 
Windows Required Reading
===========================================================================
 
   Last revised on 2009 MAR 26 by E. D. Wright.
 
 
Abstract
--------------------------------------------------------
 
   SPICE Toolkits define the SPICE data type "window" for use in the
   manipulation of continuous intervals on the real line. Corresponding
   window functions in CSPICE are designed to simplify the use of this data
   type.
 
 
Introduction
--------------------------------------------------------
 
   By definition, a window is a double precision SPICE cell that contains
   zero or more intervals.
 
   An interval is an ordered pair of numbers,
 
      [ a(i), b(i) ]
 
   such that
 
      a(i)  <  b(i)
            -
 
   The intervals within a window are both ordered and disjoint. That is,
   the beginning of each interval is greater than the end of the previous
   interval:
 
      b(i)  <  a(i+1)
 
   This restriction is enforced primarily because it allows efficient
   window operations.
 
   The intervals stored in windows typically represent intervals of time
   (seconds, days, or centuries past a reference epoch). However, windows
   can represent any kinds of intervals.
 
 
Basic Concepts
 
   ``size'' - The maximum number of elements storable in a SPICE cell.
   Cells are created with a constant size.
 
   ``cardinality - The number of elements stored in a cell. ``cardinality''
   describes how much of ``size'' is used. ``cardinality'' satisfies the
   relationship:
 
         cardinality < size
                     -
 
   ``measure'' - the measure of an interval is the length of the interval,
   that is the difference of its endpoints:
 
      m(i) = b(i) - a(i)
 
   Note that the singleton interval
 
      [ a(i), a(i) ]
 
   has measure zero. The window
 
      [1,2], [4,9], [16,16]
 
   contains intervals of measure 1, 5, and 0 respectively.
 
   The concept of measure extends readily to the gaps between adjacent
   intervals. In the example above, the window contains gaps of measure 2
   and 7. Intervals separated by gaps of measure zero or less are said to
   overlap. Overlapping intervals created by the window functions are
   merged as soon as they are created.
 
   Finally, the measure of a window is the sum of the measures of its
   intervals. In the example above, the measure of the window is 6. Note
   that a floating point window containing only singletons has measure
   zero.
 
 
The window data type
--------------------------------------------------------
 
   SPICE windows are a form of the SPICE cell data type [1], so any
   restriction that applies to the use of cells applies to the use of
   windows as well. The implementation of a SPICE window differs according
   to the Toolkit language.
 
   The CSPICE implementation of SPICE windows consists of SpiceDouble cells
   created and initialized to empty using the macro SPICEDOUBLE_CELL.
 
   Example:
 
      #include "SpiceUsr.h"
 
         #define MAXSIZ       8
 
         /*
         Define a cell 'win' to use as a window.
 
         This window can hold eight (MAXSIZ = 8) SpiceDouble values,
         thus four intervals.
         */
 
         SPICEDOUBLE_CELL ( win, MAXSIZ );
 
   The SPICEDOUBLE_CELL macro (defined in SpiceCell.h) creates a structure
   of type SpiceCell having cell name `win', maximum cell cardinality
   `MAXSIZ'.
 
   The cell size, `MAXSIZ', must have an even value to use the cell as a
   window. Two values defines a window interval, so a window of N intervals
   requires a cell of size 2*N.
 
   SPICE windows, when appearing as arguments to the CSPICE window
   functions, always have type (SpiceCell *). The SpiceCell structure has
   members (also defined in SpiceCell.h):
 
         dtype      Data type of cell: character, integer, or double
                    precision.
 
                    'dtype' has type SpiceCellDataType.
 
         length     For character cells, the declared length of the
                    cell's string array. Unused as zero for non-string
                    cells.
 
         size       The maximum number of data items that can be stored in
                    the cell's data array.
 
         card       The cell's "cardinality": the number of data items
                    currently present in the cell.
 
         isSet      Boolean flag indicating whether the cell is a CSPICE
                    set. Sets have no duplicate data items, and their data
                    items are stored in increasing order.
 
         adjust     Boolean flag indicating whether the cell's data area
                    has adjustable size.  (Adjustable size cell data areas
                    are not currently implemented.)
 
         init       Boolean flag indicating whether the cell has been
                    initialized.
 
         base       is a void pointer to the associated data array.  base
                    points to the start of the control area of this array.
 
         data       is a void pointer to the first data slot in the
                    associated data array. This slot is the element
                    following the control area.
 
   Any of the general cell functions in CSPICE may be used with windows.
   For example, copyd_c may be used to copy the contents of one window into
   another, and the function cardd_c may be used to determine the number of
   endpoints (that is, twice the number of intervals) in a window.
 
   All errors are reported via standard CSPICE error handling.
 
   With the exception of the initialization functions, all window functions
   assume that input cells do contain valid windows---that is, ordered and
   distinct sets of endpoints. The windows subsystem may not signal errors
   resulting from attempts to operate on invalid windows.
 
 
References
--------------------------------------------------------
 
         [1]   CELLS Required Reading (cells.req).
 
 
Window Functions
===========================================================================
 
   The window functions in CSPICE fall into the following categories.
 
       --   Initialization
 
       --   Unary
 
       --   Binary
 
       --   Complement
 
       --   Comparison
 
 
Initialization Functions
--------------------------------------------------------
 
   wnvald_c takes as input a double precision SPICE cell containing pairs
   of endpoints and validates it to form a window. The calling sequence is
   shown below.
 
      wnvald_c ( size, n, &window )
 
   On input, `window' is a cell of size `size' containing `n' endpoints.
   During validation, the intervals are ordered, and overlapping intervals
   are merged. On output, the cardinality of `window' is the number of
   endpoints remaining, and the window is ready for use with any of the
   window functions.
 
   Because validation is done in place, there is no chance of overflow.
   However, other errors may be detected. For example, if the left endpoint
   of any interval is greater than the corresponding right endpoint,
   wnvald_c signals an error.
 
   Validation is primarily useful for ordering and merging intervals added
   to a cell by appndd_c, or directly assigned to the cell.
 
   Building a large window is done most efficiently by assigning the window
   elements and then calling wnvald_c. Building up the window by repeated
   insertion requires repeated ordering operations; wnvald_c does a single
   ordering operation.
 
 
Unary Functions
--------------------------------------------------------
 
   Each unary function operates on a single window. Six unary operations
   are supported, each of which alters the contents of the input window.
   The calling sequences are shown below.
 
      wncond_c ( left, right, &window )      { Contract }
 
      wnexpd_c ( left, right, &window )      { Expand }
 
      wnextd_c ( side, &window )             { Extract }
 
      wnfild_c ( small, &window )            { Fill }
 
      wnfltd_c ( small, &window )            { Filter }
 
      wninsd_c ( left, right, &window )      { Insert }
 
   Each of the unary window functions works in place. That is, only one
   window, `window', appears in each calling sequence, serving as both
   input and output. Windows whose original contents need to be preserved
   should be copied prior to calling any of the unary functions.
 
   wninsd_c inserts the interval whose endpoints are `left' and `right'
   into `window'. If the input interval overlaps any of the intervals in
   the window, the intervals are merged. Thus, the cardinality of `window'
   can actually decrease as the result of an insertion.
 
   wnexpd_c and wncond_c expand (lengthen) and contract (shorten) each of
   the intervals in `window'. The adjustments are not necessarily
   symmetric. That is, wnexpd_c works by subtracting `left' units from the
   left endpoint of each interval and adding `right' units to the right
   endpoint of each interval. wncond_c is the same as `exp' with the signs
   of the arguments reversed, and is primarily provided for clarity in
   coding. (Expansion by negative increments is a messy concept.) Intervals
   are merged when expansion causes them to overlap. Intervals are dropped
   when they are contracted by amounts greater than their measures.
 
   wnfltd_c and wnfild_c remove small intervals and small gaps between
   adjacent intervals. Both functions take as input a minimum measure,
   `small'. wnfltd_c filters out (drops) intervals with measures less than
   or equal to `small', while wnfild_c merges adjacent intervals separated
   by gaps with measures less than or equal to `small'.
 
   Depending on the value of `side', wnextd_c extracts the left or right
   endpoints of each interval in `window'. The resulting window contains
   only the singleton intervals
 
      [ a(1), a(1) ], ..., [ a(n), a(n) ]
 
   or
 
      [ b(1), b(1) ], ..., [ b(n), b(n) ]
 
 
Binary Functions
--------------------------------------------------------
 
   Binary functions operate on two input windows to produce a third
   (distinct) output window. Three major binary operations are supported.
   The calling sequences are shown below.
 
      wnunid_c ( &a, &b, &c )      { Union }
 
      wnintd_c ( &a, &b, &c )      { Intersection }
 
      wndifd_c ( &a, &b, &c )      { Difference }
 
   In contrast with the unary functions, none of the binary functions work
   in place. The output window, `c', must be distinct from both of the
   input windows, `a' and `b'. We will have more to say about this later
   on.
 
   wnunid_c places the union of `a' and `b' into `c'. The union of two
   windows contains every point that is contained in the first window, or
   in the second window, or in both windows.
 
   wnintd_c places the intersection of `a' and `b' into `c'. The
   intersection of two windows contains every point that is contained in
   the first window AND in the second.
 
   wndifd_c places the difference of `a' and `b' into `c'. The difference
   of two windows contains every point that is contained in the first
   window, but NOT in the second.
 
   In each case, if the output window, `c', is not large enough to hold the
   result of the operation, as many intervals as will fit are inserted into
   the window, and a SPICE error is signaled. (You can control the effect
   of this error on your program; refer to Error Required Reading.)
 
   In each of the binary functions, the output window must be distinct from
   both of the input windows. All three of the binary operations can, in
   principle, be performed in place, but not all can be performed
   efficiently. Consequently, for the sake of consistency, none of the
   functions work in place. For example, the following calls are invalid.
 
      wnintd_c ( &a, &b, &a );
 
      wnintd_c ( &a, &b, &b );
 
 
   In each of the examples above, whether or not the function signals an
   error, the results will almost certainly be wrong. Nearly the same
   effect can be achieved, however, by placing the result into a temporary
   window, which can be immediately copied back into one of the input
   windows, as shown below.
 
      wnintd_c ( &a, &b, &temp );
 
      copy_c ( &temp, &a );
 
 
Complement Functions
--------------------------------------------------------
 
   wncomd_c determines the complement of a window with respect to an
   interval. The calling sequence is
 
      wncomd_c ( left, right, &a, &c )     { Complement }
 
   As with the binary functions, the output window, `c', must be distinct
   from the input window, `a'.
 
   Mathematically, the complement of a window contains those points that
   are not contained in the window. That is, the complement of the set of
   closed intervals
 
      [ a(1), b(1) ], [ a(2), b(2) ], ..., [ a(n), b(n) ]
 
   is the set of open intervals
 
      ( -inf, a(1) ), ( b(1), a(2) ), ..., ( b(n), +inf )
 
   Not all computer languages offer a satisfactory way to represent
   infinity, so wncomd_c must take the complement with respect to a finite
   interval.
 
   Since the results of a window function must be another window, wncomd_c
   returns the closure of the set theoretical complement. In short, the
   double precision complement of the window
 
      [ a(1), b(1) ], [ a(2), b(2) ], ..., [ a(n), b(n) ]
 
   with respect to the interval from `left' to `right' is the intersection
   of the windows
 
      ( -inf, a(1) ], [ b(1), a(2) ], ..., [ b(n), +inf )
 
   and [ `left', `right' ].
 
   Intervals of measure zero (singleton intervals) in the original window
   are replaced by gaps of measure zero, which are filled. Thus,
   complementing a window twice does not necessarily yield the original
   window.
 
 
Comparison Functions
--------------------------------------------------------
 
   Comparison functions allow the contents of windows to be compared
   against the contents of other windows. There are four comparison
   functions: three logical functions and one function. The calling
   sequences are shown below.
 
      wnelmd_c ( point, &window )                      { Element }
 
      wnincd_c ( left, right, &window )                { Inclusion }
 
      wnreld_c ( &a, op, &b )                          { Relation }
 
      wnsumd_c ( &win, &meas, &avg, &stddev,
                       &shortest, &longest)            { Summary }
 
   wnelmd_c returns true if the input point, `point', is an element of the
   input window, `window'---that is, whenever the point lies within one of
   the intervals of the window.
 
   Similarly, wnincd_c is true whenever the input interval, from `left' to
   `right', is included in the input window, `window'---that is, whenever
   the interval lies entirely within one of the intervals of the window.
 
   wnreld_c is true whenever a specified relationship between the input
   windows, `a' and `b', is satisfied. Each relationship corresponds to a
   comparison operator, `op'. The complete set of operators recognized by
   wnreld_c is shown below.
 
      "="          is equal to (contains the same intervals as)
      "<>"         is not equal to
      "<="         is a subset of
      "<"          is a proper subset of
      ">="         is a superset of
      ">"          is a proper superset of
 
   For example, the expression
 
      wnreld_c ( &needed, "<=", &avail )
 
 
   is true whenever the window `needed' is a subset of the window `avail'.
   One window is a subset of another window if each of the intervals in the
   first window is included in one of the intervals in the second window.
   In addition, the first window is a proper subset of the second if the
   second window contains at least one point not contained in the first
   window. The following pairs of expressions are equivalent.
 
      wnreld_c ( &a, ">", &b )
      wnreld_c ( &b, "<", &a )
 
      wnreld_c ( &a, ">=", &b )
      wnreld_c ( &b, "<=", &a )
 
   wnsumd_c provides a summary of the input window, `window'. It computes
   the measure of the window, `meas', and the average, `avg', and standard
   deviation, `stddev', of the measures of the individual intervals in the
   window. It also returns the indices of the left endpoints of the
   shortest and longest intervals in the window. All of these quantities
   and indices are zero if the window contains no intervals.
 
   The following describes the relation of `shortest' and `longest' to the
   window data:
 
   The left endpoint of the shortest interval has value:
 
      SPICE_CELL_ELEM_D(window, shortest)
 
   The right endpoint of the shortest interval has value:
 
      SPICE_CELL_ELEM_D(window, shortest+1)
 
   The left endpoint of the longest interval has value:
 
      SPICE_CELL_ELEM_D(window, longest)
 
   The right endpoint of the longest interval has value:
 
      SPICE_CELL_ELEM_D(window, longest+1)
 
   These use:
 
      SPICE_CELL_ELEM_D(cell,i)
 
   a macro that returns the ith data element of 'cell'.
 
 
Summary
===========================================================================
 
   The following is a summary of the window functions in CSPICE. (DP =
   double precision)
 
            wncard_c - Cardinality of a DP window
 
            wncomd_c - Complement a DP window
 
            wncond_c - Contract the intervals of a DP window
 
            wndifd_c - Difference two DP windows
 
            wnelmd_c - Element of a DP window
 
            wnexpd_c - Expand the intervals of a DP window
 
            wnextd_c - Extract the endpoints from a DP window
 
            wnfetd_c - Fetch an interval from a DP window
 
            wnfild_c - Fill small gaps in a DP window
 
            wnfltd_c - Filter small intervals from a DP window
 
            wnincd_c - Included in a DP window
 
            wninsd_c - Insert an interval into a DP window
 
            wnintd_c - Intersect two DP windows
 
            wnreld_c - Compare two DP windows
 
            wnsumd_c - Summary of a DP window
 
            wnunid_c - Union two DP windows
 
            wnvald_c - Validate a DP window
 
 
Appendix: Document Revision History
===========================================================================
 
 
Febuary 6, 2009 (EDW)
 
   Document edited to conform to current NAIF format for Required Reading
   Documents.
 
 
September 3, 2002 (NAIF)
 
   First release of CSPICE version of this document.
 
