[1] J. Inoguchi, K. Kajiwara, N. Matsuura and Y. Ohta, Discrete mKdV and discrete sine-Gordon flows on discrete space curves, J. Phys. A: Math. Theor. 47 (2014), 235202.
[2] H. Furuhata, Surfaces in centroaffine geometry, RIMS kokyuroku, 1623(2009), 1–11.
[3] H. Park, K. Kajiwara, T. Kurose and N. Matsuura, Defocusing mKdV flow on centroaffine plane curves, JSIAM Lett. 10 (2018), 25–28.
[4] K. -S. Chou and C. Qu, Integrable equations arising from motions of plane cuves, Phys. D., 162 (2002), 9–33.
[5] H. Park, Explicit Formulas for Integrable Deformations of Plane Curves in Various Geometries, Master’s thesis, Kyushu University, 2018.
[6] A. Fujioka and T. Kurose, Hamiltonian formalism for the higher KdV flows on the space of closed complex equicentroaffine curves, Int. J. Geom. Meth- ods Mod. Phys. 07 (2010), 165.
[7] U. Pinkall, Hamiltonian flows on the space of star-shaped curves, Results Math. 27 (1995), 328–332.
[8] N. Matsuura, Discrete KdV and discrete modified KdV equations arising from motions of planar discrete curves, Int. Math. Res. Notices. 2012 (2012), 1681–1698.
[9] C. Rogers and W. K. Schief, Ba¨cklund and Darboux Transformations: Ge- ometry and Modern Applications in Soliton Theory, Cambridge Texts in Ap- plied Mathematics (Cambridge: Cambridge University Press, 2002).
[10] A. I. Bobenko and Y. B. Suris, Discrete Differential Geometry, American Mathematics Society, Providence, RI, 2008.
[11] S. Kaji, K. Kajiwara and H. Park Linkage Mechanisms Governed by Inte- grable Deformations of Discrete Space Curves, in: Nonlinear Systems and Their Remarkable Mathematical Structure, Vol. 2, CRC Press, 2019
[12] R. E. Goldstein and D. M. Petrich, The Korteweg-de Vreis hierarchy as dy- namics of closed curves in the plane, Phys. Rev. Lett. 67 (1991), 3203–3206.
[13] M. Hisakado, K. Nakayama and M. Wadati, Motion of discrete curves in the plane, J. Phys. Soc. Jpn. 64 (1995), 2390–2393.
[14] A. Doliwa and P. M. Santini, The integrable dynamic of a discrete curve, in: Symmetries and Integrability of Difference Equations, eds. D. Levi, L. Vinet and P. Winternitz, CRM Proceedings and Lecture Notes 9 (Providence: American Mathematical Society, 1996), 91–102.
[15] A. Doliwa and P. M. Santini, Geometry of discrete curves and lattices and in- tegrable difference equations, in: Discrete Integrable Geometry and Physics, eds. A. Bobenko and R. Seiler, Oxford Lecture Series in Mathematics and Its Applications 16 (Oxford: Clarendon Press, 1999), 139–154.
[16] U. Pinkall, B. Springborn and S. Weißmann, A new doubly discrete analogue of smoke ring flow and the real time simulation of fluid flow, J. Phys. A: Math. Theor. 40 (2007), 12563–12576.
[17] J. Inoguchi, K. Kajiwara, N. Matsuura and Y. Ohta, Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves, J. Phys. A: Math. Theor. 45 (2012) 045206.
[18] J. Inoguchi, K. Kajiwara, N. Matsuura and Y. Ohta, Motion and Ba¨cklund transformations of discrete plane curves, Kyushu. J. Math. 66 (2012) 303– 324.
[19] S. Hirose, J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta, Discrete local induction equation, J. Integrable Syst. 4(1) (2019), xyz003, 43.
[20] K. -S. Chou and C. -Z. Qu, Integrable equations arising from motions of plane curves, Phys. D. 162 (2002), 9–33.
[21] K. -S. Chou and C. -Z. Qu, Integrable equations arising from motions of plane curves. II, J. Nonlinear Sci. 13 (2003), 487–517.
[22] K. -S. Chou and C. -Z. Qu, Motions of curves in similarity geometries and Burgers-mKdV hierarchies, Chaos Solitons Fractals. 19 (2004), 47–53.
[23] K. Kajiwara, T. Kuroda and N. Matsuura, Isogonal deformation of discrete plane curves and discrete Burgers hierarchy, Pac. J. Math. Ind. (2016), 8:3.
[24] R. M. Miura, Korteweg-de Vries Equation and Generalizations. I. A Re- markable Explicit Nonlinear Transformation, J. Math. Phys. 9 (1968), 1202– 1204.
[25] L. M. Bates and O. M. Melko On curves of constant torsion I, J. Geom.104(2) (2013), 213–227.
[26] M. Bergou, M. Wardetzky, S. Robinson, B. Audoly and E. Grinspun, Dis- crete elastic rods, ACM Trans. Graph., 27(3) (2008), Article 63
[27] M. Boiti, F. Pempinelli and B. Prinari, Integrable discretization of the sine- Gordon equation, Inverse Problems 18(5) (2002), 1309–1324.
[28] R. Byrnes, Metamorphs: Transforming Mathematical Surprises, Tarquin Pubns, 1999.
[29] A. M. Calini, T. A. Ivey, Ba¨cklund transformations and knots of constant torsion, J. Knot Theory Ram., 7 (1998), 719–746.
[30] A. M. Calini an T. A. Ivey, Topology and sine-Gordon evolution of constant torsion curves, Phys. Lett. A, 254(3–4) (1999), 170–178.
[31] Z. You and Y. Chen, Motion Structures: Deployable Structural Assemblies of Mechanisms, Taylor & Francis, 2011.
[32] G. Darboux, Lec¸cons sur la The´orie Ge´ne´rale des Surfaces, Gauthier- Villars, 1917.
[33] J. Denavit and R. S. Hartenberg, A kinematic notation for lower-pair mecha- nisms based on matrices, Trans ASME J. Appl. Mech. 23 (1955), 215–221.
[34] A. Doliwa and P. M. Santini, An elementary geometric characterization of the integrable motions of a curve, Phys. Lett. A, 185 (1994), 373–384.
[35] A. Doliwa and P. M. Santini, Integrable dynamics of a discrete curve and the Ablowitz-Ladik hierarchy, J. Math. Phys. 36 (1995), 1259–1273.
[36] M. Farber, Invitation to Topological Robotics, EMS Zurich, 2008.
[37] P. W. Fowler and S. D. Guest, A symmetry analysis of mechanisms in rotating rings of tetrahedra, Proc. R. Soc. A 461 (2005), 1829–1846.
[38] E. C. Freuder, Synthesizing constraint expressions, Commun. ACM 21(11) (1978), 958–966.
[39] H. Hasimoto, A soliton on a vortex filament, J. Fluid. Mech. 51 (1972), 477– 485.
[40] M. Hisakado and M. Wadati, Moving discrete curve and geometric phase, Phys. Lett. A, 214 (1996), 252–258.
[41] T. Hoffmann, Discrete Differential Geometry of Curves and Surfaces, MI Lecture Notes vol. 18, Kyushu University, 2009.
[42] R. M. Miura, Korteweg-de Vries Equation and Generalizations. I. A Re- markable Explicit Nonlinear Transformation, J. Math. Phys. 9 (1968), 1202– 1204.
[43] T. A. Ivey, Minimal Curves of Constant Torsion, Proc. AMS, 128(7) (2000), 2095–2103.
[44] T. Jorda´n, C. Kira´ly and S. Tanigawa, Generic global rigidity of body-hinge frameworks, J. Comb. Theory B, 117 (2016), 59–76.
[45] S. Kaji, A closed linkage mechanism having the shape of a discrete Mo¨bius strip, the Japan Society for Precision Engineering Spring Meeting Sympo- sium Extended Abstracts, 62–65, 2018. The original is in Japanese but an English translation is available at arXiv:1909.02885.
[46] S. Kaji, Geometry of the moduli space of a closed linkage: a Maple code, available at https://github.com/shizuo-kaji/Kaleidocycle
[47] S. Kaji, S. Scho¨nke, M. Grunwald and E. Fried, Mo¨bius Kaleidocycle, patent filed, JP2018-033395, 2018.
[48] M. Kapovich and J. Millson, Universality theorems for configuration spaces of planar linkages, Topology 41(6) (2002), 1051–1107.
[49] N. Katoh and S. Tanigawa, A proof of the molecular conjecture, Discrete Comput Geom., 45 (2011), 647–700.
[50] K. Klenin and J. Langowski J, Computation of writhe in modeling of super- coiled DNA, Biopolymers, 54 (2000), 307–317.
[51] G. L. Lamb Jr., Solitons and the motion of helical curves, Phys. Rev. Lett.37 (1976), 235–237.
[52] J. Langer and R. Perline, Curve motion inducing modified Korteweg-de Vries systems, Phys. Lett. A, 239 (1998), 36–40.
[53] J. Langer and D. A. Singer, Lagrangian aspects of the Kirchhoff elastic rod, SIAM Review 38(4) (1996), 605–618.
[54] S. M. LaValle, Planning algorithms, Cambridge University Press, 2006.
[55] M. L. S. Magalha˜es and M. Pollicott, Geometry and dynamics of planar linkages, Comm. Math. Phys., 317(3) (2013), 615–634.
[56] A. F. Mo¨bius, Lehrbuch der Statik, 2, Leipzig, 1837.
[57] M. S. Moses and M. K. Ackerman and G. S. Chirikjian, ORIGAMI ROTORS: Imparting continuous rotation to a moving platform using compliant flexure hinges, Proc. IDETC/CIE 2013.
[58] A. Mu¨ller, Representation of the kinematic topology of mechanisms for kine- matic analysis, Mech. Sci., 6 (2015), 1–10.
[59] A. Mu¨ller, Local kinematic analysis of closed-loop linkages – mobility, sin- gularities, and shakiness, J. Mechanisms Robotics 8(4) (2016), 041013.
[60] K. Nakayama, Elementary vortex filament model of the discrete nonlinear Schro¨dinger equation, J. Phys. Soc. Jpn. 76 (2007), 074003.
[61] K. Nakayama, H. Segur and M. Wadati, Integrability and the motions of curves, Phys. Rev. Lett. 69 (1992), 2603–2606.
[62] K. Nishinari, A discrete model of an extensible string in three-dimensional space, J. Appl. Mech. 66 (1999), 695–701.
[63] S. J. Orfanidis S J, Discrete sine-Gordon equations, Phys. Rev. D, 18(10) (1978), 3822–3827.
[64] S. J. Orfanidis, Sine-Gordon equation and nonlinear σ model on a lattice, Phys. Rev. D, 18(10) (1978), 3828–3832.
[65] C. Rogers and W. K. Schief, Ba¨cklund and Darboux Transformations: Ge- ometry and Modern Applications in Soliton Theory, Cambridge University Press, Cambridge, 2002.
[66] K. Sato and R. Tanaka, Solitons in one-dimensional mechanical linkagePhys. Rev. E, 98 (2018).
[67] D. Schattschneider and W. M. Walker, M. C. Escher Kaleidocycles, Pomegranate Communications: Rohnert Park, CA, 1987. (TASCHEN; Reprint edition, 2015).
[68] A. J. Sommese, J. D. Hauenstein, D. J. Bates and C. W. Wampler, Numeri- cally Solving Polynomial Systems with Bertini, Software, Environments, and Tools, Vol. 25, SIAM, Philadelphia, PA, 2013.
[69] L. J. Weiner, Closed curves of constant torsion, Arch. Math. (Basel) 25(1974), 313–317.
[70] L. J. Weiner, Closed curves of constant torsion II, Proc. AMS, 67(2) (1977).
論文の公開元へ
