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