Abstract
For a regular plane curve, an involute of it is the trajectory described by the end of a stretched string unwinding from a point of the curve. Even for a regular curve, the involute always has a singularity. By using a moving frame along the front and the curvature of the Legendre immersion in the unit tangent bundle, we define an involute of the front in the Euclidean plane and give properties of it. We also consider a relationship between evolutes and involutes of fronts without inflection points. As a result, the evolutes and the involutes of fronts without inflection points are corresponding to the differential and the integral of the curvature of the Legendre immersion.
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We would like to thank Professor Goo Ishikawa for valuable comments and helpful discussions. We also would like to thank the referee for helpful comments to improve the original manuscript. M. Takahashi was partially supported by JSPS KAKENHI Grant Number No. 26400078.
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Fukunaga, T., Takahashi, M. Involutes of fronts in the Euclidean plane. Beitr Algebra Geom 57, 637–653 (2016). https://doi.org/10.1007/s13366-015-0275-1
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DOI: https://doi.org/10.1007/s13366-015-0275-1