drdgeo_c

 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 )

```

#### Abstract

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

```
None.
```

```
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) )
```

```
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
```

```
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 = dlon_dt ( t );
geov = dlat_dt ( t );
geov = 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 );
```

```
None.
```

```
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
```
`Wed Apr  5 17:54:32 2017`