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

```

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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 procedures in Icy 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 procedures 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 Icy implementation of SPICE windows consists of double precision cells created and initialized to empty using the cspice_celld function.

Example:

```      ;;
;; Define the cell 'win' to use as a window. Select a value
;; for the cell size.
;;
MAXSIZ = 8

;;
;; This window can hold eight (MAXSIZ = 8) double precision
;; values, thus four intervals.
;;
win = cspice_celld( MAXSIZ )
```
The cspice_celld call creates a cell as an anonymous IDL structure assigned to variable `win' with maximum cardinality `MAXSIZE'.

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 Icy window functions, must have been created from the cspice_celld call. The cell structure has fields corresponding to a CSPICE SpiceCell format:

```      OFFSET     A 9-vector of IDL_MEMINT offsets for each of the
structure fields with respect to the beginning of the
structure.

Element 0 of the array contains the offset to the DTYPE
field; element 1 of the array contains the offset to
the LENGTH field, etc.

This array has no function in a window sense.
The Icy windows implementation uses the 'OFFSET'
values to locate data fields within the IDL structure.

DTYPE      Data type of cell: an integer scalar defining the type
of data stored in the cell data array.

LENGTH     For character cells, the integer scalar declared length
of the cell's string array.

SIZE       An integer scalar defining the maximum number of data
items that can be stored in the cell's data array.

CARD       The cell's "cardinality": the integer scalar defining
the number of data items currently present in the cell

Note CARD <= SIZE.

This field has value 0 after cell creation.

ISSET      A boolean scalar flag indicating whether the cell is a
SPICE set. Sets have no duplicate data items, and their
data items are stored in increasing order.

ADJUST     A boolean scalar flag indicating whether the cell's
data areas are not currently implemented.

INIT       A boolean scalar flag indicating whether the cell is
initialized.

BASE       The array of type DTYPE with 'SIZE' + 'DATA' elements
(an N-vector, N = SIZE + DATA)

DATA       The offset from the first element of the 'BASE' array
to the cell data. This field has value
SPICE_CELL_CTRLSZ (defined in SpiceCel.h), referred to
as the control segment size.
```
Any of the general cell procedures in Icy may be used with windows. For example, the function cspice_cardd may be used to determine the number of endpoints (that is, twice the number of intervals) in a window.

All errors are reported using standard Icy error handling.

With the exception of the initialization procedures, all window procedures 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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[1]

[2]

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# Window Procedures

The window procedures in Icy fall into the following categories.

-- Initialization

-- Unary

-- Binary

-- Complement

-- Comparison

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## Initialization Procedures

cspice_wnvald 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.

```   cspice_wnvald, 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 procedures.

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, cspice_wnvald signals an error.

Validation is primarily useful for ordering and merging intervals added to a cell by cspice_appndd, or directly assigned to the cell.

Building a large window is done most efficiently by assigning the window elements and then calling cspice_wnvald. Building up the window by repeated insertion requires repeated ordering operations; cspice_wnvald does a single ordering operation.

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## Unary Procedures

Each unary procedure 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.

```   cspice_wncond, left, right, window     { Contract }

cspice_wnexpd, left, right, window     { Expand }

cspice_wnextd, side, window            { Extract }

cspice_wnfild, small, window           { Fill }

cspice_wnfltd, small, window           { Filter }

cspice_wninsd, left, right, window     { Insert }
```
Each of the unary window procedures 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 procedures.

cspice_wninsd 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.

cspice_wnexpd and cspice_wncond expand (lengthen) and contract (shorten) each of the intervals in `window'. The adjustments are not necessarily symmetric. That is, cspice_wnexpd works by subtracting `left' units from the left endpoint of each interval and adding `right' units to the right endpoint of each interval. cspice_wncond 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.

cspice_wnfltd and cspice_wnfild remove small intervals and small gaps between adjacent intervals. Both procedures take as input a minimum measure, `small'. cspice_wnfltd filters out (drops) intervals with measures less than or equal to `small', while cspice_wnfild merges adjacent intervals separated by gaps with measures less than or equal to `small'.

Depending on the value of `side', cspice_wnextd 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 Procedures

Binary procedures operate on two input windows to produce a third (distinct) output window. Three major binary operations are supported. The calling sequences are shown below.

```   cspice_wnunid, a, b, c       { Union }

cspice_wnintd, a, b, c       { Intersection }

cspice_wndifd, a, b, c       { Difference }
```
In contrast with the unary procedures, none of the binary procedures 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.

cspice_wnunid 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.

cspice_wnintd 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.

cspice_wndifd 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 procedures, 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 procedures work in place. For example, the following calls are invalid.

```   cspice_wnintd, a, b, a

cspice_wnintd, a, b, b

```
In each of the examples above, whether or not the procedure 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.

```   cspice_wnintd, a, b, temp

a = temp
```

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## Complement Procedures

cspice_wncomd determines the complement of a window with respect to an interval. The calling sequence is

```   cspice_wncomd, left, right, a, c     { Complement }
```
As with the binary procedures, 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 cspice_wncomd must take the complement with respect to a finite interval.

Since the results of a window procedure must be another window, cspice_wncomd 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 Procedures

Comparison procedures allow the contents of windows to be compared against the contents of other windows. There are four comparison procedures: three logical functions and one procedure. The calling sequences are shown below.

```   cspice_wnelmd( point, window )                    { Element }

cspice_wnincd( left, right, window )              { Inclusion }

cspice_wnreld( a, op, b )                         { Relation }

cspice_wnsumd, window, meas,     avg,  \$
stddev, shortest, longest          { Summary }
```
cspice_wnelmd 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, cspice_wnincd 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.

cspice_wnreld 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 cspice_wnreld 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

```   cspice_wnreld( 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.

```   cspice_wnreld( a, '>', b )
cspice_wnreld( a, '<', b )

cspice_wnreld( a, '>=', b )
cspice_wnreld( a, '<=', b )
```
cspice_wnsumd 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 folowing describes the relation of `shortest' and `longest' to the window data:

```      data_start = window.data
data_end   = data_start + window.card - 1
array      = window.base[ data_start:data_end]
```
The left endpoint of the shortest interval has value:

```   array[shortest]
```
The right endpoint of the shortest interval has value:

```   array[shortest+1]
```
The left endpoint of the longest interval has value:

```   array[longest]
```
The right endpoint of the longest interval has value:

```   array[longest+1]
```

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# Summary

The following is a summary of the window procedures in Icy. (DP = double precision)

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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. Added descriptions of Icy and Mice window implementations.