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axisar_c
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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

   void axisar_c ( ConstSpiceDouble  axis   [3],
                   SpiceDouble       angle,
                   SpiceDouble       r      [3][3]  ) 

Abstract

 
   Construct a rotation matrix that rotates vectors by a specified 
   angle about a specified axis. 
 

Required_Reading

 
   ROTATION 
 

Keywords

 
   MATRIX 
   ROTATION 
 

Brief_I/O

 
   Variable  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. 
 

Parameters

 
   None. 
 

Exceptions

 
   Error free. 
 
   1)  If axis is the zero vector, the rotation generated is the 
       identity.  This is consistent with the specification of vrotv. 
 

Files

 
   None. 
 

Particulars

 
   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 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

 
   1)  A matrix that rotates vectors by pi/2 radians about the z-axis 
       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 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) 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     ); 
 

Restrictions

 
   None. 
 

Literature_References

 
   None. 
 

Author_and_Institution

 
   N.J. Bachman   (JPL) 
 

Version

 
   -CSPICE Version 1.0.0, 18-JUN-1999 (NJB)

Index_Entries

 
   axis and angle to rotation 
 
Wed Apr  5 17:54:28 2017