How regular is the evolute of a plane curve?

2025/11/30 by Thomas, Pascal J., Nikolov, Nikolai
#53A04 #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2512.00848

Abstract

We study the relationship between the smoothness of a plane curve and that of its evolute, especially in the cases where the parent curve is no more two or three times continuously differentiable, and exhibit the same kind of apparent improvement in regularity: in the generic local situation, the evolute has one order of regularity less than the parent curve.

Citations

• Boundary regularity for the distance functions, and the eikonal equation
• Return of the plane evolute
• Osculating curves: around the Tait-Kneser Theorem

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