Abstract
This paper proposes a new definition of “length” of an arbitrary given continuous curveC in the plane, which is not a single quantity but a functionL(ρ;C) of the precision ρ of measurement. ThisL(ρ;C) is monotone non-increasing in ρ, and coincides with the ordinary definitionL when ρ=0. A practical procedure for approximately calculating thisL(ρ;C) is also proposed. As illustrative examples, a theoretical calculation of the expected “length” of loci of the twodimensional Brownian motion, as well as numerical measurements and analyses of the “lengths” of coastal lines of the main islands of Japan and those of the truncated Weierstrass functions are given. ThisL(ρ;C) will give a solution to what is called the “paradox of length”.
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Kishimoto, K., Iri, M. A practical approach to the definition and measurement of “length”. Japan J. Appl. Math. 6, 179–207 (1989). https://doi.org/10.1007/BF03167878
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DOI: https://doi.org/10.1007/BF03167878