Index Page drdgeo_c

A B C D E F G H I J K L M N O P Q R S T U V W X

Procedure Abstract Required_Reading Keywords Brief_I/O Detailed_Input | Detailed_Output Parameters Exceptions Files Particulars Examples | Restrictions Literature_References Author_and_Institution Version Index_Entries

#### Procedure

       void drdgeo_c ( SpiceDouble    lon,
                       SpiceDouble    lat,
                       SpiceDouble    alt,
                       SpiceDouble    re,
                       SpiceDouble    f,
                       SpiceDouble    jacobi[3][3] )

#### Abstract

       This routine computes the Jacobian of the transformation from
       geodetic to rectangular coordinates.

#### Required_Reading

       None.

#### Keywords

       COORDINATES
       DERIVATIVES
       MATRIX

#### Brief_I/O

       Variable  I/O  Description
       --------  ---  --------------------------------------------------
       lon        I   Geodetic longitude of point (radians).
       lat        I   Geodetic latitude of point (radians).
       alt        I   Altitude of point above the reference spheroid.
       re         I   Equatorial radius of the reference spheroid.
       f          I   Flattening coefficient.
       jacobi     O   Matrix of partial derivatives.

#### Detailed_Input

       lon        Geodetic longitude of point (radians).

       lat        Geodetic latitude  of point (radians).

       alt        Altitude of point above the reference spheroid.

       re         Equatorial radius of the reference spheroid.

       f          Flattening coefficient = (re-rp) / re,  where rp is
                  the polar radius of the spheroid.  (More importantly
                  rp = re*(1-f).)

#### Detailed_Output

       jacobi     is the matrix of partial derivatives of the conversion
                  between geodetic and rectangular coordinates.  It
                  has the form

                     .-                             -.
                     |  dx/dlon   dx/dlat  dx/dalt   |
                     |  dy/dlon   dy/dlat  dy/dalt   |
                     |  dz/dlon   dz/dlat  dz/dalt   |
                     `-                             -'

                  evaluated at the input values of lon, lat and alt.

                  The formulae for computing x, y, and z from
                  geodetic coordinates are given below.

                     x = [alt +        re/g(lat,f)]*cos(lon)*cos(lat)

                     y = [alt +        re/g(lat,f)]*sin(lon)*cos(lat)

                                        2
                     z = [alt + re*(1-f) /g(lat,f)]*         sin(lat)

                  where

                     re is the polar radius of the reference spheroid.

                     f  is the flattening factor (the polar radius is
                     obtained by multiplying the equatorial radius by 1-f).

                     g( lat, f ) is given by

                            2             2     2
                        sqrt ( cos (lat) + (1-f) * sin (lat) )

#### Parameters

       None.

#### Exceptions

       1) If the flattening coefficient is greater than or equal to
          one, the error SPICE(VALUEOUTOFRANGE) is signaled.

       2) If the equatorial radius is non-positive, the error
          SPICE(BADRADIUS) is signaled.

#### Files

       None.

#### Particulars

       It is often convenient to describe the motion of an object in
       the geodetic coordinate system.  However, when performing
       vector computations its hard to beat rectangular coordinates.

       To transform states given with respect to geodetic coordinates
       to states with respect to rectangular coordinates, one makes use
       of the Jacobian of the transformation between the two systems.

       Given a state in geodetic coordinates

            ( lon, lat, alt, dlon, dlat, dalt )

       the velocity in rectangular coordinates is given by the matrix
       equation:

                      t          |                                 t
          (dx, dy, dz)   = jacobi|             * (dlon, dlat, dalt)
                                 |(lon,lat,alt)

       This routine computes the matrix

                |
          jacobi|
                |(lon,lat,alt)

#### Examples

       Suppose that one has a model that gives radius, longitude and
       latitude as a function of time (lon(t), lat(t), alt(t) ) for
       which the derivatives ( dlon/dt, dlat/dt, dalt/dt ) are
       computable.

       To find the velocity of the object in bodyfixed rectangular
       coordinates, one simply multiplies the Jacobian of the
       transformation from geodetic to rectangular coordinates,
       evaluated at (lon(t), lat(t), alt(t) ), by the vector of
       derivatives of the geodetic coordinates.

       In code this looks like:

          #include "SpiceUsr.h"
               .
               .
               .
          /.
          Load the derivatives of lon, lat, and alt into the
          geodetic velocity vector GEOV.
          ./
          geov[0] = dlon_dt ( t );
          geov[1] = dlat_dt ( t );
          geov[2] = dalt_dt ( t );

          /.
          Determine the Jacobian of the transformation from
          geodetic to rectangular coordinates at the geodetic
          coordinates of time t.
          ./
          drdgeo_c ( lon(t), lat(t), alt(t), re, f, jacobi );

          /.
          Multiply the Jacobian on the right by the geodetic
          velocity to obtain the rectangular velocity recv.
          ./
          mxv_c ( jacobi, geov, recv );

#### Restrictions

       None.

#### Literature_References

       None.

#### Author_and_Institution

       W.L. Taber     (JPL)
       N.J. Bachman   (JPL)

#### Version

       -CSPICE Version 1.0.0, 20-JUL-2001 (WLT) (NJB)

#### Index_Entries

       Jacobian of rectangular w.r.t. geodetic coordinates

#### Link to routine drdgeo_c source file drdgeo_c.c

    Wed Apr  5 17:54:32 2017