WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 15, 2016
Characterization of Lemniscate-Like Curves Generated by Linkages
Authors: , ,
Abstract: Cassini Ovals is a well-known family of curves characterized by the product of distances from two fixed points called foci. One special case is the Lemniscate of Bernoulli, a figure-8 curve which can be drawn by many ways. This paper focuses on the family of lemniscate-like curves especially generated by three-bar linkage systems where the marker is the midpoint of the middle rod. An algebraic characterization of such family of curves is investigated using the distances from the foci. It turns out that the corresponding Cartesian equation is of the form of the Hippopede defined as the intersection of a torus and a plane. A geometric construction showing the connection between three-bar linkages and lemniscate-like curves leads to another representation using polar equations. A parametric representation is also given to illustrate the family of curves using an EXCEL worksheet. Finally, another interesting family of curves so called skewed lemniscate-like curves is constructed and characterized algebraically as a natural generalization where the marker can be located on any fixed point on the middle rod.
Search Articles
Keywords: Lemniscate of Bernoulli, lemniscate-like curve, Hippopede, characterization, three-bar linkage system, skewed lemniscate-like curve, Cassini Ovals
Pages: 528-534
WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 15, 2016, Art. #50