Abstract
In 1954 Steinhaus raised the question of whether a rectifiable curve is characterized by its projections. A projection onto a lineG at the pointp counts the number of points in the set which lie on the line which is perpendicular toG and passes throughp. We prove this is so, and give a method to reconstruct a closed connected rectifiable set from its projections.
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H. T. Croft, K. J. Falconer, R. K. Guy,Unsolved Problems in Geometry, Springer-Verlag, New York, 1991.
K. J. Falconer,The Geometry of Fractal Sets, Cambridge University Press, New York, 1985.
H. Fast, Inversion of the Crofton transform for sets in the plane,Real Anal. Exchange,19(1), (1994), 59–80.
J. Favard, Definition de longeur et de l'aire,C. R. Acad. Sci. Paris,194 (1932), 334–346.
H. Federer,Geometric Measure Theory, Springer-Verlag, New York, 1969.
I. M. Gelfand, M. M. Smirnov, Crofton densities and nonlocal differentials, inThe Gelfand Mathematical Seminars, 1990–1992, L. Corwin, I. Gelfand, J. Lepowsky, eds., Birkhäuser, Boston, 1993.
F. Morgan,Geometric Measure Theory: A Beginner's Guide, Academic Press, New York, 1988.
T. J. Richardson, Total curvature and intersection tomography,Adv. in Math., to appear.
L. A. Santaló,Integral Geometry and Geometric Probability, Addison-Wesley, Reading, MA, 1976.
S. Sherman, A comparison of linear measures in the plane.Duke Math. J.,9 (1942), 1–9.
H. Steinhaus, Length, shape, and area.Colloq. Math.,3 (1954), 1–13.
W. P. Ziemer,Weakly Differentiable Functions, Springer-Verlag, New York, 1989.
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Richardson, T.J. Planar rectifiable curves are determined by their projections. Discrete Comput Geom 16, 21–31 (1996). https://doi.org/10.1007/BF02711131
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DOI: https://doi.org/10.1007/BF02711131