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SATELLITE CURVE

Homemade name.

 
Cartesian parametrization:
that comes from .
Algebraic curve when k is rational.

The satellite curves are the various trajectories of a point M on a given great circle of a sphere rotating around one of its axes, while M has a uniform motion along the circle.
These curves can also be seen as the trajectories of points on a circle in uniform rotation around an axis, this axis being itself in uniform rotation around an axis passing by the center of the circle.

The name of satellite curve comes from the fact that the trajectory, in the frame associated to the Earth, of a satellite in uniform circular rotation around the center of the Earth is such a curve: see for example this website or this one, or this book (page 181, example page 198).

In the above parametrization, the sphere centered on O turns around Oz and the plane of the circle forms an angle with respect to xOy; k is the ratio of the speed of rotation of M on the circle over the speed of rotation of the sphere around its axis.

Special cases of satellite curves include:
    - the clelias when the great circle meets the axis of rotation of the sphere ().
    - the spherical helices when (in the second definition above, the circle rolls without slipping on a fixed circle); it is the case where the curve has cuspidal points.

Examples:
k = 1 (rotation of the satellite equal to that of the sphere)

The curve is none other than the hippopede of Eudoxus, that corresponds to geostationary trajectories.

k =1/2 (speed of rotation of the satellite equal to half that of the sphere)
Note the cusp corresponding to the spherical helix
and the one corresponding to the clelia (passage by the poles)
k = 2 (speed of rotation of the satellite equal to twice that of the sphere)
The curve is none other than that of the seam line of a tennis ball, at least as long as it does not have double points.
Note that there is a clelia, but no spherical helix.

The linearization of the expressions of x and y above show that the projections on xOy of the satellite curves are
    - for k > 1: hypotrochoids with parameter , which yields
    - for k < 1: epitrochoids with parameter , which yields .
See the Capareda curves for a presentation of the satellite curves linked to this planar projection.

Some examples for :

q = 4, k = 2

q = 6, k = 3/2

q = 8,  k = 4/3

q = 3, k = 3

q = 8/3 , k = 4

q = 4, k = 1/2

q = 6, k = 2/3

q = 8,  k = 3/4

q = 3, k = 3

q = 8/3 , k = 4

Note that the satellite curves are not spherical trochoids, except in the case of spherical helices, that also are spherical cycloids.
However, the spherical trochoids and the satellite curves can be reunited in the family of the curves that are the trajectories of a point M of a fixed circle on a sphere in rotation around one of its axes, M having a uniform motion on the circle.

The general Cartesian parametrization of these curves is:
that comes from ; a is the distance to the axis, b the radius of the circle, its inclination, k the ratio of speeds; the radius of the sphere, centered on O, is .
 
 
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© Robert FERRÉOL  2018