drdsph_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 drdsph_c ( SpiceDouble    r,
SpiceDouble    colat,
SpiceDouble    lon,
SpiceDouble    jacobi )
```

#### Abstract

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

```
None.
```

```
COORDINATES
DERIVATIVES
MATRIX

```

#### Brief_I/O

```
Variable  I/O  Description
--------  ---  --------------------------------------------------
r          I   Distance of a point from the origin.
colat      I   Angle of the point from the positive z-axis.
lon        I   Angle of the point from the xy plane.
jacobi     O   Matrix of partial derivatives.
```

#### Detailed_Input

```
r          Distance of a point from the origin.

colat      Angle between the point and the positive z-axis, in

lon        Angle of the point from the xz plane in radians.
The angle increases in the counterclockwise sense
```

#### Detailed_Output

```
jacobi     is the matrix of partial derivatives of the conversion
between spherical and rectangular coordinates,
evaluated at the input coordinates.  This matrix has
the form

.-                                 -.
|  dx/dr     dx/dcolat     dx/dlon  |
|                                   |
|  dy/dr     dy/dcolat     dy/dlon  |
|                                   |
|  dz/dr     dz/dcolat     dz/dlon  |
`-                                 -'

evaluated at the input values of r, lon and lat.
Here x, y, and z are given by the familiar formulae

x = r*cos(lon)*sin(colat)
y = r*sin(lon)*sin(colat)
z = r*cos(colat)
```

```
None.
```

```
Error free.
```

```
None.
```

#### Particulars

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

To transform states given with respect to spherical coordinates
to states with respect to rectangular coordinates, one uses
the Jacobian of the transformation between the two systems.

Given a state in spherical coordinates

( r, colat, lon, dr, dcolat, dlon )

the velocity in rectangular coordinates is given by the matrix
equation:
t          |                                   t
(dx, dy, dz)   = jacobi|              * (dr, dcolat, dlon )
|(r,colat,lon)

This routine computes the matrix

|
jacobi|
|(r,colat,lon)
```

#### Examples

```
Suppose that one has a model that gives the radius, colatitude
and longitude as a function of time (r(t), colat(t), lon(t)),
for which the derivatives ( dr/dt, dcolat/dt, dlon/dt ) are
computable.

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

In code this looks like:

#include "SpiceUsr.h"
.
.
.
/.
Load the derivatives of r, colat, and lon into the
spherical velocity vector sphv.
./
sphv = dr_dt     ( t );
sphv = dcolat_dt ( t );
sphv = dlon_dt   ( t );

/.
Determine the Jacobian of the transformation from
cylindrical to rectangular at the coordinates at the
given cylindrical coordinates at time t.
./
drdsph_c ( r(t), colat(t), lon(t), jacobi );

/.
Multiply the Jacobian on the left by the spherical
velocity to obtain the rectangular velocity recv.
./
mxv_c ( jacobi, sphv, recv );
```

```
None.
```

```
None.
```

#### Author_and_Institution

```
W.L. Taber     (JPL)
I.M. Underwood (JPL)
N.J. Bachman   (JPL)
```

#### Version

```
-CSPICE Version 1.0.0, 19-JUL-2001 (WLT) (IMU) (NJB)
```

#### Index_Entries

```
Jacobian of rectangular w.r.t. spherical coordinates
```
`Wed Apr  5 17:54:32 2017`