# Evolving Affine Evolutoids

## TL;DR

This paper explores how affine evolutoids, which are envelopes of affine normal lines to a plane curve, transition into affine tangents as the slope parameter varies, using singularity theory to analyze the process.

## Contribution

It introduces a singularity theory framework to describe the evolution of affine evolutoids into affine tangents as the slope changes from 0 to 1.

## Key findings

- Existence of a critical slope where singularities occur.
- Description of singularity evolution on the discriminant surface.
- Guarantee of the first slope value with singularities.

## Abstract

The envelope of straight lines affine normal to a plane curve C is its affine evolute; the envelope of the affine lines tangent to C is the original curve, together with the entire affine tangent line at each inflexion of C. In this paper, we consider plane curves without inflexions. We use some techniques of singularity theory to explain how the first envelope turns into the second, as the (constant) slope between the set of lines forming the envelope and the set of affine tangents to C changes from 0 to 1. In particular, we guarantee the existence of the first slope for which singularities occur. Moreover, we explain how these singularities evolve in the discriminant surface.