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SPHERICAL CURVE

Spherical parametrization: .
Cartesian parametrization: . Intrinsic equation: . |

A *spherical* curve is a curve traced on a sphere.

Necessary and sufficient conditions:

- curve the normal planes of which pass by a fixed point (therefore, the polar developable of which is a cone).

- curve the osculating sphere of which has constant radius, hence the above intrinsic equation.

Examples:

1) Algebraic spherical curves

- the circles (degree 2), that are, in the case of the *great circles*, the geodesics of the sphere on which they are traced.

- the spherical biquadratics (degree 4), intersections between a sphere and a quadric, including the sphero-cylindrical curves.

- the seam line of a tennis ball (degree 6).

2) Transcendental spherical curves (apart from some special cases)

- the clelias, or spherical spirals,

- the rhumb lines of the sphere (constant angle with the meridians),

- the spherical helices (constant angle with a diameter),

- the spherical cycloids, which include the previous curves as special cases,

- the spherical trochoids, which include the previous curves as special cases,

- the satellite curves, which include the clelias and the spherical helices as special cases,

- the curves of the spherical pendulum, and more generally, the festoons of spinning tops,

- the spherical catenaries,

- the spherical brachistochrones,

- the spherical pursuit curves,

- the spherical sinusoids,

- Seiffert's spherical spiral,

- a curve associated to the spiral of the hyperbolic tangent.

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© Robert FERRÉOL 2018