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Tortuosity is a property of curve being tortuous (twisted; having many turns). There have been several attempts to quantify this property. Tortuosity is commonly used to describe diffusion in porous media,〔Epstein, N. (1989), On tortuosity and the tortuosity factor in flow and diffusion through porous media, Chem. Eng. Sci., 44(3), 777– 779. ()〕 such as soils and snow.〔Kaempfer, T. U., M. Schneebeli, and S. A. Sokratov (2005), A microstructural approach to model heat transfer in snow, Geophys. Res. Lett., 32, L21503,()〕 ==Tortuosity in 2-D== Subjective estimation (sometimes aided by optometric grading scales〔Richard M. Pearson. Optometric Grading Scales for use in everyday practice. Optometry Today, Vol. 43, No. 20, 2003, ISSN 0268-5485 ()〕) is often used. The simplest mathematical method to estimate tortuosity is the arc-chord ratio: the ratio of the length of the curve (''L'') to the distance between the ends of it (''C''): : Arc-chord ratio equals 1 for a straight line and is infinite for a circle. Another method, proposed in 1999,〔William E. Hart, Michael Goldbaum, Brad Cote, Paul Kube, Mark R. Nelson. Automated measurement of retinal vascular tortuosity. International Journal of Medical Informatics, Vol. 53, No. 2-3, p. 239-252, 1999 ()〕 is to estimate the tortuosity as integral of square (or module) of curvature. Dividing the result by length of curve or chord has also been tried. In 2002 several Italian scientists〔Enrico Grisan, Marco Foracchia, Alfredo Ruggeri. A novel method for automatic evaluation of retinal vessel tortuosity. Proceedings of the 25th Annual International Conference of the IEEE EMBS, Cancun, Mexico, 2003 ()〕 proposed one more method. At first, the curve is divided into several (''N'') parts with constant sign of curvature (using hysteresis to decrease sensitivity to noise). Then the arc-chord ratio for each part is found and the tortuosity is estimated by: : In this case tortuosity of both straight line and circle is estimated to be 0. In 1993〔M. Mächler, Very smooth nonparametric curve estimation by penalizing change of curvature, Technical Report 71, ETH Zurich, May 1993 ()〕 Swiss mathematician Martin Mächler proposed an analogy: it’s relatively easy to drive a bicycle or a car in a trajectory with a constant curvature (an arc of a circle), but it’s much harder to drive where curvature changes. This would imply that roughness (or tortuosity) could be measured by relative change of curvature. In this case the proposed "local" measure was derivative of logarithm of curvature: : However, in this case tortuosity of a straight line is left undefined. In 2005 it was proposed to measure tortuosity by an integral of square of derivative of curvature, divided by the length of a curve:〔Patasius, M.; Marozas, V.; Lukosevicius, A.; Jegelevicius, D.. Evaluation of tortuosity of eye blood vessels using the integral of square of derivative of curvature // EMBEC'05: proceedings of the 3rd IFMBE European Medical and Biological Engineering Conference, November 20–25, 2005, Prague. - ISSN 1727-1983. - Prague. - 2005, Vol. 11, p. ()〕 : In this case tortuosity of both straight line and circle is estimated to be 0. Fractal dimension has been used to quantify tortuosity.〔Caldwell, I. R., & Nams, V. O. (2006). A compass without a map: tortuosity and orientation of eastern painted turtles (''Chrysemys picta picta'') released in unfamiliar territory. ''Canadian Journal of Zoology'', 84(8), 1129-1137. ()〕 The fractal dimension in 2D for a straight line is 1 (the minimal value), and ranges up to 2 for a plane-filling curve or Brownian motion.〔Benhamou, S. (2004). How to reliably estimate the tortuosity of an animal's path: straightness, sinuosity, or fractal dimension?. ''Journal of theoretical biology'', 229(2), 209-220.〕 In most of these methods digital filters and approximation by splines can be used to decrease sensitivity to noise. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Tortuosity」の詳細全文を読む スポンサード リンク
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