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ALYSOID
Curve studied by Cesàro
in 1886.
From Greek Alusion "little chain". 
Intrinsic equation 1: ,
b
non zero.
Intrinsic equation 2: , . Cartesian parametrization: , () Transcendental curve. 
Alysoids are curves such that when we let them run on a straight line, the centre of curvature of the curve at the contact point describes a parabola of axis perpendicular to the line, not meeting this line (here, the parabola ); see at curve of Mannheim.  
When a = b (k = 1), we obtain the catenary:. 

When a = 2b (k = 1/2), we obtain the parametrization: 

In the case where b would be zero, we would obtain a different curve, particular case of pseudospiral of Pirondini, named "antiloga", whose characteristics follow:
Cartesian parametrization:
(). 

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© Robert FERRÉOL 2016