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 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

```   dvcrss_c ( Derivative of Vector cross product )

void dvcrss_c ( ConstSpiceDouble s1  [6],
ConstSpiceDouble s2  [6],
SpiceDouble      sout[6] )

```

#### Abstract

```   Compute the cross product of two 3-dimensional vectors
and the derivative of this cross product.
```

```   None.
```

#### Keywords

```   DERIVATIVE
VECTOR

```

#### Brief_I/O

```   VARIABLE  I/O  DESCRIPTION
--------  ---  --------------------------------------------------
s1         I   Left hand state for cross product and derivative.
s2         I   Right hand state for cross product and derivative.
sout       O   State associated with cross product of positions.
```

#### Detailed_Input

```   s1          is any state vector. Typically, this might represent the
apparent state of a planet or the Sun, which defines the
orientation of axes of some coordinate system.

s2          is any state vector.
```

#### Detailed_Output

```   sout        is the state associated with the cross product of the
position components of `s1' and `s2'. In other words, if
`s1' = (p1,v1) and `s2' = (p2,v2) then `sout' is
( p1xp2, d/dt( p1xp2 ) ).
```

#### Parameters

```   None.
```

#### Exceptions

```   Error free.

1)  If `s1' and `s2' are large in magnitude (taken together,
their magnitude surpasses the limit allowed by the
computer) then it may be possible to generate a
floating point overflow from an intermediate
computation even though the actual cross product and
derivative may be well within the range of double
precision numbers.

dvcrss_c does NOT check the magnitude of `s1' or `s2' to
insure that overflow will not occur.
```

#### Files

```   None.
```

#### Particulars

```   dvcrss_c calculates the three-dimensional cross product of two
vectors and the derivative of that cross product according to
the definition. The components of this state are stored
in a local buffer vector until the calculation is complete.
Thus sout may overwrite 's1' or 's2'  without interfering with
intermediate computations.
```

#### Examples

```   The numerical results shown for these examples may differ across
platforms. The results depend on the SPICE kernels used as
input, the compiler and supporting libraries, and the machine
specific arithmetic implementation.

1) Compute the cross product of two 3-dimensional vectors
and the derivative of this cross product.

Example code begins here.

/.
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{

/.
Local variables
./
SpiceDouble          sout   [6];

SpiceInt             i;

/.
Set `s1' and `s2' vectors.
./
SpiceDouble          s1     [2][6] = {
{0.0, 1.0, 0.0, 1.0, 0.0, 0.0},
{5.0, 5.0, 5.0, 1.0, 0.0, 0.0}  };
SpiceDouble          s2     [2][6] = {
{ 1.0,  0.0,  0.0, 1.0, 0.0, 0.0},
{-1.0, -1.0, -1.0, 2.0, 0.0, 0.0}  };

/.
For each vector `s1' and `s2', compute their cross product
and its derivative.
./
for ( i = 0; i < 2; i++ )
{

dvcrss_c ( s1[i], s2[i], sout );

printf( "S1  : %6.1f %6.1f %6.1f %6.1f %6.1f %6.1f\n",
s1[i][0], s1[i][1], s1[i][2],
s1[i][3], s1[i][4], s1[i][5] );
printf( "S2  : %6.1f %6.1f %6.1f %6.1f %6.1f %6.1f\n",
s2[i][0], s2[i][1], s2[i][2],
s2[i][3], s2[i][4], s2[i][5] );
printf( "SOUT: %6.1f %6.1f %6.1f %6.1f %6.1f %6.1f\n",
sout[0], sout[1], sout[2],
sout[3], sout[4], sout[5] );
printf( "\n" );

}

return ( 0 );
}

When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:

S1  :    0.0    1.0    0.0    1.0    0.0    0.0
S2  :    1.0    0.0    0.0    1.0    0.0    0.0
SOUT:    0.0    0.0   -1.0    0.0    0.0   -1.0

S1  :    5.0    5.0    5.0    1.0    0.0    0.0
S2  :   -1.0   -1.0   -1.0    2.0    0.0    0.0
SOUT:    0.0    0.0    0.0    0.0   11.0  -11.0

2) One can construct non-inertial coordinate frames from apparent
positions of objects or defined directions. However, if one
wants to convert states in this non-inertial frame to states
in an inertial reference frame, the derivatives of the axes of
the non-inertial frame are required.

Define a reference frame with the apparent direction of the
Sun as seen from Earth as the primary axis X. Use the Earth
pole vector to define with the primary axis the XY plane of
the frame, with the primary axis Y pointing in the direction
of the pole.

Use the meta-kernel shown below to load the required SPICE
kernels.

KPL/MK

This meta-kernel is intended to support operation of SPICE
example programs. The kernels shown here should not be
assumed to contain adequate or correct versions of data
required by SPICE-based user applications.

In order for an application to use this meta-kernel, the
kernels referenced here must be present in the user's
current working directory.

The names and contents of the kernels referenced
by this meta-kernel are as follows:

File name                     Contents
---------                     --------
de421.bsp                     Planetary ephemeris
pck00008.tpc                  Planet orientation and
naif0009.tls                  Leapseconds

\begindata

'pck00008.tpc',
'naif0009.tls'  )

\begintext

End of meta-kernel

Example code begins here.

/.
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{

/.
Local variables
./
SpiceDouble          et;
SpiceDouble          lt;
SpiceDouble          state  [6];
SpiceDouble          tmpsta [6];
SpiceDouble          trans  [6][6];
SpiceDouble          x_new  [6];
SpiceDouble          y_new  [6];
SpiceDouble          z_new  [6];
SpiceDouble          zinert [6];

/.
Define the earth body-fixed pole vector (z). The pole
has no velocity in the Earth fixed frame IAU_EARTH.
./
SpiceDouble          z      [6] = { 0.0, 0.0, 1.0, 0.0, 0.0, 0.0 };

/.
Load SPK, PCK, and LSK kernels, use a meta kernel for
convenience.
./

/.
Calculate the state transformation between IAU_EARTH and
J2000 at an arbitrary epoch.
./
str2et_c ( "Jan 1, 2009", &et );
sxform_c ( "IAU_EARTH", "J2000", et, trans );

/.
Transform the earth pole vector from the IAU_EARTH frame
to J2000.
./
mxvg_c ( trans, z, 6, 6, zinert );

/.
Calculate the apparent state of the Sun from Earth at
the epoch `et' in the J2000 frame.
./
spkezr_c ( "Sun", et, "J2000", "lt+s", "Earth", state, &lt );

/.
Define the X axis of the new frame to aligned with
the computed state. Calculate the state's unit vector
and its derivative to get the X axis and its
derivative.
./
dvhat_c ( state, x_new );

/.
Define the Z axis of the new frame as the cross product
between the computed state and the Earth pole.
Calculate the Z direction in the new reference frame,
then calculate the this direction's unit vector and its
derivative to get the Z axis and its derivative.
./
dvcrss_c ( state, zinert, tmpsta );
dvhat_c ( tmpsta, z_new );

/.
As for `z_new', calculate the Y direction in the new
reference frame, then calculate this direction's unit
vector and its derivative to get the Y axis and its
derivative.
./
ducrss_c ( z_new, state, tmpsta );
dvhat_c ( tmpsta, y_new );

/.
Display the results.
./
printf( "New X-axis:\n" );
printf( "   position: %15.12f %15.12f %15.12f\n",
x_new[0], x_new[1], x_new[2] );
printf( "   velocity: %15.12f %15.12f %15.12f\n",
x_new[3], x_new[4], x_new[5] );
printf( "New Y-axis:\n" );
printf( "   position: %15.12f %15.12f %15.12f\n",
y_new[0], y_new[1], y_new[2] );
printf( "   velocity: %15.12f %15.12f %15.12f\n",
y_new[3], y_new[4], y_new[5] );
printf( "New Z-axis:\n" );
printf( "   position: %15.12f %15.12f %15.12f\n",
z_new[0], z_new[1], z_new[2] );
printf( "   velocity: %15.12f %15.12f %15.12f\n",
z_new[3], z_new[4], z_new[5] );

return ( 0 );
}

When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:

New X-axis:
position:  0.183446637633 -0.901919663328 -0.391009273602
velocity:  0.000000202450  0.000000034660  0.000000015033
New Y-axis:
position:  0.078846540163 -0.382978080242  0.920386339077
velocity:  0.000000082384  0.000000032309  0.000000006387
New Z-axis:
position: -0.979862518033 -0.199671507623  0.000857203851
velocity:  0.000000044531 -0.000000218531 -0.000000000036

Note that these vectors define the transformation between the
new frame and J2000 at the given `et':

.-            -.
|       :      |
|   R   :  0   |
M = | ......:......|
|       :      |
| dRdt  :  R   |
|       :      |
`-            -'

with

r    = { {x_new[0], y_new[0], z_new[0]},
{x_new[1], y_new[1], z_new[1]},
{x_new[2], y_new[2], z_new[2]} }

dRdt = { {x_new[3], y_new[3], z_new[3]},
{x_new[4], y_new[4], z_new[4]},
{x_new[5], y_new[5], z_new[5]} }
```

#### Restrictions

```   None.
```

#### Literature_References

```   None.
```

#### Author_and_Institution

```   J. Diaz del Rio     (ODC Space)
E.D. Wright         (JPL)
```

#### Version

```   -CSPICE Version 1.0.1, 06-JUL-2021 (JDR)

code examples.

-CSPICE Version 1.0.0, 23-NOV-2009 (EDW)
```

#### Index_Entries

```   Compute the derivative of a cross product
```
`Fri Dec 31 18:41:05 2021`