Cylindrical equation: , hence the parametrization: . Radius of curvature: . Normal radius of curvature: . Geodesic radius of curvature: .

A cylindrical curve is a curve traced on a cylinder of revolution.

Examples:
1) Algebraic cylindrical curves
    - degree 1: straight lines
    - degree 2: circles (case f = constant), and more generally ellipses (sections by planes)
    - degree 3: section of the cylinder by a ruled quadric, with a common generatrix; example: the horopter curve.
    - degree 4: other sections by quadrics; examples: bicylindrical curves, horse fetters (including the Viviani curve), cylindroconical curves and the pancake curve
    - degree 8 : the Archytas curve (section by a torus)

2) Transcendental cylindrical curves
    - the circular helix (case where f is a linear function)
    - cylindrical catenaries
    - the pseudogeodesic of the cylinder, special case of the previous curves.

3) Families of cylindrical curves (that are algebraic for certain values of the parameters)
    - cylindrical sine waves (case where f is sinusoidal), including ellipses, the Viviani curve and the pancake curve
    - cylindrical tangent waves (case where f is tangentoidal), including the horopter curve
 

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© Robert FERRÉOL  2018