void eraNumat(double epsa, double dpsi, double deps, double rmatn[3][3])
/*
** - - - - - - - - -
** e r a N u m a t
** - - - - - - - - -
**
** Form the matrix of nutation.
**
** Given:
** epsa double mean obliquity of date (Note 1)
** dpsi,deps double nutation (Note 2)
**
** Returned:
** rmatn double[3][3] nutation matrix (Note 3)
**
** Notes:
**
**
** 1) The supplied mean obliquity epsa, must be consistent with the
** precession-nutation models from which dpsi and deps were obtained.
**
** 2) The caller is responsible for providing the nutation components;
** they are in longitude and obliquity, in radians and are with
** respect to the equinox and ecliptic of date.
**
** 3) The matrix operates in the sense V(true) = rmatn * V(mean),
** where the p-vector V(true) is with respect to the true
** equatorial triad of date and the p-vector V(mean) is with
** respect to the mean equatorial triad of date.
**
** Called:
** eraIr initialize r-matrix to identity
** eraRx rotate around X-axis
** eraRz rotate around Z-axis
**
** Reference:
**
** Explanatory Supplement to the Astronomical Almanac,
** P. Kenneth Seidelmann (ed), University Science Books (1992),
** Section 3.222-3 (p114).
**
** Copyright (C) 2013-2014, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
/* Build the rotation matrix. */
eraIr(rmatn);
eraRx(epsa, rmatn);
eraRz(-dpsi, rmatn);
eraRx(-(epsa + deps), rmatn);
return;
}
void
palDeuler( const char *order, double phi, double theta, double psi,
double rmat[3][3] ) {
int i = 0;
double rotations[3];
/* Initialise rmat */
eraIr( rmat );
/* copy the rotations into an array */
rotations[0] = phi;
rotations[1] = theta;
rotations[2] = psi;
/* maximum three rotations */
while (i < 3 && order[i] != '\0') {
switch (order[i]) {
case 'X':
case 'x':
case '1':
eraRx( rotations[i], rmat );
break;
case 'Y':
case 'y':
case '2':
eraRy( rotations[i], rmat );
break;
case 'Z':
case 'z':
case '3':
eraRz( rotations[i], rmat );
break;
default:
/* break out the loop if we do not recognize something */
i = 3;
}
/* Go to the next position */
i++;
}
return;
}
void eraC2teqx(double rbpn[3][3], double gst, double rpom[3][3],
double rc2t[3][3])
/*
** - - - - - - - - - -
** e r a C 2 t e q x
** - - - - - - - - - -
**
** Assemble the celestial to terrestrial matrix from equinox-based
** components (the celestial-to-true matrix, the Greenwich Apparent
** Sidereal Time and the polar motion matrix).
**
** Given:
** rbpn double[3][3] celestial-to-true matrix
** gst double Greenwich (apparent) Sidereal Time
** rpom double[3][3] polar-motion matrix
**
** Returned:
** rc2t double[3][3] celestial-to-terrestrial matrix (Note 2)
**
** Notes:
**
** 1) This function constructs the rotation matrix that transforms
** vectors in the celestial system into vectors in the terrestrial
** system. It does so starting from precomputed components, namely
** the matrix which rotates from celestial coordinates to the
** true equator and equinox of date, the Greenwich Apparent Sidereal
** Time and the polar motion matrix. One use of the present function
** is when generating a series of celestial-to-terrestrial matrices
** where only the Sidereal Time changes, avoiding the considerable
** overhead of recomputing the precession-nutation more often than
** necessary to achieve given accuracy objectives.
**
** 2) The relationship between the arguments is as follows:
**
** [TRS] = rpom * R_3(gst) * rbpn * [CRS]
**
** = rc2t * [CRS]
**
** where [CRS] is a vector in the Geocentric Celestial Reference
** System and [TRS] is a vector in the International Terrestrial
** Reference System (see IERS Conventions 2003).
**
** Called:
** eraCr copy r-matrix
** eraRz rotate around Z-axis
** eraRxr product of two r-matrices
**
** Reference:
**
** McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003),
** IERS Technical Note No. 32, BKG (2004)
**
** Copyright (C) 2013, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double r[3][3];
/* Construct the matrix. */
eraCr(rbpn, r);
eraRz(gst, r);
eraRxr(rpom, r, rc2t);
return;
}
void eraC2ixys(double x, double y, double s, double rc2i[3][3])
/*
** - - - - - - - - - -
** e r a C 2 i x y s
** - - - - - - - - - -
**
** Form the celestial to intermediate-frame-of-date matrix given the CIP
** X,Y and the CIO locator s.
**
** Given:
** x,y double Celestial Intermediate Pole (Note 1)
** s double the CIO locator s (Note 2)
**
** Returned:
** rc2i double[3][3] celestial-to-intermediate matrix (Note 3)
**
** Notes:
**
** 1) The Celestial Intermediate Pole coordinates are the x,y
** components of the unit vector in the Geocentric Celestial
** Reference System.
**
** 2) The CIO locator s (in radians) positions the Celestial
** Intermediate Origin on the equator of the CIP.
**
** 3) The matrix rc2i is the first stage in the transformation from
** celestial to terrestrial coordinates:
**
** [TRS] = RPOM * R_3(ERA) * rc2i * [CRS]
**
** = RC2T * [CRS]
**
** where [CRS] is a vector in the Geocentric Celestial Reference
** System and [TRS] is a vector in the International Terrestrial
** Reference System (see IERS Conventions 2003), ERA is the Earth
** Rotation Angle and RPOM is the polar motion matrix.
**
** Called:
** eraIr initialize r-matrix to identity
** eraRz rotate around Z-axis
** eraRy rotate around Y-axis
**
** Reference:
**
** McCarthy, D. D., Petit, G. (eds.), IERS Conventions (2003),
** IERS Technical Note No. 32, BKG (2004)
**
** Copyright (C) 2013-2014, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double r2, e, d;
/* Obtain the spherical angles E and d. */
r2 = x*x + y*y;
e = (r2 != 0.0) ? atan2(y, x) : 0.0;
d = atan(sqrt(r2 / (1.0 - r2)));
/* Form the matrix. */
eraIr(rc2i);
eraRz(e, rc2i);
eraRy(d, rc2i);
eraRz(-(e+s), rc2i);
return;
}
void eraBp00(double date1, double date2,
double rb[3][3], double rp[3][3], double rbp[3][3])
/*
** - - - - - - - -
** e r a B p 0 0
** - - - - - - - -
**
** Frame bias and precession, IAU 2000.
**
** Given:
** date1,date2 double TT as a 2-part Julian Date (Note 1)
**
** Returned:
** rb double[3][3] frame bias matrix (Note 2)
** rp double[3][3] precession matrix (Note 3)
** rbp double[3][3] bias-precession matrix (Note 4)
**
** Notes:
**
** 1) The TT date date1+date2 is a Julian Date, apportioned in any
** convenient way between the two arguments. For example,
** JD(TT)=2450123.7 could be expressed in any of these ways,
** among others:
**
** date1 date2
**
** 2450123.7 0.0 (JD method)
** 2451545.0 -1421.3 (J2000 method)
** 2400000.5 50123.2 (MJD method)
** 2450123.5 0.2 (date & time method)
**
** The JD method is the most natural and convenient to use in
** cases where the loss of several decimal digits of resolution
** is acceptable. The J2000 method is best matched to the way
** the argument is handled internally and will deliver the
** optimum resolution. The MJD method and the date & time methods
** are both good compromises between resolution and convenience.
**
** 2) The matrix rb transforms vectors from GCRS to mean J2000.0 by
** applying frame bias.
**
** 3) The matrix rp transforms vectors from J2000.0 mean equator and
** equinox to mean equator and equinox of date by applying
** precession.
**
** 4) The matrix rbp transforms vectors from GCRS to mean equator and
** equinox of date by applying frame bias then precession. It is
** the product rp x rb.
**
** 5) It is permissible to re-use the same array in the returned
** arguments. The arrays are filled in the order given.
**
** Called:
** eraBi00 frame bias components, IAU 2000
** eraPr00 IAU 2000 precession adjustments
** eraIr initialize r-matrix to identity
** eraRx rotate around X-axis
** eraRy rotate around Y-axis
** eraRz rotate around Z-axis
** eraCr copy r-matrix
** eraRxr product of two r-matrices
**
** Reference:
** "Expressions for the Celestial Intermediate Pole and Celestial
** Ephemeris Origin consistent with the IAU 2000A precession-
** nutation model", Astron.Astrophys. 400, 1145-1154 (2003)
**
** n.b. The celestial ephemeris origin (CEO) was renamed "celestial
** intermediate origin" (CIO) by IAU 2006 Resolution 2.
**
** Copyright (C) 2013-2017, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
/* J2000.0 obliquity (Lieske et al. 1977) */
const double EPS0 = 84381.448 * ERFA_DAS2R;
double t, dpsibi, depsbi, dra0, psia77, oma77, chia,
dpsipr, depspr, psia, oma, rbw[3][3];
/* Interval between fundamental epoch J2000.0 and current date (JC). */
t = ((date1 - ERFA_DJ00) + date2) / ERFA_DJC;
/* Frame bias. */
eraBi00(&dpsibi, &depsbi, &dra0);
/* Precession angles (Lieske et al. 1977) */
psia77 = (5038.7784 + (-1.07259 + (-0.001147) * t) * t) * t * ERFA_DAS2R;
oma77 = EPS0 + ((0.05127 + (-0.007726) * t) * t) * t * ERFA_DAS2R;
chia = ( 10.5526 + (-2.38064 + (-0.001125) * t) * t) * t * ERFA_DAS2R;
/* Apply IAU 2000 precession corrections. */
eraPr00(date1, date2, &dpsipr, &depspr);
psia = psia77 + dpsipr;
oma = oma77 + depspr;
/* Frame bias matrix: GCRS to J2000.0. */
eraIr(rbw);
eraRz(dra0, rbw);
eraRy(dpsibi*sin(EPS0), rbw);
eraRx(-depsbi, rbw);
eraCr(rbw, rb);
/* Precession matrix: J2000.0 to mean of date. */
eraIr(rp);
eraRx(EPS0, rp);
eraRz(-psia, rp);
eraRx(-oma, rp);
eraRz(chia, rp);
/* Bias-precession matrix: GCRS to mean of date. */
eraRxr(rp, rbw, rbp);
return;
}
void eraC2tcio(double rc2i[3][3], double era, double rpom[3][3],
double rc2t[3][3])
/*
** - - - - - - - - - -
** e r a C 2 t c i o
** - - - - - - - - - -
**
** Assemble the celestial to terrestrial matrix from CIO-based
** components (the celestial-to-intermediate matrix, the Earth Rotation
** Angle and the polar motion matrix).
**
** Given:
** rc2i double[3][3] celestial-to-intermediate matrix
** era double Earth rotation angle (radians)
** rpom double[3][3] polar-motion matrix
**
** Returned:
** rc2t double[3][3] celestial-to-terrestrial matrix
**
** Notes:
**
** 1) This function constructs the rotation matrix that transforms
** vectors in the celestial system into vectors in the terrestrial
** system. It does so starting from precomputed components, namely
** the matrix which rotates from celestial coordinates to the
** intermediate frame, the Earth rotation angle and the polar motion
** matrix. One use of the present function is when generating a
** series of celestial-to-terrestrial matrices where only the Earth
** Rotation Angle changes, avoiding the considerable overhead of
** recomputing the precession-nutation more often than necessary to
** achieve given accuracy objectives.
**
** 2) The relationship between the arguments is as follows:
**
** [TRS] = RPOM * R_3(ERA) * rc2i * [CRS]
**
** = rc2t * [CRS]
**
** where [CRS] is a vector in the Geocentric Celestial Reference
** System and [TRS] is a vector in the International Terrestrial
** Reference System (see IERS Conventions 2003).
**
** Called:
** eraCr copy r-matrix
** eraRz rotate around Z-axis
** eraRxr product of two r-matrices
**
** Reference:
**
** McCarthy, D. D., Petit, G. (eds.), 2004, IERS Conventions (2003),
** IERS Technical Note No. 32, BKG
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double r[3][3];
/* Construct the matrix. */
eraCr(rc2i, r);
eraRz(era, r);
eraRxr(rpom, r, rc2t);
return;
}
void eraFw2m(double gamb, double phib, double psi, double eps,
double r[3][3])
/*
** - - - - - - - -
** e r a F w 2 m
** - - - - - - - -
**
** Form rotation matrix given the Fukushima-Williams angles.
**
** Given:
** gamb double F-W angle gamma_bar (radians)
** phib double F-W angle phi_bar (radians)
** psi double F-W angle psi (radians)
** eps double F-W angle epsilon (radians)
**
** Returned:
** r double[3][3] rotation matrix
**
** Notes:
**
** 1) Naming the following points:
**
** e = J2000.0 ecliptic pole,
** p = GCRS pole,
** E = ecliptic pole of date,
** and P = CIP,
**
** the four Fukushima-Williams angles are as follows:
**
** gamb = gamma = epE
** phib = phi = pE
** psi = psi = pEP
** eps = epsilon = EP
**
** 2) The matrix representing the combined effects of frame bias,
** precession and nutation is:
**
** NxPxB = R_1(-eps).R_3(-psi).R_1(phib).R_3(gamb)
**
** 3) Three different matrices can be constructed, depending on the
** supplied angles:
**
** o To obtain the nutation x precession x frame bias matrix,
** generate the four precession angles, generate the nutation
** components and add them to the psi_bar and epsilon_A angles,
** and call the present function.
**
** o To obtain the precession x frame bias matrix, generate the
** four precession angles and call the present function.
**
** o To obtain the frame bias matrix, generate the four precession
** angles for date J2000.0 and call the present function.
**
** The nutation-only and precession-only matrices can if necessary
** be obtained by combining these three appropriately.
**
** Called:
** eraIr initialize r-matrix to identity
** eraRz rotate around Z-axis
** eraRx rotate around X-axis
**
** Reference:
**
** Hilton, J. et al., 2006, Celest.Mech.Dyn.Astron. 94, 351
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
/* Construct the matrix. */
eraIr(r);
eraRz(gamb, r);
eraRx(phib, r);
eraRz(-psi, r);
eraRx(-eps, r);
return;
}
void eraPb06(double date1, double date2,
double *bzeta, double *bz, double *btheta)
/*
** - - - - - - - -
** e r a P b 0 6
** - - - - - - - -
**
** This function forms three Euler angles which implement general
** precession from epoch J2000.0, using the IAU 2006 model. Frame
** bias (the offset between ICRS and mean J2000.0) is included.
**
** Given:
** date1,date2 double TT as a 2-part Julian Date (Note 1)
**
** Returned:
** bzeta double 1st rotation: radians cw around z
** bz double 3rd rotation: radians cw around z
** btheta double 2nd rotation: radians ccw around y
**
** Notes:
**
** 1) The TT date date1+date2 is a Julian Date, apportioned in any
** convenient way between the two arguments. For example,
** JD(TT)=2450123.7 could be expressed in any of these ways,
** among others:
**
** date1 date2
**
** 2450123.7 0.0 (JD method)
** 2451545.0 -1421.3 (J2000 method)
** 2400000.5 50123.2 (MJD method)
** 2450123.5 0.2 (date & time method)
**
** The JD method is the most natural and convenient to use in
** cases where the loss of several decimal digits of resolution
** is acceptable. The J2000 method is best matched to the way
** the argument is handled internally and will deliver the
** optimum resolution. The MJD method and the date & time methods
** are both good compromises between resolution and convenience.
**
** 2) The traditional accumulated precession angles zeta_A, z_A,
** theta_A cannot be obtained in the usual way, namely through
** polynomial expressions, because of the frame bias. The latter
** means that two of the angles undergo rapid changes near this
** date. They are instead the results of decomposing the
** precession-bias matrix obtained by using the Fukushima-Williams
** method, which does not suffer from the problem. The
** decomposition returns values which can be used in the
** conventional formulation and which include frame bias.
**
** 3) The three angles are returned in the conventional order, which
** is not the same as the order of the corresponding Euler
** rotations. The precession-bias matrix is
** R_3(-z) x R_2(+theta) x R_3(-zeta).
**
** 4) Should zeta_A, z_A, theta_A angles be required that do not
** contain frame bias, they are available by calling the ERFA
** function eraP06e.
**
** Called:
** eraPmat06 PB matrix, IAU 2006
** eraRz rotate around Z-axis
**
** Copyright (C) 2013-2014, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double r[3][3], r31, r32;
/* Precession matrix via Fukushima-Williams angles. */
eraPmat06(date1, date2, r);
/* Solve for z. */
*bz = atan2(r[1][2], r[0][2]);
/* Remove it from the matrix. */
eraRz(*bz, r);
/* Solve for the remaining two angles. */
*bzeta = atan2 (r[1][0], r[1][1]);
r31 = r[2][0];
r32 = r[2][1];
*btheta = atan2(-ERFA_DSIGN(sqrt(r31 * r31 + r32 * r32), r[0][2]),
r[2][2]);
return;
}
