Mathematics:
Radial Coordinates
We can also add the displacement vector (formula3)
In the same way to this main formula, secondary radial formulas can be added to carry out measurement and control of satellites that rotate around the orbital ones or particles.
Time is an important parameter for this systems of coordinates, because with him we can build multiple types of structures and spatial bodies.
The O coordinate sould be positive when its movement takes the clock sense of spining. Negative in opposite sense.
The H coordinate sould be positive when it is mesuare from O to H.
Negative when it is measure from O in opposite sense to H.
As we can see radial coordinates are not related with Cartesian coordinates, as spherical coordinates usually make.
Measurements. -
When we proceed to adjust the situation of the orbital one, we put on as index the current value of the H coordinate and in the sub-index the value of O coordinate.
To this departure position as you can see in the second formula, the angular movements that the orbital one goes taking is added to the index and sub-index receptively; also the new distances that the R vector goes taking is added to R.
Applying different angular speeds and different distances to R during a period of time, the formula goes giving us in each moment the position of the orbital one on the sphere around the centre C.
The particle -P- ( like if were a pencil ) will draw and build us the orbits and spatial figures that we make with radial coordinates.
Characteristics. -
With the mentioned formula of radial coordinates (and adding them new functions to each one of the parameters according to the measure to carry out) many gauging can be obtained, as can be:
> To situate and follow to any orbital one and its satellites.
> Creation of spirals on any point of the sphere.
> Creation of circumferences and ellipses so much on C or on any point of the sphere.
> Geometric drawings : Cylinders, cones, springs, screws etc.
> Polygons and polyhedrons
> Lathe
> Demarcation of orbits around C.
> Creation of energy orbits (quantum mechanics)
> Creation of harmonic or oscillatory movements on the orbits.
Etc.
In the following pages we will see easy development of these examples.