| axisar_c |
|
Table of contents
Procedure
axisar_c ( Axis and angle to rotation )
void axisar_c ( ConstSpiceDouble axis [3],
SpiceDouble angle,
SpiceDouble r [3][3] )
AbstractConstruct a rotation matrix that rotates vectors by a specified angle about a specified axis. Required_ReadingROTATION KeywordsMATRIX ROTATION Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- axis I Rotation axis. angle I Rotation angle, in radians. r O Rotation matrix corresponding to `axis' and `angle'. Detailed_Input
axis,
angle are, respectively, a rotation axis and a rotation
angle. `axis' and `angle' determine a coordinate
transformation whose effect on any vector `v' is to
rotate `v' by `angle' radians about the vector `axis'.
Detailed_Output
r is a rotation matrix representing the coordinate
transformation determined by `axis' and `angle': for
each vector `v', r*v is the vector resulting from
rotating `v' by `angle' radians about `axis'.
ParametersNone. Exceptions
Error free.
1) If `axis' is the zero vector, the rotation generated is the
identity. This is consistent with the specification of vrotv_c.
FilesNone. Particularsaxisar_c can be thought of as a partial inverse of raxisa_c. axisar_c really is a `left inverse': the code fragment raxisa_c ( r, axis, &angle ); axisar_c ( axis, angle, r ); preserves `r', except for round-off error, as long as `r' is a rotation matrix. On the other hand, the code fragment axisar_c ( axis, angle, r ); raxisa_c ( r, axis, &angle ); preserves `axis' and `angle', except for round-off error, only if `angle' is in the range (0, pi). So axisar_c is a right inverse of raxisa_c only over a limited domain. 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 a matrix that rotates vectors by pi/2 radians about
the Z-axis, and compute the rotation axis and angle based on
that matrix.
Example code begins here.
/.
Program axisar_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"
int main( )
{
/.
Local variables
./
SpiceDouble angle;
SpiceDouble angout;
SpiceDouble axis [3];
SpiceDouble axout [3];
SpiceDouble rotmat [3][3];
SpiceInt i;
/.
Define an axis and an angle for rotation.
./
axis[0] = 0.0;
axis[1] = 0.0;
axis[2] = 1.0;
angle = halfpi_c();
/.
Determine the rotation matrix.
./
axisar_c ( axis, angle, rotmat );
/.
Now calculate the rotation axis and angle based on
`rotmat' as input.
./
raxisa_c ( rotmat, axout, &angout );
/.
Display the results.
./
printf( "Rotation matrix:\n" );
printf( "\n" );
for ( i = 0; i < 3; i++ )
{
printf( "%10.5f %9.5f %9.5f\n",
rotmat[i][0], rotmat[i][1], rotmat[i][2] );
}
printf( "\n" );
printf( "Rotation axis : %9.5f %9.5f %9.5f\n",
axout[0], axout[1], axout[2] );
printf( "Rotation angle (deg): %9.5f\n", angout * dpr_c() );
return ( 0 );
}
When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:
Rotation matrix:
0.00000 -1.00000 0.00000
1.00000 0.00000 0.00000
0.00000 0.00000 1.00000
Rotation axis : 0.00000 0.00000 1.00000
Rotation angle (deg): 90.00000
2) Linear interpolation between two rotation matrices.
Let r(t) be a time-varying rotation matrix; `r' could be
a C-matrix describing the orientation of a spacecraft
structure. Given two points in time `t1' and `t2' at which
r(t) is known, and given a third time `t3', where
t1 < t3 < t2,
we can estimate r[t3 - 1] by linear interpolation. In other
words, we approximate the motion of `r' by pretending that
`r' rotates about a fixed axis at a uniform angular rate
during the time interval [t1, t2]. More specifically, we
assume that each column vector of `r' rotates in this
fashion. This procedure will not work if `r' rotates through
an angle of pi radians or more during the time interval
[t1, t2]; an aliasing effect would occur in that case.
Example code begins here.
/.
Program axisar_ex2
./
#include <stdio.h>
#include "SpiceUsr.h"
int main( )
{
/.
Local variables
./
SpiceDouble angle;
SpiceDouble axis [3];
SpiceDouble delta [3][3];
SpiceDouble frac;
SpiceDouble q [3][3];
SpiceDouble r1 [3][3];
SpiceDouble r2 [3][3];
SpiceDouble r3 [3][3];
SpiceDouble t1;
SpiceDouble t2;
SpiceDouble t3;
SpiceInt i;
/.
Lets assume that r(t) is the matrix that rotates
vectors by pi/2 radians about the Z-axis every
minute.
Let
r1 = r[t1 - 1], for t1 = 0", and
r2 = r[t2 - 1], for t1 = 60".
Define both matrices and times.
./
axis[0] = 0.0;
axis[1] = 0.0;
axis[2] = 1.0;
t1 = 0.0;
t2 = 60.0;
t3 = 30.0;
ident_c ( r1 );
axisar_c ( axis, halfpi_c(), r2 );
mxmt_c ( r2, r1, q );
raxisa_c ( q, axis, &angle );
/.
Find the fraction of the total rotation angle that `r'
rotates through in the time interval [t1, t3].
./
frac = ( t3 - t1 ) / ( t2 - t1 );
/.
Finally, find the rotation `delta' that r(t) undergoes
during the time interval [t1, t3], and apply that
rotation to `r1', yielding r[t3 - 1], which we'll call `r3'.
./
axisar_c ( axis, frac * angle, delta );
mxm_c ( delta, r1, r3 );
/.
Display the results.
./
printf( "Time (s) : %9.5f\n", t3 );
printf( "Rotation axis : %9.5f %9.5f %9.5f\n",
axis[0], axis[1], axis[2] );
printf( "Rotation angle (deg): %9.5f\n", frac * angle * dpr_c() );
printf( "Rotation matrix :\n" );
printf( "\n" );
for ( i = 0; i < 3; i++ )
{
printf( "%10.5f %9.5f %9.5f\n", r3[i][0], r3[i][1], r3[i][2] );
}
return ( 0 );
}
When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:
Time (s) : 30.00000
Rotation axis : 0.00000 0.00000 1.00000
Rotation angle (deg): 45.00000
Rotation matrix :
0.70711 -0.70711 0.00000
0.70711 0.70711 0.00000
0.00000 0.00000 1.00000
RestrictionsNone. Literature_ReferencesNone. Author_and_InstitutionN.J. Bachman (JPL) J. Diaz del Rio (ODC Space) Version
-CSPICE Version 1.0.1, 06-JUL-2021 (JDR)
Edited the header to comply with NAIF standard. Added complete
code examples based on existing code fragments.
-CSPICE Version 1.0.0, 18-JUN-1999 (NJB)
Index_Entriesaxis and angle to rotation |
Fri Dec 31 18:41:01 2021