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raxisa_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

raxisa_c ( Rotation axis of a matrix )

void raxisa_c ( ConstSpiceDouble     matrix,
SpiceDouble          axis  ,
SpiceDouble        * angle       )

Abstract

Compute the axis of the rotation given by an input matrix
and the angle of the rotation about that axis.

ROTATION

ANGLE
MATRIX
ROTATION

Brief_I/O

VARIABLE  I/O  DESCRIPTION
--------  ---  --------------------------------------------------
matrix     I   3x3 rotation matrix in double precision.
axis       O   Axis of the rotation.
angle      O   Angle through which the rotation is performed.

Detailed_Input

matrix      is a 3x3 rotation matrix in double precision.

Detailed_Output

axis        is a unit vector pointing along the axis of the rotation.
In other words, `axis' is a unit eigenvector of the input
matrix, corresponding to the eigenvalue 1. If the input
matrix is the identity matrix, `axis' will be the vector
(0, 0, 1). If the input rotation is a rotation by pi
radians, both `axis' and -axis may be regarded as the
axis of the rotation.

angle       is the angle between `v' and matrix*v for any non-zero
vector `v' orthogonal to `axis'.  `angle' is given in
radians. The angle returned will be in the range from 0

None.

Exceptions

1)  If the input matrix is not a rotation matrix (where a fairly
loose tolerance is used to check this), an error is signaled
by a routine in the call tree of this routine.

2)  If the input matrix is the identity matrix, this routine
returns an angle of 0.0, and an axis of ( 0.0, 0.0, 1.0 ).

None.

Particulars

Every rotation matrix has an axis `a' such any vector `v'
parallel to that axis satisfies the equation

v = matrix * v

This routine returns a unit vector `axis' parallel to the axis of
the input rotation matrix. Moreover for any vector `w' orthogonal
to the axis of the rotation, the two vectors

axis,
w x (matrix*w)

(where "x" denotes the cross product operation)

will be positive scalar multiples of one another (at least
to within the ability to make such computations with double
precision arithmetic, and under the assumption that `matrix'
does not represent a rotation by zero or pi radians).

The angle returned will be the angle between `w' and matrix*w
for any vector orthogonal to `axis'.

If the input matrix is a rotation by 0 or pi radians some
choice must be made for the axis returned. In the case of
a rotation by 0 radians, `axis' is along the positive z-axis.
In the case of a rotation by 180 degrees, two choices are
possible. The choice made this routine is unspecified.

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) Given an axis and an angle of rotation about that axis,
determine the rotation matrix. Using this matrix as input,
compute the axis and angle of rotation, and verify that
the later are equivalent by subtracting the original matrix
and the one resulting from using the computed axis and angle
of rotation on the axisar_c call.

Example code begins here.

/.
Program raxisa_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{

/.
Local variables
./
SpiceDouble          angle;
SpiceDouble          angout;
SpiceDouble          axout  ;
SpiceDouble          r      ;
SpiceDouble          rout   ;

SpiceInt             i;

/.
Define an axis and an angle for rotation.
./
SpiceDouble          axis    = { 1.0, 2.0, 3.0 };

angle = 0.1 * twopi_c ( );

/.
Determine the rotation matrix.
./
axisar_c ( axis, angle, r );

/.
Now calculate the rotation axis and angle based on the
matrix as input.
./
raxisa_c ( r, axout, &angout );

printf( "Axis : %11.8f %11.8f %11.8f\n",
axout, axout, axout );
printf( "Angle: %11.8f\n", angout );
printf( " \n" );

/.
Now input the `axout' and `angout' to axisar_c to
compare against the original rotation matrix `r'.
./
printf( "Difference between input and output matrices:\n" );

axisar_c ( axout, angout, rout );

for ( i = 0; i < 3; i++ )
{

printf( "%20.16f %19.16f %19.16f\n",
rout[i] - r[i],
rout[i] - r[i],
rout[i] - r[i] );

}

return ( 0 );
}

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

Axis :  0.26726124  0.53452248  0.80178373
Angle:  0.62831853

Difference between input and output matrices:
-0.0000000000000001  0.0000000000000000  0.0000000000000000
0.0000000000000001 -0.0000000000000001  0.0000000000000000
0.0000000000000000  0.0000000000000001  0.0000000000000000

Note, the zero matrix is accurate to round-off error. A
numerical demonstration of equality.

2) This routine can be used to numerically approximate the
instantaneous angular velocity vector of a rotating object.

Suppose that R(t) is the rotation matrix whose columns
represent the inertial pointing vectors of the body-fixed axes
of an object at time t.

Then the angular velocity vector points along the vector given
by:

T
limit  axis( R(t+h)R )
h-->0

And the magnitude of the angular velocity at time t is given
by:

T
d angle ( R(t+h)R(t) )
----------------------   at   h = 0
dh

This code example computes the instantaneous angular velocity
vector of the Earth at 2000 Jan 01 12:00:00 TDB.

Use the PCK kernel below to load the required triaxial
ellipsoidal shape model and orientation data for the Earth.

pck00010.tpc

Example code begins here.

/.
Program raxisa_ex2
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{

/.
Local variables
./
SpiceDouble          angle;
SpiceDouble          angvel ;
SpiceDouble          axis   ;
SpiceDouble          infrot ;
SpiceDouble          h;
SpiceDouble          rt     ;
SpiceDouble          rth    ;
SpiceDouble          t;

/.
Load a PCK file containing a triaxial
ellipsoidal shape model and orientation
data for the Earth.
./
furnsh_c ( "pck00010.tpc" );

/.
Load time into the double precision variable `t'
and the delta time (1 ms) into the double precision
variable `h'
./
t = 0.0;
h = 1e-3;

/.
Get the rotation matrices from IAU_EARTH to J2000
at `t' and t+h.
./
pxform_c ( "IAU_EARTH", "J2000", t,   rt  );
pxform_c ( "IAU_EARTH", "J2000", t+h, rth );

/.
Compute the infinitesimal rotation R(t+h)R(t)**T
./
mxmt_c ( rth, rt, infrot );

/.
Compute the `axis' and `angle' of the infinitesimal rotation
./
raxisa_c ( infrot, axis, &angle );

/.
Scale `axis' to get the angular velocity vector
./
vscl_c ( angle/h, axis, angvel );

/.
Output the results.
./
printf( "Instantaneous angular velocity vector:\n" );
printf( "%14.10f %14.10f %14.10f\n",
angvel, angvel, angvel );

return ( 0 );
}

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

Instantaneous angular velocity vector:
0.0000000000   0.0000000000   0.0000729212

Restrictions

1)  If the input matrix is not a rotation matrix but is close enough
to pass the tests this routine performs on it, no error will be
signaled, but the results may have poor accuracy.

2)  The input matrix is taken to be an object that acts on (rotates)
vectors---it is not regarded as a coordinate transformation. To
find the axis and angle of a coordinate transformation, input
the transpose of that matrix to this routine.

None.

Author_and_Institution

N.J. Bachman        (JPL)
J. Diaz del Rio     (ODC Space)
W.L. Taber          (JPL)
F.S. Turner         (JPL)

Version

-CSPICE Version 1.0.2, 05-JUL-2021 (JDR)

Edited the header to comply with NAIF standard.