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CYLINDRICAL CURVE

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