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On the elliptic curves arising from 4-bar mechanisms. (English) Zbl 1254.14036

C. G. Gibson and P. E. Newstead [Acta Appl. Math. 7, 113–135 (1986; Zbl 0629.70002)] investigated the algebraic geometry of the residual curve (configuration curve minus spurious components) of a planar 4-bar mechanism in the constructable case, where at least one real configuration exists. One result states that the residual curve and the mechanism’s Darboux cubic are isomorphic over the complex numbers.
In the article under review, the authors ask whether a real isomorphism between residual curve and Darboux cubic exists. In the generic non-constructable case this is not the case. For the constructable case, the question remains open.

MSC:

14H52 Elliptic curves
14P25 Topology of real algebraic varieties
70B15 Kinematics of mechanisms and robots

Citations:

Zbl 0629.70002

Software:

SINGULAR
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References:

[1] C.G. Gibson, Elementary geometry of algebraic curves , Cambridge University Press, Cambridge, 1998. · Zbl 0997.14500
[2] C.G. Gibson and P.E. Newstead, On the Geometry of the planar \(4\)-bar mechanism , Acta Appl. Math. 7 (1986), 113-135. · Zbl 0629.70002
[3] M. Kapovich and J. Millson, On the moduli space of polygons in the euclidean plane , J. Diff. Geom. 42 (1995), 133-164. · Zbl 0847.51026
[4] Singular: A computer algebra system for polynomial computations , Version 3-0-2, by G.M. Greuel, G. Pfister and H. Schoenemann BF Math. Univ. D- 67653 , Kaiserslautern. \noindentstyle
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