Index Page Windows Required Reading

## Table of Contents

       Windows Required Reading
          Abstract
          Introduction
             Basic Concepts
          The window data type
          References
       Window Functions
          Initialization Functions
          Unary Functions
          Binary Functions
          Complement Functions
          Comparison Functions
       Summary
       Appendix: Document Revision History
             Febuary 6, 2009 (EDW)
             September 3, 2002 (NAIF)

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# Windows Required Reading

Last revised on 2010 MAY 18 by B. V. Semenov.

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## 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.

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## 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.

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### 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.

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## 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.

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## References

[1]
  CELLS Required Reading (cells.req).

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# Window Functions

The window functions in CSPICE fall into the following categories.

`--` Initialization

`--` Unary

`--` Binary

`--` Complement

`--` Comparison

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## 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.

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## 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) ]

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## 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 );

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## 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.

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## 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'.

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# 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

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# Appendix: Document Revision History

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### Febuary 6, 2009 (EDW)

Document edited to conform to current NAIF format for Required Reading Documents.

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### September 3, 2002 (NAIF)

First release of CSPICE version of this document.