Procedure

   SpiceDouble vsepg_c ( ConstSpiceDouble * v1,
                         ConstSpiceDouble * v2,
                         SpiceInt           ndim )

Abstract

   vsepg_c finds the separation angle in radians between two double
   precision vectors of arbitrary dimension. This angle is defined
   as zero if either vector is zero.

Required_Reading

   None.

Keywords

    ANGLE
    VECTOR

Brief_I/O

    VARIABLE  I/O  DESCRIPTION
    --------  ---  --------------------------------------------------
     v1        I     First vector.
     v2        I     Second vector.
     ndim      I     The number of elements in v1 and v2.

Detailed_Input

    v1      is any double precision vector of arbitrary dimension.
    v2      is also a double precision vector of arbitrary dimension.
            v1 or v2 or both may be the zero vector.
    ndim    is the dimension of the both of the input vectors
            v1 and v2.

Detailed_Output

    vsepg_c the angle between v1 and v2 expressed in radians.
            vsepg_c is strictly non-negative.  For input vectors of
            four or more dimensions, the angle is defined as the
            generalization of the definition for three dimensions.
            If either v1 or v2 is the zero vector, then vsepg_c is
            defined to be 0 radians.

Parameters

    None.

Exceptions

   Error free.

Files

    None

Particulars

    In four or more dimensions this angle does not have a physically
    realizable interpretation.  However, the angle is defined as
    the generalization of the following definition which is valid in
    three or two dimensions:

    In the plane, it is a simple matter to calculate the angle
    between two vectors once the two vectors have been made to be
    unit length.  Then, since the two vectors form the two equal
    sides of an isosceles triangle, the length of the third side
    is given by the expression

          length = 2.0 * sine ( vsepg/2.0 )

    The length is given by the magnitude of the difference of the
    two unit vectors

          length = norm ( u1 - u2 )

    Once the length is found, the value of vsepg_c may be calculated
    by inverting the first expression given above as

          vsepg_c = 2.0 * arcsine ( length/2.0 )

    This expression becomes increasingly unstable when vsepg_c gets
    larger than pi/2 or 90 degrees.  In this situation (which is
    easily detected by determining the sign of the dot product of
    v1 and v2) the supplementary angle is calculated first and
    then vsepg_c is given by

          vsepg_c = pi - SUPPLEMENTARY_ANGLE

Examples

    The following table gives sample values for v1, v2 and vsepg_c
    implied by the inputs.

    v1               v2               ndim          vsepg_c
    -----------------------------------------------------------------
    (1, 0, 0, 0)     (1, 0, 0, 0)       4           0.0
    (1, 0, 0)        (0, 1, 0)          3           pi/2 (=1.71...)
    (3, 0)           (-5, 0)            2           pi   (=3.14...)

Restrictions

    The user is required to insure that the input vectors will not
    cause floating point overflow upon calculation of the vector
    dot product since no error detection or correction code is
    implemented.  In practice, this is not a significant
    restriction.

Literature_References

    None

Author_and_Institution

   C.A. Curzon     (JPL)
   K.R. Gehringer  (JPL)
   H.A. Neilan     (JPL)
   W.L. Taber      (JPL)
   E.D. Wright     (JPL)

Version

   -CSPICE Version 1.0.0, 29-JUN-1999

Index_Entries

   angular separation of n-dimensional vectors

Link to routine vsepg_c source file vsepg_c.c