void axisar_c ( ConstSpiceDouble axis [3],
SpiceDouble angle,
SpiceDouble r [3][3] )
Construct a rotation matrix that rotates vectors by a specified
angle about a specified axis.
ROTATION
MATRIX
ROTATION
Variable I/O Description
  
axis I Rotation axis.
angle I Rotation angle, in radians.
r O Rotation matrix corresponding to axis and angle.
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.
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.
None.
Error free.
1) If axis is the zero vector, the rotation generated is the
identity. This is consistent with the specification of vrotv.
None.
axisar_c can be thought of as a partial inverse of raxisa_c.
axisar_c is really is a `left inverse': the code fragment
raxisa_c ( r, axis, &angle );
axisar_c ( axis, angle, r );
preserves r, except for roundoff 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 roundoff 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.
1) A matrix that rotates vectors by pi/2 radians about the zaxis
can be found using the code fragment
axis[0] = 0.
axis[1] = 0.
axis[2] = 1.
axisar_c ( axis, halfpi_c(), r );
The returned matrix r will equal
+ +
 0 1 0 
 
 1 0 0 .
 
 0 0 1 
+ +
2) Linear interpolation between two rotation matrices:
Let r(t) be a timevarying rotation matrix; r could be
a Cmatrix 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) 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.
If we let
r1 = r(t1)
r2 = r(t2), and
1
q = r2 * r1 ,
then the rotation axis and angle of q define the rotation
that each column of r(t) undergoes from time t1 to time
t2. Since r(t) is orthogonal, we can find q using the
transpose of r1. We find the rotation axis and angle via
raxisa_c.
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), which we'll call r3.
axisar_c ( axis, frac * angle, delta );
mxm_c ( delta, r1, r3 );
None.
None.
N.J. Bachman (JPL)
CSPICE Version 1.0.0, 18JUN1999 (NJB)
axis and angle to rotation
