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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
When $d=0$ this curve is a nephroid, when $d=3r/2$ it is a Cayley sextic.
📌 3 Key Takeaways
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Objective & Hypothesis:In $\mathbb{R}^2$, let $\Gamma$ be a fixed circle with centre $C$ and radius $r$, and $\ell$ a straight line at distance $d$ of $C$.
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Key Findings & Data:We study the curve which is the envelope of the circles whose centre lies on $\Gamma$ and which are tangent to $\ell$.
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Conclusion & Impact:When $d=0$ this curve is a nephroid, when $d=3r/2$ it is a Cayley sextic.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Envelope · circles · tangent line
Abstract
In $\mathbb{R}^2$, let $\Gamma$ be a fixed circle with centre $C$ and radius $r$, and $\ell$ a straight line at distance $d$ of $C$. We study the curve which is the envelope of the circles whose centre lies on $\Gamma$ and which are tangent to $\ell$. When $d=0$ this curve is a nephroid, when $d=3r/2$ it is a Cayley sextic.
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Cite This Article
Vieli, F. J. G. (2026). Deforming the Nephroid into the Cayley Sextic and Beyond. International Electronic Journal of Geometry, Volume: 19 Issue: 1. https://doi.org/10.36890/iejg.1793372