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SPHERICAL CURVE

Différential equation : .
Spherical parametrization: . Cartesian parametrization: . Intrinsic equation: hence (see the notations). |

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 and the spherical
ellipse.

- 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