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#### 11.2.1 Astronomical Coordinate Calculations

The frame bias matrix `rb` transforms vectors from Geocentric Celestial Reference System to the J2000.0 mean equator and equinox by applying frame bias. It does not depend on the date.

The precession matrix `rp` transforms vectors from the J2000.0 mean equator and equinox to the mean equator and equinox of the date by applying precession.

The bias-precession matrix `rbp` transforms vectors from Geocentric Celestial Reference System to the mean equator and equinox of the date by applying frame bias and then precession. `rbp = rp # rb`

The bias-precession-nutation matrix `rbpn`...

The Celestial Intermediate Pole coordinates `x`, `y` are components of the Celestial Intermediate Pole unit vector in the Geocentric Celestial Reference System.

The Celestial Intermediate Origin locator `s` positions the Celestial Intermediate Origin on the equator of the Celestial Intermediate Pole.

Given celestial coordinates `crs` (in the Geocentric Celestial Reference System), the celestial-to-intermediate-frame-of-date matrix `rc2i`, the polar motion matrix `rpom`, and Earth Rotation Angle `era`, the terrestrial coordinates `trs` (in the International Terrestrial Reference System) are found through

`trs = rpom # R_3(era) # rc2i # crs`

The celestial-to-terrestrial matrix `rc2t` translates coordinate from the Geocentric Celestial Reference System to the International Terrestrial Reference System:

`trs = rc2t # crs`

`rc2t = rpom # R_3(era) # rc2i`

Given celestial coordinates `crs` (in the Geocentric Celestial Reference System), the celestial-to-true matrix `rbpn`, the polar motion matrix `rpom`, and Greenwich apparent Sidereal Time `gst`, the terrestrial coordinates `trs` (in the International Terrestrial Reference System) are found through

`trs = rpom # R_3(gst) # rbpn # crs`

The celestial-to-terrestrial matrix `rc2t` translates coordinate from the Geocentric Celestial Reference System to the International Terrestrial Reference System:

`trs = rc2t # crs`

`rc2t = rpom # R_3(gst) # rbpn`

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