A noncircular oval with convex unit-tangent iterates
Abstract
For an oriented regular plane curve γ, define cal Tγ=γ+τ, where τ is its unit tangent. We construct a noncircular curve Γ0 such that every iterate cal TnΓ0, n≥0, is a smooth embedded closed curve with everywhere positive curvature. This answers a question of Tabachnikov. The construction starts with a noncompact convex curve C, asymptotic at its two ends to parallel lines, satisfying cal T C=C+(V,0) for some V>0. From C we construct long closed convex curves. After matching perimeters, we compare each curve with a convex curve whose unit-tangent image is the next curve in the sequence. In suitable arclength coordinates, their curvature functions differ by an exponentially small amount in L1. Uniform inverse estimates turn these approximate relations into an exact sequence Γn+1=cal TΓn. Each Γn lies in a strip of width bounded independently of n, whereas the radii of iterated circles tend to infinity.
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BibTeX
@misc{menezes2026noncircular,
title = {{A noncircular oval with convex unit-tangent iterates}},
author = {Dean Matthew Menezes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10796},
eprint = {2610.10796},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10796},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A noncircular oval with convex unit-tangent iterates
AU - Dean Matthew Menezes
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10796
UR - https://arxiv.org/abs/2610.10796
AB - For an oriented regular plane curve γ, define cal Tγ=γ+τ, where τ is its unit tangent. We construct a noncircular curve Γ_0 such that every iterate cal T^nΓ_0, n≥0, is a smooth embedded closed curve with everywhere positive curvature. This answers a question of Tabachnikov. The construction starts with a noncompact convex curve C, asymptotic at its two ends to parallel lines, satisfying cal T C=C+(V,0) for some V>0. From C we construct long closed convex curves. After matching perimeters, we compare each curve with a convex curve whose unit-tangent image is the next curve in the sequence. In suitable arclength coordinates, their curvature functions differ by an exponentially small amount in L^1. Uniform inverse estimates turn these approximate relations into an exact sequence Γ_n+1=cal TΓ_n. Each Γ_n lies in a strip of width bounded independently of n, whereas the radii of iterated circles tend to infinity.
ER -