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Index Page surfpv_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

       surfpv_c ( Surface point and velocity )

       void surfpv_c ( ConstSpiceDouble      stvrtx[6],
                       ConstSpiceDouble      stdir [6],
                       SpiceDouble           a,
                       SpiceDouble           b,
                       SpiceDouble           c,
                       SpiceDouble           stx   [6],
                       SpiceBoolean        * found      )

#### Abstract

       Find the state (position and velocity) of the surface intercept
       defined by a specified ray, ray velocity, and ellipsoid.

#### Required_Reading

       None.

#### Keywords

       ELLIPSOID
       GEOMETRY

#### Brief_I/O

       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       stvrtx     I   State of ray's vertex.
       stdir      I   State of ray's direction vector.
       a          I   Length of ellipsoid semi-axis along the X-axis.
       b          I   Length of ellipsoid semi-axis along the Y-axis.
       c          I   Length of ellipsoid semi-axis along the Z-axis.
       stx        O   State of surface intercept.
       found      O   Flag indicating whether intercept state was found.

#### Detailed_Input

       stvrtx      is the state of a ray's vertex. The first three
                   components of `stvrtx' are the vertex's x, y, and z
                   position components; the vertex's x, y, and z
                   velocity components follow.

                   The reference frame relative to which `stvrtx' is
                   specified has axes aligned with with those of a
                   triaxial ellipsoid. See the description below of
                   the arguments `a', `b', and `c'.

                   The vertex may be inside or outside of this
                   ellipsoid, but not on it, since the surface
                   intercept is a discontinuous function at
                   vertices on the ellipsoid's surface.

                   No assumption is made about the units of length
                   and time, but these units must be consistent with
                   those of the other inputs.

       stdir       is the state of the input ray's direction vector.
                   The first three components of `stdir' are a non-zero
                   vector giving the x, y, and z components of the
                   ray's direction; the direction vector's x, y, and
                   z velocity components follow.

                   `stdir' is specified relative to the same reference
                   frame as is `stvrtx'.

       a,
       b,
       c           are, respectively, the lengths of a triaxial
                   ellipsoid's semi-axes lying along the x, y, and
                   z axes of the reference frame relative to which
                   `stvrtx' and `stdir' are specified.

#### Detailed_Output

       stx         is the state of the intercept of the input ray on
                   the surface of the input ellipsoid. The first
                   three components of `stx' are the intercept's x, y,
                   and z position components; the intercept's x, y,
                   and z velocity components follow.

                   `stx' is specified relative to the same reference
                   frame as are `stvrtx' and `stdir'.

                   `stx' is defined if and only if both the intercept
                   and its velocity are computable, as indicated by
                   the output argument `found'.

                   The position units of `stx' are the same as those of
                   `stvrtx', `stdir', and `a', `b', and `c'. The time units are
                   the same as those of `stvrtx' and `stdir'.

       found       is a logical flag indicating whether `stx' is
                   defined. `found' is SPICETRUE if and only if both the
                   intercept and its velocity are computable. Note
                   that in some cases the intercept may computable
                   while the velocity is not; this can happen for
                   near-tangency cases.

#### Parameters

       None.

#### Exceptions

       1)  If the input ray's direction vector is the zero vector, an
           error is signaled by a routine in the call tree of this
           routine.

       2)  If any of the ellipsoid's axis lengths is nonpositive, an
           error is signaled by a routine in the call tree of this
           routine.

       3)  If the vertex of the ray is on the ellipsoid, the error
           SPICE(INVALIDVERTEX) is signaled by a routine in the call tree
           of this routine.

#### Files

       None.

#### Particulars

       The position and velocity of the ray's vertex as well as the
       ray's direction vector and velocity vary with time. The
       inputs to surfpv_c may be considered the values of these
       vector functions at a particular time, say t0. Thus

          State of vertex:           stvrtx = ( V(t0), V'(t0) )

          State of direction vector: stdir  = ( D(t0), D'(t0) )

       To determine the intercept point, W(t0), we simply compute the
       intersection of the ray originating at V(t0) in the direction of
       D(t0) with the ellipsoid

            2      2      2
           x      y      z
          --- +  --- +  ---  =  1
            2      2      2
           A      B      C

       W(t) is the path of the intercept point along the surface of
       the ellipsoid. To determine the velocity of the intercept point,
       we need to take the time derivative of W(t), and evaluate it at
       t0. Unfortunately W(t) is a complicated expression, and its
       derivative is even more complicated.

       However, we know that the derivative of W(t) at t0, W'(t0), is
       tangent to W(t) at t0. Thus W'(t0) lies in the plane that is tangent
       to the ellipsoid at t0. Let X(t) be the curve in the tangent plane
       that represents the intersection of the ray emanating from V(t0)
       with direction D(t0) with that tangent plane.

          X'(t0) = W'(t0)

       The expression for X'(t) is much simpler than that of W'(t);
       surfpv_c evaluates X'(t) at t0.

       Derivation of X(t) and X'(t)
       ----------------------------------------------------------------

       W(t0) is the intercept point. Let N be a surface normal at I(t0).
       Then the tangent plane at W(t0) is the set of points X(t) such
       that

          < X(t) - I(t0), N > = 0

       X(t) can be expressed as the vector sum of the vertex
       and some scalar multiple of the direction vector,

          X(t) = V(t) + s(t) * D(t)

       where s(t) is a scalar function of time. The derivative of
       X(t) is given by

          X'(t) = V'(t)  +  s(t) * D'(t)  +  s'(t) * D(t)

       We have V(t0), V'(t0), D(t0), D'(t0), W(t0), and N, but to
       evaluate X'(t0), we need s(t0) and s'(t0). We derive an
       expression for s(t) as follows.

       Because X(t) is in the tangent plane, it must satisfy

          < X(t) - W(t0), N > = 0.

       Substituting the expression for X(t) into the equation above
       gives

          < V(t) + s(t) * D(t) - W(t0), N > = 0.

       Thus

          < V(t) - W(t0), N >  +  s(t) * < D(t), N > = 0,

       and
                       < V(t) - W(t0), N >
          s(t)  =  -  ---------------------
                           < D(t), N >

       The derivative of s(t) is given by

          s'(t) =

              < D(t),N > * < V'(t),N >  -  < V(t)-W(t0),N > * < D'(t),N >
          -  -------------------------------------------------------------
                                               2
                                    < D(t), N >

#### Examples

       The numerical results shown for this example 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) Illustrate the role of the ray vertex velocity and
          ray direction vector velocity via several simple cases. Also
          show the results of a near-tangency computation.

          Example code begins here.

          /.
             Program surfpv_ex1
          ./
          #include <stdio.h>
          #include "SpiceUsr.h"

          int main()
          {

             SpiceBoolean            found;

             SpiceDouble             a;
             SpiceDouble             b;
             SpiceDouble             c;
             SpiceDouble             stvrtx [6];
             SpiceDouble             stdir  [6];
             SpiceDouble             stx    [6];

             SpiceInt                i;

             a = 1.0;
             b = 2.0;
             c = 3.0;

             printf ( "\nEllipsoid radii: \n"
                      "     a = %f\n"
                      "     b = %f\n"
                      "     c = %f\n",
                      a,
                      b,
                      c                      );

             for ( i = 0;  i < 3;  i++ )
             {
                if ( i == 0 )
                {
                   printf ( "\n%s\n\n",
                            "Case 1: Vertex varies, direction is constant"  );

                   stvrtx[0] =  2.0;
                   stvrtx[1] =  0.0;
                   stvrtx[2] =  0.0;
                   stvrtx[3] =  0.0;
                   stvrtx[4] =  0.0;
                   stvrtx[5] =  3.0;

                   stdir[0]  = -1.0;
                   stdir[1]  =  0.0;
                   stdir[2]  =  0.0;
                   stdir[3]  =  0.0;
                   stdir[4]  =  0.0;
                   stdir[5]  =  0.0;
                }
                else if ( i == 1 )
                {
                   printf ( "\n%s\n\n",
                            "Case 2: Vertex and direction both vary"  );

                   stvrtx[0] =  2.0;
                   stvrtx[1] =  0.0;
                   stvrtx[2] =  0.0;
                   stvrtx[3] =  0.0;
                   stvrtx[4] =  0.0;
                   stvrtx[5] =  3.0;

                   stdir[0]  = -1.0;
                   stdir[1]  =  0.0;
                   stdir[2]  =  0.0;
                   stdir[3]  =  0.0;
                   stdir[4]  =  0.0;
                   stdir[5]  =  4.0;
                }
                else
                {
                   printf ( "\n%s\n\n",
                            "Case 3: Vertex and direction both vary; "
                            "near-tangent case"                       );

                   stvrtx[2] =  c - 1.e-15;
                   stvrtx[5] =  1.e299;
                   stdir[5]  =  1.e299;
                }

                printf ( "Vertex:\n"
                         " %20.12e %20.12e %20.12e\n",
                         stvrtx[0],
                         stvrtx[1],
                         stvrtx[2]                    );

                printf ( "Vertex velocity:\n"
                         " %20.12e %20.12e %20.12e\n",
                         stvrtx[3],
                         stvrtx[4],
                         stvrtx[5]                    );

                printf ( "Direction:\n"
                         " %20.12e %20.12e %20.12e\n",
                         stdir[0],
                         stdir[1],
                         stdir[2]                     );

                printf ( "Direction velocity:\n"
                         " %20.12e %20.12e %20.12e\n",
                         stdir[3],
                         stdir[4],
                         stdir[5]                     );

                surfpv_c ( stvrtx, stdir, a, b, c, stx, &found );

                if ( !found)
                {
                   printf ( " No intercept state found.\n" );
                }
                else
                {
                   printf ( "Intercept:\n"
                            " %20.12e %20.12e %20.12e\n",
                            stx[0],
                            stx[1],
                            stx[2]                        );

                   printf ( "Intercept velocity:\n"
                            " %20.12e %20.12e %20.12e\n\n",
                            stx[3],
                            stx[4],
                            stx[5]                        );
                }
             }

             return ( 0 );
          }

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

          Ellipsoid radii:
               a = 1.000000
               b = 2.000000
               c = 3.000000

          Case 1: Vertex varies, direction is constant

          Vertex:
             2.000000000000e+00   0.000000000000e+00   0.000000000000e+00
          Vertex velocity:
             0.000000000000e+00   0.000000000000e+00   3.000000000000e+00
          Direction:
            -1.000000000000e+00   0.000000000000e+00   0.000000000000e+00
          Direction velocity:
             0.000000000000e+00   0.000000000000e+00   0.000000000000e+00
          Intercept:
             1.000000000000e+00   0.000000000000e+00   0.000000000000e+00
          Intercept velocity:
             0.000000000000e+00   0.000000000000e+00   3.000000000000e+00

          Case 2: Vertex and direction both vary

          Vertex:
             2.000000000000e+00   0.000000000000e+00   0.000000000000e+00
          Vertex velocity:
             0.000000000000e+00   0.000000000000e+00   3.000000000000e+00
          Direction:
            -1.000000000000e+00   0.000000000000e+00   0.000000000000e+00
          Direction velocity:
             0.000000000000e+00   0.000000000000e+00   4.000000000000e+00
          Intercept:
             1.000000000000e+00   0.000000000000e+00   0.000000000000e+00
          Intercept velocity:
             0.000000000000e+00   0.000000000000e+00   7.000000000000e+00

          Case 3: Vertex and direction both vary; near-tangent case

          Vertex:
             2.000000000000e+00   0.000000000000e+00   3.000000000000e+00
          Vertex velocity:
             0.000000000000e+00   0.000000000000e+00  1.000000000000e+299
          Direction:
            -1.000000000000e+00   0.000000000000e+00   0.000000000000e+00
          Direction velocity:
             0.000000000000e+00   0.000000000000e+00  1.000000000000e+299
          Intercept:
             2.580956827952e-08   0.000000000000e+00   3.000000000000e+00
          Intercept velocity:
           -3.874532036208e+306   0.000000000000e+00  2.999999974190e+299

#### Restrictions

       None.

#### Literature_References

       None.

#### Author_and_Institution

       N.J. Bachman        (JPL)
       J. Diaz del Rio     (ODC Space)
       J.E. McLean         (JPL)
       W.L. Taber          (JPL)

#### Version

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

           Edited the header to comply with NAIF standard.
           Reformatted example's output to comply with maximum line length
           for header comments.

           Updated -Exceptions section.

       -CSPICE Version 1.0.1, 22-JAN-2009 (NJB) (JEM) (WLT)

           Corrected header typo.

       -CSPICE Version 1.0.0, 05-JAN-2009 (NJB) (JEM) (WLT)

#### Index_Entries

       ellipsoid surface point and velocity

#### Link to routine surfpv_c source file surfpv_c.c

    Fri Dec 31 18:41:13 2021