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The Total Squared Curvature of Curves and Approximation by Piecewise Circular Curves

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Abstract

For a smooth curve in the Euclidean spaces, the total squared curvature is defined as the integral of the square of the curvature. If one takes three points on the curve which are close to one another, the reciprocal of the radius of the circle passing through those points approximates the curvature. We use this to approximate the total squared curvature and study its convergence rate.

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References

  1. Bryant R., Griffiths P.: Reduction for constrained variational problems and \({\int{\frac{\kappa^2}{2}}ds}\) . Amer. J. Math. 108, 525–570 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Enomoto K.: Approximation of length of curves in Riemannian manifolds by geodesic polygons. Yokohama Math. J. 34, 53–58 (1986)

    MathSciNet  MATH  Google Scholar 

  3. Enomoto K.: Closed curves in E 2 and S 2 whose total squared curvatures are almost minimal. Geom. Dedicata 65, 193–201 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Enomoto K., Itoh J., Sinclair R.: The total absolute curvature of open curves in E 3. Illinois J. Math. 52, 47–76 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Fenchel W.: Über Krümmung und Windung geschlossener Raumkurven. Math. Ann. 101, 238–252 (1929)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gleason A.: A curvature formula. Am. J. Math. 101, 86–93 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gorai Y., Tasaki H., Yamakawa M.: Convergence rates of approximate sums of the areas of surfaces of revolution. Tsukuba J. Math. 33, 281–297 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Langer J., Singer D.: The total squared curvature of closed curves. J. Differ. Geom. 20, 1–22 (1984)

    MathSciNet  MATH  Google Scholar 

  9. Milnor J.: On the total curvature of knots. Ann. Math. 52, 248–257 (1953)

    Article  MathSciNet  Google Scholar 

  10. Truesdell C.: The influence of elasticity on analysis; the classical heritage. Bull. Am. Math. Soc. 9, 293–310 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Walther A.: Differenzengeometrisches zur Krümmung einer Raumkurve. Monatsh. Math. Phys. 44, 280–284 (1936)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Industrial Science and Technology, Tokyo University of Science, Oshamambe, Hokkaido, 049–3514, Japan

    Kazuyuki Enomoto

  2. 59–6, Aza Horizoe, Arai, Wakabayashi-ku, Sendai, Miyagi, 984–0032, Japan

    Mio Okura

Authors
  1. Kazuyuki Enomoto
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  2. Mio Okura
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Correspondence to Kazuyuki Enomoto.

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Enomoto, K., Okura, M. The Total Squared Curvature of Curves and Approximation by Piecewise Circular Curves. Results. Math. 64, 215–228 (2013). https://doi.org/10.1007/s00025-013-0310-1

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  • DOI: https://doi.org/10.1007/s00025-013-0310-1

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