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Index Page pjelpl_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
       pjelpl_c ( Project ellipse onto plane )

       void pjelpl_c ( ConstSpiceEllipse  * elin,
                       ConstSpicePlane    * plane,
                       SpiceEllipse       * elout  )
#### Abstract
       Project an ellipse onto a plane, orthogonally.
#### Required_Reading
       ELLIPSES
       PLANES
#### Keywords
       ELLIPSE
       GEOMETRY
       MATH
#### Brief_I/O
       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       elin       I   A SPICE ellipse to be projected.
       plane      I   A plane onto which elin is to be projected.
       elout      O   A SPICE ellipse resulting from the projection.
#### Detailed_Input
       elin,
       plane       are, respectively, a SPICE ellipse and a
                   SPICE plane. The geometric ellipse represented
                   by elin is to be orthogonally projected onto the
                   geometric plane represented by plane.
#### Detailed_Output
       elout       is a SPICE ellipse that represents the geometric
                   ellipse resulting from orthogonally projecting the
                   ellipse represented by inel onto the plane
                   represented by plane.
#### Parameters
       None.
#### Exceptions
       1)  If the input plane is invalid, an error is signaled by a
           routine in the call tree of this routine.

       2)  The input ellipse may be degenerate--its semi-axes may be
           linearly dependent. Such ellipses are allowed as inputs.

       3)  The ellipse resulting from orthogonally projecting the input
           ellipse onto a plane may be degenerate, even if the input
           ellipse is not.
#### Files
       None.
#### Particulars
       Projecting an ellipse orthogonally onto a plane can be thought of
       finding the points on the plane that are `under' or `over' the
       ellipse, with the `up' direction considered to be perpendicular
       to the plane. More mathematically, the orthogonal projection is
       the set of points Y in the plane such that for some point X in
       the ellipse, the vector Y - X is perpendicular to the plane.
       The orthogonal projection of an ellipse onto a plane yields
       another ellipse.
#### Examples
       1)  With  center  = { 1.,  1.,  1. },
                 vect1   = { 2.,  0.,  0. },
                 vect2   = { 0.,  1.,  1. },
                 normal  = { 0.,  0.,  1. }

           the code fragment

                 nvc2pl_c ( normal,  0.,      plane           );
                 cgv2el_c ( center,  vect1,   vect2,   elin   );
                 pjelpl_c ( elin,    plane,   elout           );
                 el2cgv_c ( elout,   prjctr,  prjmaj,  prjmin );

           returns

                 prjctr  = { 1.,  1.,  0. },
                 prjmaj  = { 2.,  0.,  0. },
                 prjmin  = { 0.,  1.,  0. }

       2)  With  vect1   = { 2.,  0.,  0. },
                 vect2   = { 1.,  1.,  1. },
                 center  = { 0.,  0.,  0. },
                 normal  = { 0.,  0.,  1. },

           the code fragment

                 nvc2pl_c ( normal,  0.,      plane           );
                 cgv2el_c ( center,  vect1,   vect2,   elin   );
                 pjelpl_c ( elin,    plane,   elout           );
                 el2cgv_c ( elout,   prjctr,  prjmaj,  prjmin );

           returns

                 prjctr  = { 0.,  0.,  0. };

                 prjmaj  = { -2.227032728823213,
                             -5.257311121191336e-1,
                              0.                  };

                 prjmin  = {  2.008114158862273e-1,
                             -8.506508083520399e-1,
                              0.                  };

       3)    An example of actual use: Suppose we wish to compute the
             distance from an ellipsoid to a line. Let the line be
             defined by a point P and a direction vector DIRECT; the
             line is the set of points

                P   +   t * DIRECT,

             where t is any real number. Let the ellipsoid have semi-
             axis lengths A, B, and C.

             We can reduce the problem to that of finding the distance
             between the line and an ellipse on the ellipsoid surface by
             considering the fact that the surface normal at the nearest
             point to the line will be orthogonal to DIRECT; the set of
             surface points where this condition holds lies in a plane,
             and hence is an ellipse on the surface. The problem can be
             further simplified by projecting the ellipse orthogonally
             onto the plane defined by

                < X, DIRECT >  =  0.

             The problem is then a two dimensional one: find the
             distance of the projected ellipse from the intersection of
             the line and this plane (which is necessarily one point).
             A `paraphrase' of the relevant code is:

                #include "SpiceUsr.h"
                     .
                     .
                     .
                /.
                Step 1. Find the candidate ellipse cand.
                          normal is a normal vector to the plane
                          containing the candidate ellipse. The
                          ellipse must exist, since it's the
                          intersection of an ellipsoid centered at
                          the origin and a plane containing the
                          origin. For this reason, we don't check
                          inedpl_c's "found flag" found below.
                ./

                normal[0]  =  direct[0] / (a*a);
                normal[1]  =  direct[1] / (b*b);
                normal[2]  =  direct[2] / (c*c);

                nvc2pl_c ( normal, 0., &candpl );

                inedpl_c ( a, b, c, &candpl, cand, &found );

                /.
                Step 2. Project the candidate ellipse onto a
                          plane orthogonal to the line. We'll
                          call the plane prjpl and the
                          projected ellipse prjel.
                ./
                nvc2pl_c ( direct,  0.,     &prjpl );
                pjelpl_c ( &cand,   &prjpl, &prjel );

                /.
                Step 3. Find the point on the line lying in the
                          projection plane, and then find the
                          near point pjnear on the projected
                          ellipse. Here prjpt is the point on the
                          input line that lies in the projection
                          plane. The distance between prjpt and
                          pjnear is dist.
                ./

                vprjp_c  ( linept,  &prjpl,  prjpt          );
                npelpt_c ( &prjel,   prjpt,  pjnear,  &dist );

                /.
                Step 4. Find the near point pnear on the
                         ellipsoid by taking the inverse
                         orthogonal projection of PJNEAR; this is
                         the point on the candidate ellipse that
                         projects to pjnear. Note that the output
                         dist was computed in step 3.

                         The inverse projection of pjnear is
                         guaranteed to exist, so we don't have to
                         check found.
                ./
                vprjpi_c ( pjnear, &prjpl, &candpl, pnear, &found );

                /.
                The value of dist returned is the distance we're looking
                for.

                The procedure described here is carried out in the routine
                npedln_c.
                ./
#### Restrictions
       None.
#### Literature_References
       None.
#### Author_and_Institution
       N.J. Bachman        (JPL)
       J. Diaz del Rio     (ODC Space)
#### Version
       -CSPICE Version 1.0.1, 24-AUG-2021 (JDR)

           Edited the header to comply with NAIF standard.

       -CSPICE Version 1.0.0, 02-SEP-1999 (NJB)
#### Index_Entries
       project ellipse onto plane
#### Link to routine pjelpl_c source file pjelpl_c.c

    Fri Dec 31 18:41:10 2021