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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
       termpt_c ( Terminator points on an extended object )

       void termpt_c ( ConstSpiceChar      * method,
                       ConstSpiceChar      * ilusrc,
                       ConstSpiceChar      * target,
                       SpiceDouble           et,
                       ConstSpiceChar      * fixref,
                       ConstSpiceChar      * abcorr,
                       ConstSpiceChar      * corloc,
                       ConstSpiceChar      * obsrvr,
                       ConstSpiceDouble      refvec[3],
                       SpiceDouble           rolstp,
                       SpiceInt              ncuts,
                       SpiceDouble           schstp,
                       SpiceDouble           soltol,
                       SpiceInt              maxn,
                       SpiceInt              npts  [],
                       SpiceDouble           points[][3],
                       SpiceDouble           epochs[],
                       SpiceDouble           trmvcs[][3]  )
#### Abstract
       Find terminator points on a target body. The caller specifies
       half-planes, bounded by the illumination source center-target
       center vector, in which to search for terminator points.

       The terminator can be either umbral or penumbral. The umbral
       terminator is the boundary of the region on the target surface
       where no light from the source is visible. The penumbral
       terminator is the boundary of the region on the target surface
       where none of the light from the source is blocked by the target
       itself.

       The surface of the target body may be represented either by a
       triaxial ellipsoid or by topographic data.
#### Required_Reading
       CK
       DSK
       FRAMES
       NAIF_IDS
       PCK
       SPK
       TIME
#### Keywords
       DSK
       GEOMETRY
       SHADOW
       TERMINATOR
#### Brief_I/O
       VARIABLE  I/O  DESCRIPTION
       --------  ---  --------------------------------------------------
       method     I   Computation method.
       ilusrc     I   Illumination source.
       target     I   Name of target body.
       et         I   Epoch in ephemeris seconds past J2000 TDB.
       fixref     I   Body-fixed, body-centered target body frame.
       abcorr     I   Aberration correction.
       corloc     I   Aberration correction locus.
       obsrvr     I   Name of observing body.
       refvec     I   Reference vector for cutting half-planes.
       rolstp     I   Roll angular step for cutting half-planes.
       ncuts      I   Number of cutting half-planes.
       schstp     I   Angular step size for searching.
       soltol     I   Solution convergence tolerance.
       maxn       I   Maximum number of entries in output arrays.
       npts       O   Counts of terminator points corresponding to cuts.
       points     O   Terminator points.
       epochs     O   Times associated with terminator points.
       trmvcs     O   Terminator vectors emanating from the observer.
#### Detailed_Input
       method      is a short string providing parameters defining
                   the computation method to be used. In the syntax
                   descriptions below, items delimited by angle brackets
                   "<>" are to be replaced by actual values. Items
                   delimited by brackets "[]" are optional.

                   `method' may be assigned the following values:

                      "<shadow>/<curve type>/<shape specification>"

                   An example of such a string is

                      "UMBRAL/TANGENT/DSK/UNPRIORITIZED"

                   In the `method' string

                      <shadow> may be either of the strings

                         "UMBRAL"    indicates the terminator is the
                                     boundary of the portion of the surface
                                     that receives no light from the
                                     illumination source. The shape of the
                                     source is modeled as a sphere. See the
                                     -Particulars section below for details.

                         "PENUMBRAL" indicates the terminator is the
                                     boundary of the portion of the surface
                                     that receives all possible light from
                                     the illumination source. The shape of
                                     the source is modeled as a sphere.

                                     The penumbral terminator bounds the
                                     portion of the surface that is not
                                     subject to self-occultation of light
                                     from the illumination source. Given
                                     that the light source is modeled as a
                                     sphere, from any target surface point
                                     nearer to the source than the
                                     penumbral terminator, the source
                                     appears to be a lit disc. See the
                                     -Particulars section below for details.

                      <curve type> may be either of the strings

                         "TANGENT"   for topographic (DSK) target models
                                     indicates that a terminator point is
                                     defined as the point of tangency, on
                                     the surface represented by the
                                     specified data, of a line also tangent
                                     to the illumination source.

                                     For ellipsoidal target models, a
                                     terminator point is a point of
                                     tangency of a plane that is also
                                     tangent to the illumination source.
                                     See the -Particulars section below for
                                     details.

                                     Terminator points are generated within
                                     a specified set of "cutting"
                                     half-planes that have as an edge the
                                     line containing the illumination
                                     source center-target center vector.
                                     Multiple terminator points may be
                                     found within a given half-plane, if
                                     the target body shape allows for this.

                                     This is the highest-accuracy method
                                     supported by this routine. It
                                     generally executes much more slowly
                                     than the GUIDED method described
                                     below.

                         "GUIDED"    indicates that terminator points are
                                     "guided" so as to lie on rays
                                     emanating from the target body's
                                     center and passing through the
                                     terminator on the target body's
                                     reference ellipsoid. The terminator
                                     points are constrained to lie on the
                                     target body's surface. As with the
                                     "TANGENT" method (see above), cutting
                                     half-planes are used to generate
                                     terminator points.

                                     The GUIDED method produces a unique
                                     terminator point for each cutting
                                     half-plane. If multiple terminator
                                     point candidates lie in a given
                                     cutting half-plane, the outermost one
                                     is chosen.

                                     This method may be used only with the
                                     CENTER aberration correction locus
                                     (see the description of `corloc' below).

                                     Terminator points generated by this
                                     method are approximations; they are
                                     generally not true ray-surface tangent
                                     points. However, these approximations
                                     can be generated much more quickly
                                     than tangent points.

                      <shape specification> may be either of the strings

                         "DSK/UNPRIORITIZED[/SURFACES = <surface list>]"

                            The DSK option indicates that terminator point
                            computation is to use topographic data provided
                            by DSK files (abbreviated as "DSK data" below)
                            to model the surface of the target body.

                            The surface list specification is optional. The
                            syntax of the list is

                               <surface 1> [, <surface 2>...]

                            If present, it indicates that data only for the
                            listed surfaces are to be used; however, data
                            need not be available for all surfaces in the
                            list. If the list is absent, loaded DSK data
                            for any surface associated with the target body
                            are used.

                            The surface list may contain surface names or
                            surface ID codes. Names containing blanks must
                            be delimited by double quotes, for example

                               "SURFACES = \"Mars MEGDR 128 PIXEL/DEG\""

                            If multiple surfaces are specified, their names
                            or IDs must be separated by commas.

                            See the -Particulars section below for details
                            concerning use of DSK data.

                         "ELLIPSOID"

                            The ELLIPSOID shape option generates terminator
                            points on the target body's reference
                            ellipsoid. When the ELLIPSOID shape is
                            selected, The TANGENT curve option may be used
                            with any aberration correction locus, while the
                            GUIDED option may be used only with the CENTER
                            locus (see the description of `corloc' below).

                            When the locus is set to "CENTER", the
                            "TANGENT" and "GUIDED" curve options produce
                            the same results.

                      Neither case nor white space are significant in
                      `method', except within double-quoted strings. For
                      example, the string " eLLipsoid/tAnGenT " is valid.

                      Within double-quoted strings, blank characters are
                      significant, but multiple consecutive blanks are
                      considered equivalent to a single blank. Case is
                      not significant. So

                         \"Mars MEGDR 128 PIXEL/DEG\"

                      is equivalent to

                         \" mars megdr  128  pixel/deg \"

                      but not to

                         \"MARS MEGDR128PIXEL/DEG\"

       ilusrc      is the name of the illumination source. This source
                   may be any ephemeris object. Case, blanks, and
                   numeric values are treated in the same way as for the
                   input `target'.

                   The shape of the illumination source is considered
                   to be spherical. The radius of the sphere is the
                   largest radius of the source's reference ellipsoid.

       target      is the name of the target body. The target body is
                   an extended ephemeris object.

                   The string `target' is case-insensitive, and leading
                   and trailing blanks in `target' are not significant.
                   Optionally, you may supply a string containing the
                   integer ID code for the object. For example both
                   "MOON" and "301" are legitimate strings that indicate
                   the Moon is the target body.

                   When the target body's surface is represented by a
                   tri-axial ellipsoid, this routine assumes that a
                   kernel variable representing the ellipsoid's radii is
                   present in the kernel pool. Normally the kernel
                   variable would be defined by loading a PCK file.

       et          is the epoch of participation of the observer,
                   expressed as TDB seconds past J2000 TDB: `et' is
                   the epoch at which the observer's state is computed.

                   When aberration corrections are not used, `et' is also
                   the epoch at which the position and orientation of
                   the target body are computed.

                   When aberration corrections are used, the position
                   and orientation of the target body are computed at
                   et-lt, where `lt' is the one-way light time between the
                   aberration correction locus and the observer. The
                   locus is specified by the input argument `corloc'.
                   See the descriptions of `abcorr' and `corloc' below for
                   details.

       fixref      is the name of a body-fixed reference frame centered
                   on the target body. `fixref' may be any such frame
                   supported by the SPICE system, including built-in
                   frames (documented in the Frames Required Reading)
                   and frames defined by a loaded frame kernel (FK). The
                   string `fixref' is case-insensitive, and leading and
                   trailing blanks in `fixref' are not significant.

                   The output terminator points in the array `points' and
                   the output observer-terminator vectors in the array
                   `trmvcs' are expressed relative to this reference
                   frame.

       abcorr      indicates the aberration corrections to be applied
                   when computing the target's position and orientation.
                   Corrections are applied at the location specified by
                   the aberration correction locus argument `corloc',
                   which is described below.

                   For remote sensing applications, where apparent
                   terminator points seen by the observer are desired,
                   normally either of the corrections

                      "LT+S"
                      "CN+S"

                   should be used. These and the other supported options
                   are described below. `abcorr' may be any of the
                   following:

                      "NONE"     Apply no correction. Return the
                                 geometric terminator points on the
                                 target body.

                   Let `lt' represent the one-way light time between the
                   observer and the aberration correction locus. The
                   following values of `abcorr' apply to the "reception"
                   case in which photons depart from the locus at the
                   light-time corrected epoch et-lt and *arrive* at the
                   observer's location at `et':

                      "LT"       Correct for one-way light time (also
                                 called "planetary aberration") using a
                                 Newtonian formulation. This correction
                                 yields the locus at the moment it
                                 emitted photons arriving at the
                                 observer at `et'.

                                 The light time correction uses an
                                 iterative solution of the light time
                                 equation. The solution invoked by the
                                 "LT" option uses one iteration.

                                 Both the target position as seen by the
                                 observer, and rotation of the target
                                 body, are corrected for light time. The
                                 position of the illumination source as
                                 seen from the target is corrected as
                                 well.

                      "LT+S"     Correct for one-way light time and
                                 stellar aberration using a Newtonian
                                 formulation. This option modifies the
                                 locus obtained with the "LT" option to
                                 account for the observer's velocity
                                 relative to the solar system
                                 barycenter. These corrections yield
                                 points on the apparent terminator.

                      "CN"       Converged Newtonian light time
                                 correction. In solving the light time
                                 equation, the "CN" correction iterates
                                 until the solution converges. Both the
                                 position and rotation of the target
                                 body are corrected for light time. The
                                 position of the illumination source as
                                 seen from the target is corrected as
                                 well.

                      "CN+S"     Converged Newtonian light time and
                                 stellar aberration corrections. This
                                 option produces a solution that is at
                                 least as accurate at that obtainable
                                 with the "LT+S" option. Whether the
                                 "CN+S" solution is substantially more
                                 accurate depends on the geometry of the
                                 participating objects and on the
                                 accuracy of the input data. In all
                                 cases this routine will execute more
                                 slowly when a converged solution is
                                 computed.

       corloc      is a string specifying the aberration correction
                   locus: the point or set of points for which
                   aberration corrections are performed. `corloc' may be
                   assigned the values:

                      "CENTER"

                          Light time and stellar aberration corrections
                          are applied to the vector from the observer to
                          the center of the target body. The one way
                          light time from the target center to the
                          observer is used to determine the epoch at
                          which the target body orientation is computed.

                          This choice is appropriate for small target
                          objects for which the light time from the
                          surface to the observer varies little across
                          the entire target. It may also be appropriate
                          for large, nearly ellipsoidal targets when the
                          observer is very far from the target.

                          Computation speed for this option is faster
                          than for the ELLIPSOID TERMINATOR option.

                      "ELLIPSOID TERMINATOR"

                          Light time and stellar aberration corrections
                          are applied to individual terminator points on
                          the reference ellipsoid. For a terminator
                          point on the surface described by topographic
                          data, lying in a specified cutting half-plane,
                          the unique reference ellipsoid terminator
                          point in the same half-plane is used as the
                          locus of the aberration corrections.

                          This choice is appropriate for large target
                          objects for which the light time from the
                          terminator to the observer is significantly
                          different from the light time from the target
                          center to the observer.

                          Because aberration corrections are repeated
                          for individual terminator points,
                          computational speed for this option is
                          relatively slow.

       obsrvr      is the name of the observing body. The observing body
                   is an ephemeris object: it typically is a spacecraft,
                   the earth, or a surface point on the earth. `obsrvr' is
                   case-insensitive, and leading and trailing blanks in
                   `obsrvr' are not significant. Optionally, you may
                   supply a string containing the integer ID code for
                   the object. For example both "MOON" and "301" are
                   legitimate strings that indicate the Moon is the
                   observer.

       refvec,
       rolstp,
       ncuts       are, respectively, a reference vector, a roll step
                   angle, and a count of cutting half-planes.

                   `refvec' defines the first of a sequence of cutting
                   half-planes in which terminator points are to be
                   found. Each cutting half-plane has as its edge the
                   line containing the illumination source center-target
                   center vector; the first half-plane contains `refvec'.

                   `refvec' is expressed in the body-fixed reference frame
                   designated by `fixref'.

                   `rolstp' is an angular step by which to roll the cutting
                   half-planes about the target-illumination source vector,
                   which we'll call the "axis." The ith half-plane is
                   rotated from `refvec' about the axis in the
                   counter-clockwise direction by i*rolstp. Units are
                   radians. `rolstp' should be set to

                      2*pi/ncuts

                   to generate an approximately uniform distribution of
                   points along the terminator.

                   `ncuts' is the number of cutting half-planes used to
                   find terminator points; the angular positions of
                   consecutive half-planes increase in the positive
                   (counterclockwise) sense about the axis and are
                   distributed roughly equally about that vector: each
                   half-plane has angular separation of approximately

                      `rolstp' radians

                   from each of its neighbors. When the aberration
                   correction locus is set to "CENTER", the angular
                   separation is the value above, up to round-off.
                   When the locus is "TANGENT", the separations are
                   less uniform due to differences in the aberration
                   corrections used for the respective terminator points.

       schstp,
       soltol      are used only for DSK-based surfaces. These inputs
                   are, respectively, the search angular step size and
                   solution convergence tolerance used to find tangent
                   rays and associated terminator points within each
                   cutting half plane. These values are used when the
                   `method' argument includes the TANGENT option. In this
                   case, terminator points are found by a two-step
                   search process:

                      1) Bracketing: starting with a direction having
                         sufficiently small angular separation from the
                         axis, rays emanating from the surface of the
                         illumination source are generated within the
                         half-plane at successively greater angular
                         separations from the axis, where the increment
                         of angular separation is `schstp'. The rays are
                         tested for intersection with the target
                         surface. When a transition from
                         non-intersection to intersection is found, the
                         angular separation of a tangent ray has been
                         bracketed.

                      2) Root finding: each time a tangent ray is
                         bracketed, a search is done to find the angular
                         separation from the starting direction at which
                         a tangent ray exists. The search terminates
                         when successive rays are separated by no more
                         than `soltol'. When the search converges, the
                         last ray-surface intersection point found in
                         the convergence process is considered to be a
                         terminator point.

                   `schstp' and `soltol' have units of radians.

                   Target bodies with simple surfaces---for example,
                   convex shapes---will have a single terminator point
                   within each cutting half-plane. For such surfaces,
                   `schstp' can be set large enough so that only one
                   bracketing step is taken. A value greater than pi,
                   for example 4.0, is recommended.

                   Target bodies with complex surfaces can have
                   multiple terminator points within a given cutting
                   half-plane. To find all terminator points, `schstp'
                   must be set to a value smaller than the angular
                   separation of any two terminator points in any
                   cutting half-plane, where the vertex of the angle is
                   near a point on the surface of the illumination
                   source. `schstp' must not be too small, or the search
                   will be excessively slow.

                   For both kinds of surfaces, `soltol' must be chosen so
                   that the results will have the desired precision.
                   Note that the choice of `soltol' required to meet a
                   specified bound on terminator point height errors
                   depends on the illumination source-target distance.

       maxn        is the maximum number of terminator points that can
                   be stored in the output array `points'.
#### Detailed_Output
       npts        is an array of counts of terminator points within
                   the specified set of cutting half-planes. The Ith
                   element of `npts' is the terminator point count in the
                   Ith half-plane. `npts' should be declared with length
                   at least `ncuts'.

       points      is an array containing the terminator points found
                   by this routine. Terminator points are ordered by
                   the indices of the half-planes in which they're
                   found. The terminator points in a given half-plane
                   are ordered by decreasing angular separation from
                   the illumination source-target direction; the
                   outermost terminator point in a given half-plane is
                   the first of that set.

                   The terminator points for the half-plane containing
                   `refvec' occupy array elements

                       points[ 0         ][0]          through
                       points[ npts[0]-1 ][2]

                   Terminator points for the second half plane occupy
                   elements

                      points[ npts[0]           ][0]  through
                      points[ npts[0]+npts[1]-1 ][2]

                   and so on.

                   `points' should be declared with dimensions

                      [maxn][3]

                   Terminator points are expressed in the reference
                   frame designated by `fixref'. For each terminator
                   point, the orientation of the frame is evaluated at
                   the epoch corresponding to the terminator point; the
                   epoch is provided in the output array `epochs'
                   (described below).

                   Units of the terminator points are km.

       epochs      is an array of epochs associated with the terminator
                   points, accounting for light time if aberration
                   corrections are used. `epochs' contains one element
                   for each terminator point. `epochs' should be declared
                   with length

                      maxn

                   The element

                      epochs[i]

                   is associated with the terminator point

                      points[i][j], j = 0 to 2

                   If `corloc' is set to "CENTER", all values of `epochs'
                   will be the epoch associated with the target body
                   center. That is, if aberration corrections are used,
                   and if `lt' is the one-way light time from the target
                   center to the observer, the elements of `epochs' will
                   all be set to

                      et - lt

                   If `corloc' is set to "ELLIPSOID TERMINATOR", all
                   values of `epochs' for the terminator points in a
                   given half plane will be those for the reference
                   ellipsoid terminator point in that half plane. That
                   is, if aberration corrections are used, and if lt(i)
                   is the one-way light time to the observer from the
                   reference ellipsoid terminator point in the ith half
                   plane, the elements of `epochs' for that half plane
                   will all be set to

                      et - lt[i]

       trmvcs      is an array of vectors connecting the observer to
                   the terminator points. The terminator vectors are
                   expressed in the frame designated by `fixref'. For the
                   Ith vector, the orientation of the frame is
                   evaluated at the Ith epoch provided in the output
                   array `epochs' (described above).

                   `trmvcs' should be declared with dimensions

                      [maxn][3]

                   The elements

                      trmvcs[i][j], j = 0 to 2

                   are associated with the terminator point

                      points[i][j], j = 0 to 2

                   Units of the terminator vectors are km.
#### Parameters
       None.
#### Exceptions
       1)  If the specified aberration correction is unrecognized, an
           error is signaled by a routine in the call tree of this
           routine.

       2)  If transmission corrections are commanded, the error
           SPICE(INVALIDOPTION) is signaled by a routine in the call tree
           of this routine.

       3)  If either the target or observer input strings cannot be
           converted to an integer ID code, the error
           SPICE(IDCODENOTFOUND) is signaled by a routine in the call
           tree of this routine.

       4)  If `obsrvr' and `target' map to the same NAIF integer ID code, the
           error SPICE(BODIESNOTDISTINCT) is signaled by a routine in the
           call tree of this routine.

       5)  If the input target body-fixed frame `fixref' is not recognized,
           the error SPICE(NOFRAME) is signaled by a routine in the call
           tree of this routine. A frame name may fail to be recognized
           because a required frame specification kernel has not been
           loaded; another cause is a misspelling of the frame name.

       6)  If the input frame `fixref' is not centered at the target body,
           the error SPICE(INVALIDFRAME) is signaled by a routine in the
           call tree of this routine.

       7)  If the input argument `method' is not recognized, the error
           SPICE(INVALIDMETHOD) is signaled by either this routine or a
           routine in the call tree of this routine.

       8)  If `method' contains an invalid terminator type, the error
           SPICE(INVALIDTERMTYPE) is signaled by a routine in the call
           tree of this routine.

       9)  If the target and observer have distinct identities but are
           at the same location, the error SPICE(NOSEPARATION) is
           signaled by a routine in the call tree of this routine.

       10) If insufficient ephemeris data have been loaded prior to
           calling termpt_c, an error is signaled by a routine in
           the call tree of this routine. When light time correction is
           used, sufficient ephemeris data must be available to
           propagate the states of both observer and target to the solar
           system barycenter.

       11) If the computation method requires an ellipsoidal target shape
           and triaxial radii of the target body have not been loaded
           into the kernel pool prior to calling termpt_c, an error is
           signaled by a routine in the call tree of this routine.

           When the target shape is modeled by topographic data, radii
           of the reference triaxial ellipsoid are still required if
           the aberration correction locus is ELLIPSOID TERMINATOR or if
           the terminator point generation method is GUIDED.

       12) If the target body's shape is modeled as an ellipsoid, and if
           any of the radii of the target body are non-positive, an error
           is signaled by a routine in the call tree of this routine. The
           target must be an extended body.

       13) If PCK data specifying the target body-fixed frame orientation
           have not been loaded prior to calling termpt_c, an error is
           signaled by a routine in the call tree of this routine.

       14) If `method' specifies that the target surface is represented by
           DSK data, and no DSK files are loaded for the specified
           target, an error is signaled by a routine in the call tree
           of this routine.

       15) If the array bound `maxn' is less than 1, the error
           SPICE(INVALIDSIZE) is signaled by a routine in the call tree
           of this routine.

       16) If the number of cutting half-planes specified by `ncuts' is
           negative or greater than `maxn', the error SPICE(INVALIDCOUNT)
           is signaled by a routine in the call tree of this routine.

       17) If the aberration correction locus is not recognized, the
           error SPICE(INVALIDLOCUS) is signaled by a routine in the call
           tree of this routine.

       18) If the GUIDED terminator type is used with the ELLIPSOID
           TERMINATOR aberration correction locus, the error
           SPICE(BADTERMLOCUSMIX) is signaled by a routine in the call
           tree of this routine.

       19) If the reference vector `refvec' is the zero vector, the error
           SPICE(ZEROVECTOR) is signaled by a routine in the call tree of
           this routine.

       20) If the reference vector `refvec' and the observer target vector
           are linearly dependent, the error SPICE(DEGENERATECASE) is
           signaled by a routine in the call tree of this routine.

       21) If the terminator points cannot all be stored in the output
           `points' array, the error SPICE(OUTOFROOM) is signaled by a
           routine in the call tree of this routine.

       22) If `ncuts' is greater than 1, the roll step `rolstp' must be
           positive. Otherwise, the error SPICE(INVALIDROLLSTEP) is
           signaled by a routine in the call tree of this routine.

       23) If any of the `method', `ilusrc', `target', `fixref',
           `abcorr', `corloc' or `obsrvr' input string pointers is null,
           the error SPICE(NULLPOINTER) is signaled.

       24) If any of the `method', `ilusrc', `target', `fixref',
           `abcorr', `corloc' or `obsrvr' input strings has zero length,
           the error SPICE(EMPTYSTRING) is signaled.
#### Files
       Appropriate kernels must be loaded by the calling program before
       this routine is called.

       The following data are required:

       -  SPK data: ephemeris data for the target, observer, and
          illumination source must be loaded. If aberration
          corrections are used, the states of target and observer
          relative to the solar system barycenter must be calculable
          from the available ephemeris data. Typically ephemeris data
          are made available by loading one or more SPK files via
          furnsh_c.

       -  Target body orientation data: these may be provided in a text
          or binary PCK file. In some cases, target body orientation
          may be provided by one more more CK files. In either case,
          data are made available by loading the files via furnsh_c.

       -  Shape data for the target body:

            PCK data:

               If the target body shape is modeled as an ellipsoid,
               triaxial radii for the target body must be loaded into
               the kernel pool. Typically this is done by loading a
               text PCK file via furnsh_c.

               Triaxial radii are also needed if the target shape is
               modeled by DSK data but one or both of the GUIDED
               terminator definition method or the ELLIPSOID
               TERMINATOR aberration correction locus are selected.

            DSK data:

               If the target shape is modeled by DSK data, DSK files
               containing topographic data for the target body must be
               loaded. If a surface list is specified, data for at
               least one of the listed surfaces must be loaded.

       -  Shape data for the illumination source:

            PCK data:

               Triaxial radii for the illumination source must be
               loaded into the kernel pool. Typically this is done by
               loading a text PCK file via furnsh_c.

       The following data may be required:

       -  Frame data: if a frame definition is required to convert the
          observer and target states to the body-fixed frame of the
          target, that definition must be available in the kernel
          pool. Typically the definition is supplied by loading a
          frame kernel via furnsh_c.

       -  Surface name-ID associations: if surface names are specified
          in `method', the association of these names with their
          corresponding surface ID codes must be established by
          assignments of the kernel variables

             NAIF_SURFACE_NAME
             NAIF_SURFACE_CODE
             NAIF_SURFACE_BODY

          Normally these associations are made by loading a text
          kernel containing the necessary assignments. An example
          of such a set of assignments is

             NAIF_SURFACE_NAME += 'Mars MEGDR 128 PIXEL/DEG'
             NAIF_SURFACE_CODE += 1
             NAIF_SURFACE_BODY += 499

       -  SCLK data: if the target body's orientation is provided by
          CK files, an associated SCLK kernel must be loaded.

       In all cases, kernel data are normally loaded once per program
       run, NOT every time this routine is called.
#### Particulars
       Terminator definition
       =====================

       The definitions of terminators used by this routine vary
       depending on the target surface model.

       In all cases, the surface of the illumination source is
       modeled as a sphere.

       Ellipsoidal target surface model
       --------------------------------

       The umbral terminator is the boundary of the set of target
       surface points at which the illumination source is completely
       below the local tangent plane: the entire illumination source is
       below the horizon as seen from any surface point on the far side,
       relative to the source, of the umbral terminator. At an umbral
       terminator point, the target surface tangent plane containing
       that point is tangent to the surface of the light source as well,
       and the outward normal vectors at the two points of tangency are
       parallel.

       The penumbral terminator is the boundary of the set of target
       surface points at which the illumination source is completely
       above the local tangent plane: the entire illumination source is
       above the horizon as seen from any surface point on the near
       side, relative to the source, of the penumbral terminator. At a
       penumbral terminator point, the target surface tangent plane
       containing that point is tangent to the surface of the light
       source as well, and the outward normal vectors at the two points
       of tangency are anti-parallel.

       Topographic target surface model (DSK case)
       -------------------------------------------

       The concept of a plane tangent to both a topographic target
       surface and an illumination source is problematic. If the target
       tangent point is required to lie in a given cutting half-plane
       bounded by the line containing the target-source vector, the
       desired plane may not exist. In general, planes tangent to both
       the illumination source and the target will rest upon the high
       points of the target surface.

       For topographic target surface models, this routine uses a
       modified terminator definition: terminator points are target
       surface points at which a line is tangent to both the target and
       the illumination source. The line is constrained to lie in the
       plane containing the specified cutting half-plane. The concepts
       of umbral and penumbral terminators still apply. For umbral
       terminator points, the common tangent line does not cross the
       target-source line; for penumbral points, it does.

       Note that for ellipsoids, the terminator definitions based on
       tangent lines are not equivalent to the definitions based on
       tangent planes. Typically, a plane tangent to the target
       ellipsoid at a point found by the method described above will not
       be tangent to the illumination source: it will be rotated about
       the common tangent line and "cut into" the sphere representing
       the light source. This implies that some of the source will be
       visible at umbral terminator points and some will be blocked at
       penumbral terminator points: both umbral and penumbral terminator
       points found by this method will lie in a region bounded by the
       true terminators.

       The two definitions are equivalent for spherical targets.

       Using DSK data
       ==============

       DSK loading and unloading
       -------------------------

       DSK files providing data used by this routine are loaded by
       calling furnsh_c and can be unloaded by calling unload_c or
       kclear_c. See the documentation of furnsh_c for limits on numbers
       of loaded DSK files.

       For run-time efficiency, it's desirable to avoid frequent
       loading and unloading of DSK files. When there is a reason to
       use multiple versions of data for a given target body---for
       example, if topographic data at varying resolutions are to be
       used---the surface list can be used to select DSK data to be
       used for a given computation. It is not necessary to unload
       the data that are not to be used. This recommendation presumes
       that DSKs containing different versions of surface data for a
       given body have different surface ID codes.

       DSK data priority
       -----------------

       A DSK coverage overlap occurs when two segments in loaded DSK
       files cover part or all of the same domain---for example, a
       given longitude-latitude rectangle---and when the time
       intervals of the segments overlap as well.

       When DSK data selection is prioritized, in case of a coverage
       overlap, if the two competing segments are in different DSK
       files, the segment in the DSK file loaded last takes
       precedence. If the two segments are in the same file, the
       segment located closer to the end of the file takes
       precedence.

       When DSK data selection is unprioritized, data from competing
       segments are combined. For example, if two competing segments
       both represent a surface as sets of triangular plates, the
       union of those sets of plates is considered to represent the
       surface.

       Currently only unprioritized data selection is supported.
       Because prioritized data selection may be the default behavior
       in a later version of the routine, the UNPRIORITIZED keyword is
       required in the `method' argument.

       Syntax of the `method' input argument
       -------------------------------------

       The keywords and surface list in the `method' argument
       are called "clauses." The clauses may appear in any
       order, for example

          UMBRAL/TANGENT/DSK/UNPRIORITIZED/<surface list>
          DSK/UMBRAL/TANGENT/<surface list>/UNPRIORITIZED
          UNPRIORITIZED/<surface list>/DSK/TANGENT/UMBRAL

       The simplest form of the `method' argument specifying use of
       DSK data is one that lacks a surface list, for example:

          "PENUMBRAL/TANGENT/DSK/UNPRIORITIZED"
          "UMBRAL/GUIDED/DSK/UNPRIORITIZED"

       For applications in which all loaded DSK data for the target
       body are for a single surface, and there are no competing
       segments, the above strings suffice. This is expected to be
       the usual case.

       When, for the specified target body, there are loaded DSK
       files providing data for multiple surfaces for that body, the
       surfaces to be used by this routine for a given call must be
       specified in a surface list, unless data from all of the
       surfaces are to be used together.

       The surface list consists of the string

          SURFACES =

       followed by a comma-separated list of one or more surface
       identifiers. The identifiers may be names or integer codes in
       string format. For example, suppose we have the surface
       names and corresponding ID codes shown below:

          Surface Name                              ID code
          ------------                              -------
          "Mars MEGDR 128 PIXEL/DEG"                1
          "Mars MEGDR 64 PIXEL/DEG"                 2
          "Mars_MRO_HIRISE"                         3

       If data for all of the above surfaces are loaded, then
       data for surface 1 can be specified by either

          "SURFACES = 1"

       or

          "SURFACES = \"Mars MEGDR 128 PIXEL/DEG\""

       Double quotes are used to delimit the surface name because
       it contains blank characters.

       To use data for surfaces 2 and 3 together, any
       of the following surface lists could be used:

          "SURFACES = 2, 3"

          "SURFACES = \"Mars MEGDR  64 PIXEL/DEG\", 3"

          "SURFACES = 2, Mars_MRO_HIRISE"

          "SURFACES = \"Mars MEGDR 64 PIXEL/DEG\", Mars_MRO_HIRISE"

       An example of a `method' argument that could be constructed
       using one of the surface lists above is

         "UMBRAL/TANGENT/DSK/UNPRIORITIZED/SURFACES= \"Mars MEGDR 64 PIXEL/DEG\",3"
#### Examples
       The numerical results shown for these examples 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) Find apparent terminator points on Phobos as seen from Mars.
          Use the "umbral" shadow definition.

          Due to Phobos' irregular shape, the TANGENT terminator point
          definition will be used. It suffices to compute light time and
          stellar aberration corrections for the center of Phobos, so
          the CENTER aberration correction locus will be used. Use
          converged Newtonian light time and stellar aberration
          corrections in order to model the apparent position and
          orientation of Phobos.

          For comparison, compute terminator points using both ellipsoid
          and topographic shape models.

          Use the target body-fixed +Z axis as the reference direction
          for generating cutting half-planes. This choice enables the
          user to see whether the first terminator point is near the
          target's north pole.

          For each option, use just three cutting half-planes in order
          to keep the volume of output manageable. In most applications,
          the number of cuts and the number of resulting terminator
          points would be much greater.

          Use the meta-kernel shown below to load the required SPICE
          kernels.

             KPL/MK

             File: termpt_ex1.tm

             This meta-kernel is intended to support operation of SPICE
             example programs. The kernels shown here should not be
             assumed to contain adequate or correct versions of data
             required by SPICE-based user applications.

             In order for an application to use this meta-kernel, the
             kernels referenced here must be present in the user's
             current working directory.

             The names and contents of the kernels referenced
             by this meta-kernel are as follows:

                File name                        Contents
                ---------                        --------
                de430.bsp                        Planetary ephemeris
                mar097.bsp                       Mars satellite ephemeris
                pck00010.tpc                     Planet orientation and
                                                 radii
                naif0011.tls                     Leapseconds
                phobos512.bds                    DSK based on
                                                 Gaskell ICQ Q=512
                                                 Phobos plate model
             \begindata

                KERNELS_TO_LOAD = ( 'de430.bsp',
                                    'mar097.bsp',
                                    'pck00010.tpc',
                                    'naif0011.tls',
                                    'phobos512.bds' )
             \begintext

             End of meta-kernel

          Example code begins here.

          /.
             Program termpt_ex1

             Find terminator points on Phobos as seen from Mars.

             Compute terminator points using the tangent
             definition, using the "umbral" shadow type.
             The sun is the illumination source. Perform
             aberration corrections for the target center.
             Use both ellipsoid and DSK shape models.
          ./

          #include <stdio.h>
          #include "SpiceUsr.h"

          int main()
          {
             /.
             Local constants
             ./
             #define META            "termpt_ex1.tm"
             #define MTHLEN          51
             #define NMETH            2
             #define MAXN         10000

             /.
             Local variables
             ./
             SpiceChar             * abcorr;
             SpiceChar             * corloc;
             SpiceChar             * fixref;
             SpiceChar             * ilusrc;
             SpiceChar             * obsrvr;
             SpiceChar             * target;

             SpiceChar               method [NMETH][MTHLEN] =

                               { "UMBRAL/TANGENT/ELLIPSOID",
                                 "UMBRAL/TANGENT/DSK/UNPRIORITIZED" };

             SpiceDouble             delrol;
             SpiceDouble             dist;
             SpiceDouble             et;
             SpiceDouble             lt;
             SpiceDouble             points  [MAXN][3];
             SpiceDouble             pos     [3];
             SpiceDouble             roll;
             SpiceDouble             schstp;
             SpiceDouble             soltol;
             SpiceDouble             trmvcs  [MAXN][3];
             SpiceDouble             trgeps  [MAXN];
             SpiceDouble             z       [3] = { 0.0, 0.0, 1.0 };

             SpiceInt                i;
             SpiceInt                j;
             SpiceInt                k;
             SpiceInt                ncuts;
             SpiceInt                npts    [MAXN];
             SpiceInt                start;

             /.
             Load kernel files via the meta-kernel.
             ./
             furnsh_c ( META );

             /.
             Set illumination source, target, observer,
             and target body-fixed, body-centered reference frame.
             ./
             ilusrc = "SUN";
             obsrvr = "MARS";
             target = "PHOBOS";
             fixref = "IAU_PHOBOS";

             /.
             Set aberration correction and correction locus.
             ./
             abcorr = "CN+S";
             corloc = "CENTER";

             /.
             Convert the UTC request time string seconds past
             J2000, TDB.
             ./
             str2et_c ( "2008 AUG 11 00:00:00", &et );

             /.
             Compute a set of terminator points using light
             time and stellar aberration corrections. Use
             both ellipsoid and DSK shape models. Use an
             angular step size corresponding to a height of
             about 100 meters to ensure we don't miss the
             terminator. Set the convergence tolerance to limit
             the height convergence error to about 1 meter.
             Compute 3 terminator points for each computation
             method.

             Get the approximate light source-target distance
             at ET. We'll ignore the observer-target light
             time for this approximation.
             ./

             spkpos_c ( ilusrc, et, "J2000", abcorr,
                        target, pos, &lt            );

             dist   = vnorm_c( pos );

             schstp = 1.0e-1 / dist;
             soltol = 1.0e-3 / dist;
             ncuts  = 3;

             printf ( "\n"
                      "Light source:   %s\n"
                      "Observer:       %s\n"
                      "Target:         %s\n"
                      "Frame:          %s\n"
                      "\n"
                      "Number of cuts: %d\n",
                      ilusrc,
                      obsrvr,
                      target,
                      fixref,
                      (int)ncuts            );

             delrol = twopi_c() / ncuts;

             for ( i = 0;  i < NMETH;  i++ )
             {
                termpt_c ( method[i], ilusrc, target, et,
                           fixref,    abcorr, corloc, obsrvr,
                           z,         delrol, ncuts,  schstp,
                           soltol,    MAXN,   npts,   points,
                           trgeps,    trmvcs                  );
                /.
                Write the results.
                ./
                printf ( "\n"
                         "Computation method = %s\n"
                         "Locus              = %s\n"
                         "\n",
                         method[i],
                         corloc                     );

                start = 0;

                for ( j = 0;  j < ncuts;  j++ )
                {
                   roll = j * delrol;

                   printf ( "\n"
                            "  Roll angle (deg) = %21.9f\n"
                            "     Target epoch  = %21.9f\n"
                            "     Number of terminator points at "
                            "this roll angle: %d\n",
                            roll * dpr_c(),
                            trgeps[j],
                            npts[j]                            );

                   printf ( "      Terminator points:\n" );

                   for ( k = 0;  k < npts[j];  k++ )
                   {
                      printf ( " %20.9f %20.9f %20.9f\n",
                               points[k+start][0],
                               points[k+start][1],
                               points[k+start][2]        );
                   }

                   start += npts[j];
                }
             }
             printf ( "\n" );

             return ( 0 );
          }

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

          Light source:   SUN
          Observer:       MARS
          Target:         PHOBOS
          Frame:          IAU_PHOBOS

          Number of cuts: 3

          Computation method = UMBRAL/TANGENT/ELLIPSOID
          Locus              = CENTER

            Roll angle (deg) =           0.000000000
               Target epoch  =   271684865.152078211
               Number of terminator points at this roll angle: 1
                Terminator points:
                    2.040498332          5.012722925          8.047281838

            Roll angle (deg) =         120.000000000
               Target epoch  =   271684865.152078211
               Number of terminator points at this roll angle: 1
                Terminator points:
                  -11.058054707          0.167672089         -4.782740292

            Roll angle (deg) =         240.000000000
               Target epoch  =   271684865.152078211
               Number of terminator points at this roll angle: 1
                Terminator points:
                    8.195238564         -6.093889437         -5.122310498

          Computation method = UMBRAL/TANGENT/DSK/UNPRIORITIZED
          Locus              = CENTER

            Roll angle (deg) =           0.000000000
               Target epoch  =   271684865.152078211
               Number of terminator points at this roll angle: 1
                Terminator points:
                    1.626396122          3.995432317          8.853689531

            Roll angle (deg) =         120.000000000
               Target epoch  =   271684865.152078211
               Number of terminator points at this roll angle: 1
                Terminator points:
                  -11.186659739         -0.142366278         -4.646137201

            Roll angle (deg) =         240.000000000
               Target epoch  =   271684865.152078211
               Number of terminator points at this roll angle: 1
                Terminator points:
                    9.338447077         -6.091352469         -5.960849305

       2) Find apparent terminator points on Mars as seen from the
          earth.

          Use both the "umbral" and "penumbral" shadow definitions. Use
          only ellipsoid shape models for easier comparison. Find
          distances between corresponding terminator points on the
          umbral and penumbral terminators.

          Use the ELLIPSOID TERMINATOR aberration correction locus
          in order to perform separate aberration corrections for
          each terminator point. Because of the large size of Mars,
          corrections for the target center are less accurate.

          For each option, use just three cutting half-planes, in order
          to keep the volume of output manageable. In most applications,
          the number of cuts and the number of resulting terminator
          points would be much greater.

          Use the meta-kernel shown below to load the required SPICE
          kernels.

             KPL/MK

             File: termpt_ex2.tm

             This meta-kernel is intended to support operation of SPICE
             example programs. The kernels shown here should not be
             assumed to contain adequate or correct versions of data
             required by SPICE-based user applications.

             In order for an application to use this meta-kernel, the
             kernels referenced here must be present in the user's
             current working directory.

             The names and contents of the kernels referenced
             by this meta-kernel are as follows:

                File name                        Contents
                ---------                        --------
                de430.bsp                        Planetary ephemeris
                mar097.bsp                       Mars satellite ephemeris
                pck00010.tpc                     Planet orientation and
                                                 radii
                naif0011.tls                     Leapseconds
                megr90n000cb_plate.bds           Plate model based on
                                                 MEGDR DEM, resolution
                                                 4 pixels/degree.

             \begindata

                KERNELS_TO_LOAD = ( 'de430.bsp',
                                    'mar097.bsp',
                                    'pck00010.tpc',
                                    'naif0011.tls',
                                    'megr90n000cb_plate.bds' )

             \begintext

             End of meta-kernel

          Example code begins here.

          /.
             Program termpt_ex2

             Find terminator points on Mars as seen from the
             earth.

             Use only ellipsoid shape models. Use the
             ELLIPSOID TERMINATOR aberration correction
             locus.

             Use both UMBRAL and PENUMBRAL shadow definitions.
             Compute the distances between corresponding
             umbral and penumbral terminator points.

             Check terminator points by computing solar
             incidence angles at each point.
          ./

          #include <stdio.h>
          #include <math.h>
          #include "SpiceUsr.h"

          int main()
          {
             /.
             Local constants
             ./
             #define CORLEN          21
             #define META            "termpt_ex2.tm"
             #define MTHLEN          51
             #define NMETH            2
             #define MAXN         10000

             /.
             Local variables
             ./
             SpiceChar             * abcorr;

             SpiceChar               corloc [NMETH][CORLEN] =

                               { "ELLIPSOID TERMINATOR",
                                 "ELLIPSOID TERMINATOR" };

             SpiceChar             * fixref;

             SpiceChar               ilumth [NMETH][MTHLEN] =

                               { "ELLIPSOID",
                                 "ELLIPSOID" };

             SpiceChar             * ilusrc;
             SpiceChar             * obsrvr;
             SpiceChar             * target;

             SpiceChar               method [NMETH][MTHLEN] =

                            { "UMBRAL/TANGENT/ELLIPSOID",
                              "PENUMBRAL/TANGENT/ELLIPSOID" };

             SpiceDouble             adjang;
             SpiceDouble             alt;
             SpiceDouble             angsrc;
             SpiceDouble             delrol;
             SpiceDouble             dist;
             SpiceDouble             emissn;
             SpiceDouble             et;
             SpiceDouble             f;
             SpiceDouble             ilupos  [3];
             SpiceDouble             lat;
             SpiceDouble             lon;
             SpiceDouble             lt;
             SpiceDouble             phase;
             SpiceDouble             points  [MAXN][3];
             SpiceDouble             svpnts  [MAXN][3];
             SpiceDouble             tptilu  [3];
             SpiceDouble             radii   [3];
             SpiceDouble             re;
             SpiceDouble             roll;
             SpiceDouble             rp;
             SpiceDouble             schstp;
             SpiceDouble             solar;
             SpiceDouble             soltol;
             SpiceDouble             srcrad  [3];
             SpiceDouble             srfvec  [3];
             SpiceDouble             trmvcs  [MAXN][3];
             SpiceDouble             trgepc;
             SpiceDouble             trgeps  [MAXN];
             SpiceDouble             z       [3] = { 0.0, 0.0, 1.0 };

             SpiceInt                i;
             SpiceInt                j;
             SpiceInt                k;
             SpiceInt                m;
             SpiceInt                n;
             SpiceInt                ncuts;
             SpiceInt                npts    [MAXN];
             SpiceInt                start;

             /.
             Load kernel files via the meta-kernel.
             ./
             furnsh_c ( META );

             /.
             Set illumination source, target, observer,
             and target body-fixed, body-centered reference frame.
             ./
             ilusrc = "SUN";
             obsrvr = "EARTH";
             target = "MARS";
             fixref = "IAU_MARS";

             /.
             Set the aberration correction.
             ./
             abcorr = "CN+S";

             /.
             Convert the UTC request time string seconds past
             J2000, TDB.
             ./
             str2et_c ( "2008 AUG 11 00:00:00", &et );

             /.
             Look up the target body's radii. We'll use these to
             convert Cartesian to planetographic coordinates. Use
             the radii to compute the flattening coefficient of
             the reference ellipsoid.
             ./
             bodvrd_c ( target, "RADII", 3, &n, radii );

             /.
             Compute the flattening coefficient for planetodetic
             coordinates
             ./
             re = radii[0];
             rp = radii[2];
             f  = ( re - rp ) / re;

             /.
             Get the radii of the illumination source as well.
             We'll use these radii to compute the angular radius
             of the source as seen from the terminator points.
             ./
             bodvrd_c ( ilusrc, "RADII", 3, &n, srcrad );

             /.
             Compute a set of terminator points using light time and
             stellar aberration corrections. Use both ellipsoid
             and DSK shape models.

             Get the approximate light source-target distance
             at ET. We'll ignore the observer-target light
             time for this approximation.
             ./
             spkpos_c ( ilusrc, et,     fixref, abcorr,
                        target, ilupos, &lt            );

             dist = vnorm_c( ilupos );

             /.
             Set the angular step size so that a single step will
             be taken in the root bracketing process; that's all
             that is needed since we don't expect to have multiple
             terminator points in any cutting half-plane.
             ./
             schstp = 4.0;

             /.
             Set the convergence tolerance to minimize the
             height error. We can't achieve the precision
             suggested by the formula because the sun-Mars
             distance is about 2.4e8 km. Compute 3 terminator
             points for each computation method.
             ./
             soltol = 1.e-7/dist;

             /.
             Set the number of cutting half-planes and roll step.
             ./
             ncuts  = 3;
             delrol = twopi_c() / ncuts;

             printf ( "\n"
                      "Light source:   %s\n"
                      "Observer:       %s\n"
                      "Target:         %s\n"
                      "Frame:          %s\n"
                      "\n"
                      "Number of cuts: %d\n",
                      ilusrc,
                      obsrvr,
                      target,
                      fixref,
                      (int)ncuts            );

             delrol = twopi_c() / ncuts;

             for ( i = 0;  i < NMETH;  i++ )
             {
                termpt_c ( method[i], ilusrc, target, et,
                           fixref,    abcorr, corloc, obsrvr,
                           z,         delrol, ncuts,  schstp,
                           soltol,    MAXN,   npts,   points,
                           trgeps,    trmvcs                  );
                /.
                Write the results.
                ./
                printf ( "\n\n"
                         "Computation method = %s\n"
                         "Locus              = %s\n",
                         method[i],
                         corloc[i]                  );

                start = 0;

                for ( j = 0;  j < ncuts;  j++ )
                {
                   roll = j * delrol;

                   printf ( "\n"
                            "   Roll angle (deg) = %21.9f\n"
                            "    Target epoch    = %21.9f\n"
                            "    Number of terminator points at "
                            "this roll angle: %d\n",
                            roll * dpr_c(),
                            trgeps[j],
                            npts[j]                            );

                   for ( k = 0;  k < npts[j];  k++ )
                   {
                      printf ( "    Terminator point planetodetic "
                               "coordinates:\n" );

                      m = k+start;

                      recgeo_c ( points[m], re, f,
                                 &lon,          &lat, &alt );

                      printf ( "      Longitude        (deg): %20.9f\n"
                               "      Latitude         (deg): %20.9f\n"
                               "      Altitude          (km): %20.9f\n",
                               lon * dpr_c(),
                               lat * dpr_c(),
                               alt                                     );

                      /.
                      Get illumination angles for this terminator point.
                      ./

                      illumg_c ( ilumth[i], target,  ilusrc, et,
                                 fixref,    abcorr,  obsrvr,
                                 points[m], &trgepc, srfvec,
                                 &phase,    &solar,  &emissn    );

                      printf ( "      Incidence angle  (deg): "
                               "%20.9f\n",   solar * dpr_c()    );

                      /.
                      Adjust the incidence angle for the angular
                      radius of the illumination source. Use the
                      epoch associated with the terminator point
                      for this lookup.
                      ./
                      spkpos_c ( ilusrc, trgeps[m], fixref,
                                 abcorr, target,    tptilu, &lt );

                      dist   = vnorm_c( tptilu );

                      angsrc = asin ( maxd_c( 3, srcrad[0],
                                                 srcrad[1],
                                                 srcrad[2] ) / dist );

                      if ( i == 0 )
                      {
                         /.
                         For points on the umbral terminator,
                         the ellipsoid outward normal is tilted
                         away from the terminator-source center
                         direction by the angular radius of the
                         source. Subtract this radius from the
                         illumination incidence angle to get the
                         angle between the local normal and the
                         direction to the corresponding tangent
                         point on the source.
                         ./
                         adjang = solar - angsrc;
                      }
                      else
                      {
                         /.
                         For the penumbral case, the outward
                         normal is tilted toward the illumination
                         source by the angular radius of the
                         source. Adjust the illumination
                         incidence angle for this.
                         ./
                         adjang = solar + angsrc;
                      }

                      printf ( "      Adjusted angle   (deg): "
                               "%20.9f\n",   adjang * dpr_c() );

                      if ( i == 0 )
                      {
                         /.
                         Save terminator points for comparison.
                         ./
                         vequ_c ( points[m], svpnts[m] );
                      }
                      else
                      {
                         /.
                         Compare terminator points with last
                         saved values.
                         ./
                         dist = vdist_c( points[m], svpnts[m] );

                         printf ( "      Distance offset  (km):  "
                                  "%20.9f\n",   dist             );
                      }
                   }
                   start += npts[j];
                }
             }
             printf ( "\n" );

             return ( 0 );
          }

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

          Light source:   SUN
          Observer:       EARTH
          Target:         MARS
          Frame:          IAU_MARS

          Number of cuts: 3

          Computation method = UMBRAL/TANGENT/ELLIPSOID
          Locus              = ELLIPSOID TERMINATOR

             Roll angle (deg) =           0.000000000
              Target epoch    =   271683700.369686902
              Number of terminator points at this roll angle: 1
              Terminator point planetodetic coordinates:
                Longitude        (deg):          4.189318082
                Latitude         (deg):         66.416132677
                Altitude          (km):          0.000000000
                Incidence angle  (deg):         90.163842885
                Adjusted angle   (deg):         89.999999980

             Roll angle (deg) =         120.000000000
              Target epoch    =   271683700.372003794
              Number of terminator points at this roll angle: 1
              Terminator point planetodetic coordinates:
                Longitude        (deg):        107.074551917
                Latitude         (deg):        -27.604435701
                Altitude          (km):          0.000000000
                Incidence angle  (deg):         90.163842793
                Adjusted angle   (deg):         89.999999888

             Roll angle (deg) =         240.000000000
              Target epoch    =   271683700.364983618
              Number of terminator points at this roll angle: 1
              Terminator point planetodetic coordinates:
                Longitude        (deg):        -98.695906077
                Latitude         (deg):        -27.604435700
                Altitude          (km):          0.000000000
                Incidence angle  (deg):         90.163843001
                Adjusted angle   (deg):         90.000000096

          Computation method = PENUMBRAL/TANGENT/ELLIPSOID
          Locus              = ELLIPSOID TERMINATOR

             Roll angle (deg) =           0.000000000
              Target epoch    =   271683700.369747400
              Number of terminator points at this roll angle: 1
              Terminator point planetodetic coordinates:
                Longitude        (deg):          4.189317837
                Latitude         (deg):         66.743818467
                Altitude          (km):          0.000000000
                Incidence angle  (deg):         89.836157094
                Adjusted angle   (deg):         89.999999999
                Distance offset  (km):          19.483590936

             Roll angle (deg) =         120.000000000
              Target epoch    =   271683700.372064054
              Number of terminator points at this roll angle: 1
              Terminator point planetodetic coordinates:
                Longitude        (deg):        107.404259674
                Latitude         (deg):        -27.456458359
                Altitude          (km):          0.000000000
                Incidence angle  (deg):         89.836157182
                Adjusted angle   (deg):         90.000000087
                Distance offset  (km):          19.411414247

             Roll angle (deg) =         240.000000000
              Target epoch    =   271683700.365043879
              Number of terminator points at this roll angle: 1
              Terminator point planetodetic coordinates:
                Longitude        (deg):        -99.025614323
                Latitude         (deg):        -27.456458357
                Altitude          (km):          0.000000000
                Incidence angle  (deg):         89.836156972
                Adjusted angle   (deg):         89.999999877
                Distance offset  (km):          19.411437239
#### Restrictions
       None.
#### Literature_References
       None.
#### Author_and_Institution
       N.J. Bachman        (JPL)
       J. Diaz del Rio     (ODC Space)
#### Version
       -CSPICE Version 1.0.1, 10-AUG-2021 (JDR)

           Edited the header to comply with NAIF standard.
           Added missing exceptions.

       -CSPICE Version 1.0.0, 05-APR-2017 (NJB)
#### Index_Entries
       find terminator points on target body
#### Link to routine termpt_c source file termpt_c.c

    Fri Dec 31 18:41:14 2021