Abstract
Cauchy’s “Arm Lemma” may be generalized to permit nonconvex “openings” of a planar convex chain. Although this (and further extensions) were known, no proofs have appeared in the literature. Here two induction proofs are offered. The extension can then be employed to establish that a curve that is the intersection of a plane with a convex polyhedron “develops” without self-intersection.
Supported by NSF grant CCR-9731804.
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References
M. Aigner and G.M. Ziegler. Proofs from THE BOOK. Springer-Verlag, Berlin, 1998.
S.S. Chern. Curves and surfaces in Eucidean space. In S. S. Chern, editor, Studies in Global Geometry and Analaysis, volume 4 of MAA Studies in Mathmatics, pages 16–56. Math. Assoc. Amer., 1967. Also in vol. 27 of the MAA Studies in Mathmatics, Global Differential Geometry, 1989, pp. 99—139.
R. Connelly. Rigidity and energy. Invent. Math., 66:11–33, 1982.
P. Cromwell. Polyhedra. Cambridge University Press, 1997.
H.W. Guggenheimer. Differential Geometry. McGraw-Hill, 1963.
J.E. Hopcroft, D.A. Joseph, and S.H. Whitesides. Movement problems for 2-dimensional linkages. SIAM J. Comput., 13:610–629, 1984.
J. O’Rourke. Computational Geometry in C (Second Edition). Cambridge University Press, 1998.
J. O’Rourke. On the development of the intersection of a plane with a polytope. Technical Report 068, Dept. Comput. Sci., Smith College, Northampton, MA, June 2000. LANL arXiv cs.CG/0006035 v3, http://cs.smith.edu/~orourke/papers.html.
J. O’Rourke. On the development of the intersection of a plane with a polytope. Comput. Geom. Theory Appl., 2001. Submitted.
J. O’Rourke and C. Schevon. On the development of closed convex curves on 3-polytopes. J. Geom., 13:152–157, 1989.
A. Schur. Über die Schwarzche Extremaleigenschaft des Kreises unter den Kurven konstantes Krümmung. Math. Ann., 83:143–148, 1921.
D. Singer. Geometry: Plane and Fancy. Springer-Verlag, Berlin, 1997.
I. J. Schoenberg and S.K. Zaremba. On Cauchy’s lemma concerning convex polygons. Canad. J. Math., 19:1062–1077, 1967.
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O’Rourke, J. (2001). An Extension of Cauchy’s Arm Lemma with Application to Curve Development. In: Akiyama, J., Kano, M., Urabe, M. (eds) Discrete and Computational Geometry. JCDCG 2000. Lecture Notes in Computer Science, vol 2098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47738-1_27
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DOI: https://doi.org/10.1007/3-540-47738-1_27
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