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.

© Robert FERRÉOL 2018