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Envelopes of Legendre Curves in the Unit Tangent Bundle over the Euclidean Plane

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Abstract

For singular plane curves, the classical definitions of envelopes are vague. In order to define envelopes for singular plane curves, we introduce a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane and the curvature. Then we give a definition of an envelope for the one-parameter family of Legendre curves. We investigate properties of the envelopes. For instance, the envelope is also a Legendre curve. Moreover, we consider bi-Legendre curves and give a relationship between envelopes.

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Correspondence to Masatomo Takahashi.

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This is a partially supported by JSPS KAKENHI Grant Number JP 26400078.

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Takahashi, M. Envelopes of Legendre Curves in the Unit Tangent Bundle over the Euclidean Plane. Results Math 71, 1473–1489 (2017). https://doi.org/10.1007/s00025-016-0619-7

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  • DOI: https://doi.org/10.1007/s00025-016-0619-7

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