#### Geometry for space-based radar

Figure 1: Coordinate systems for a space-based radar

#### Geometry for space-based radar

The geometry for space-based radar is characterized by three Cartesian coordinate systems.

One of them is an initial coordinate system (x_{I}; y_{I}; z_{I} ),
whose origin is at the center of Earth, and whose z_{I} axis is along the Earth’s rotation axis.
In this coordinate system, the Earth rotates and the radar satellite describes an elliptical orbit around it according to Kepler’s laws.
Its orbital period is T.

A second coordinate system (x_{E}; y_{E}; z_{E} ) uses the same origin.
The orientation of the z_{E} axis is identical to z_{I}.
However, this coordinate system is bound to the rotation of the Earth
and rotates with the angular velocity ω_{E}.
Each point on the Earth’s surface occupies a fixed position in this coordinate system,
which, with direction and distance from the origin, forms a vector p_{E}.
The magnitude of the distance from the center is ρ_{E}.
In this coordinate system, the radar satellite describes a wave-shaped orbit (see Figure 2).
Only after a large number of orbits, the satellite is again above the same point of the Earth.

Figure 2: Wave-shaped satellite orbit above the Earth’s surface

The third coordinate system (x_{A}; y_{A}; z_{A} )
is bound with its origin to the phase center of the radar antenna.
In this coordinate system the radar measures the direction and the distance of the reflecting objects.
The coordinate z_{A} points in the same direction
as the instantaneous vector R_{E}(T).
The coordinate x_{A} is perpendicular to the vector
of the tangential velocity of the satellite.
In this direction the radar measures a runtime of the echo signals and calculates a range from it.
The coordinate y_{A} lies on the tangent of motion and
a measurement result in this direction is called azimuth (or cross-range, see
Side-Looking-Airborne Radar (SLAR).