Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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INVARIANCE OF THE AREA OF OVALOIDS

  • Youngwook Kim (Department of Mathematics Korea University) ;
  • Sung-Eun Koh (Department of Mathematics Konkuk University) ;
  • Hyung Yong Lee (Department of Mathematics Korea University) ;
  • Heayong Shin (Department of Mathematics Chung-ang University) ;
  • Seong-Deog Yang (Department of Mathematics Korea University)
  • Received : 2023.11.07
  • Accepted : 2024.01.26
  • Published : 2024.07.31
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Abstract

Consider a two dimensional smooth convex body with a marked point on the boundary of it, sitting tangentially at the marked point over a base curve in 𝔼2, ℍ2 or 𝕊2 and the image of this body by the reflection with respect to the tangent line of the base curve at the marked point. When we roll these two bodies simultaneously along the base curve, the trajectories of the marked point bound a closed region. We show that the area of the closed region is independent of the shape of the base curve if the base curve is not highly curved with respect to the boundary curve of the convex body.

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Acknowledgement

Sung-Eun Koh is supported by NRF2020R1A2C1A01003666.

References

  1. T. M. Apostol and M. A. Mnatsakanian, New horizons in geometry, The Dolciani Mathematical Expositions, 47, Math. Assoc. America, Washington, DC, 2012.
  2. H. Choi, Invariance of the length and the area of cycloids, Amer. Math. Monthly 127 (2020), no. 6, 537-544. https://doi.org/10.1080/00029890.2020.1743611