tools are used to find dual curves.