Compute the axis of the rotation given by an input matrix and the angle of the rotation about that axis.
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.
matrix is a 3x3 rotation matrix in double precision.
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 to pi radians.
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.
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 bodyfixed 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 Thus to approximate the angular velocity vector the following code fragment will do [ Load t into the double precision variable t Load h into the double precision variable h Load r(t+h) into the 3 by 3 double precision array rth Load r(t) into the 3 by 3 double precision array rt . . . ] /. T Compute the infinitesimal rotation r(t+h)r(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 );
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.
1) If the input matrix is not a rotation matrix (a fairly loose tolerance is used to check this) a routine in the call tree of this routine will signal an error indicating the problem. 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 ).
N.J. Bachman (JPL) W.L. Taber (JPL) F.S. Turner (JPL)
-CSPICE Version 1.0.1, 05-JAN-2005 (NJB) (WLT) (FST) Various header updates were made to reflect changes made to the underlying SPICELIB Fortran code. Miscellaneous header corrections were made as well. -CSPICE Version 1.0.0, 31-MAY-1999 (WLT) (NJB)
rotation axis of a matrix