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SPIRAL OF THE HYPERBOLIC TANGENT
Polar equation: .
For k = 1/2:
Curvilinear abscissa: . Radius of curvature: . 

Spiral with an asymptotic circle with the above equation (animation in the header for k ranging from 0 to 3).
This curve is the projection on xOy of the spherical curve: 

SPIRAL OF THE HYPERBOLIC COTANGENT
Polar equation: .
For k = 1/2:
Radius of curvature: . 

Cf. the Poinsot spiral.
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© Robert FERRÉOL 2017