| Window Functions in CSPICE |
Table of ContentsWindow Functions in CSPICE Revisions Overview Concepts Usage Naming Conventions Functions Unary Functions Binary Functions Complement Functions Comparison Functions Initialization Functions Summary Window Functions in CSPICERevisions
First release of CSPICE version of this document. Overview
By definition, a window is a double precision CSPICE 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 window operations to be carried out efficiently. 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. Concepts
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. Usage
All errors are reported via standard CSPICE error handling. All window functions assume that input cells do in fact contain valid windows---that is, ordered and distinct sets of endpoints. Errors resulting from attempts to operate on invalid windows are not flagged. Naming Conventions
Functions
Unary Functions
CSPICE windows, when appearing as arguments to the CSPICE window functions, always have type (SpiceCell *).
wncond_c ( left, right, win ) { Contract } wnexpd_c ( left, right, win ) { Expand } wnextd_c ( side, win ) { Extract } wnfild_c ( small, win ) { Fill } wnfltd_c ( small, win ) { Filter } wninsd_c ( left, right, win ) { Insert }Note that each of the unary window functions works in place. That is, only one window, win, 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 the window win. If the input interval overlaps any of the intervals in the window, the intervals are merged. Thus, the cardinality of win can actually decrease as the result of an insertion. wnexpd_c and wncond_c expand (lengthen) and contract (shorten) each of the intervals in the window win. 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 ('L') or right ('R') endpoints each interval in the window win. 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
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 an error is signaled. 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 is 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 ( 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 )Because Fortran offers no satisfactory way to represent infinity, wncomd_c must take the complement with respect to a finite interval. In addition, because 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 ]Note that 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
wnelmd_c ( point, win ) { Element } wnincd_c ( left, right, win ) { Inclusion } wnreld_c ( a, op, b ) { Relation } wnsumd_c ( win, meas, avg, stddev, shortest, longest ) { Summary }wnelmd_c, the simplest of the three, is true whenever the input point, point, is an element of the input window, win---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, win---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 ccomparison 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, win. 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 (shortest) and longest (longest) intervals in the window. All of these quantities and indices are zero if the window contains no intervals. Initialization Functions
wnvald_c ( size, n, win ) { Validate }On input, win 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 win 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 via appndd_c, read from input files or initialized via CSPICE cell assignment macros. Note that 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. Summary
Routine Functionality
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com ( left, right, a, c ) Complement a window.
con ( left, right, win ) Contract intervals.
dif ( a, b, c ) Difference windows.
elm ( point, win ) Contains this point?
exp ( left, right, win ) Expand intervals.
ext ( side, win ) Extract endpoints.
fil ( small, win ) Fill small gaps.
flt ( small, win ) Filter small intervals.
inc ( left, right, win ) Includes this interval?
ins ( left, right, win ) Insert an interval.
int ( a, b, c ) Intersect windows.
rel ( a, rel, b ) Satisfies this relationship?
sum ( win,
meas, avg, stddev,
short, long ) Summary of window.
uni ( a, b, c ) Union windows.
val ( size, card, win ) Validate a window.
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