/[PAMELA software]/quicklook/OrbitalRate/inc/sgp4.h
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Annotation of /quicklook/OrbitalRate/inc/sgp4.h

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Revision 1.1 - (hide annotations) (download)
Tue Dec 5 19:49:14 2006 UTC (18 years ago) by pam-rm2
Branch: MAIN
CVS Tags: v2r02, v2r01, v2r00, HEAD
File MIME type: text/plain
New version of OrbitalRate quicklook.  Initial import.
Nico

1 pam-rm2 1.1 //
2     // stdafx.h
3     //
4     #ifndef sgp4_h
5     #define sgp4_h
6    
7     //#define WIN32_LEAN_AND_MEAN // Exclude rarely-used stuff from Windows headers
8     #include <stdio.h>
9     //#include <tchar.h>
10     #include <ctype.h>
11     #include <string>
12     #include <map>
13     #include <vector>
14     #include <algorithm>
15     #include <assert.h>
16     #include <time.h>
17     #include <math.h>
18    
19     using namespace std;
20     //
21     // globals.h
22     //
23    
24     const double PI = 3.141592653589793;
25     const double TWOPI = 2.0 * PI;
26     const double RADS_PER_DEG = PI / 180.0;
27    
28     const double GM = 398601.2; // Earth gravitational constant, km^3/sec^2
29     const double GEOSYNC_ALT = 42241.892; // km
30     const double EARTH_DIA = 12800.0; // km
31     const double DAY_SIDERAL = (23 * 3600) + (56 * 60) + 4.09; // sec
32     const double DAY_24HR = (24 * 3600); // sec
33    
34     const double AE = 1.0;
35     const double AU = 149597870.0; // Astronomical unit (km) (IAU 76)
36     const double SR = 696000.0; // Solar radius (km) (IAU 76)
37     const double TWOTHRD = 2.0 / 3.0;
38     const double XKMPER_WGS72 = 6378.135; // Earth equatorial radius - km (WGS '72)
39     const double F = 1.0 / 298.26; // Earth flattening (WGS '72)
40     const double GE = 398600.8; // Earth gravitational constant (WGS '72)
41     const double J2 = 1.0826158E-3; // J2 harmonic (WGS '72)
42     const double J3 = -2.53881E-6; // J3 harmonic (WGS '72)
43     const double J4 = -1.65597E-6; // J4 harmonic (WGS '72)
44     const double CK2 = J2 / 2.0;
45     const double CK4 = -3.0 * J4 / 8.0;
46     const double XJ3 = J3;
47     const double E6A = 1.0e-06;
48     const double QO = AE + 120.0 / XKMPER_WGS72;
49     const double S = AE + 78.0 / XKMPER_WGS72;
50     const double HR_PER_DAY = 24.0; // Hours per day (solar)
51     const double MIN_PER_DAY = 1440.0; // Minutes per day (solar)
52     const double SEC_PER_DAY = 86400.0; // Seconds per day (solar)
53     const double OMEGA_E = 1.00273790934; // earth rotation per sideral day
54     const double XKE = sqrt(3600.0 * GE / //sqrt(ge) ER^3/min^2
55     (XKMPER_WGS72 * XKMPER_WGS72 * XKMPER_WGS72));
56     const double QOMS2T = pow((QO - S), 4); //(QO - S)^4 ER^4
57    
58     // Utility functions
59     double sqr (const double x);
60     double Fmod2p(const double arg);
61     double AcTan (const double sinx, double cosx);
62    
63     double rad2deg(const double);
64     double deg2rad(const double);
65     //
66     // coord.h
67     //
68     // Copyright 2002-2003 Michael F. Henry
69     //
70     //////////////////////////////////////////////////////////////////////
71     // Geocentric coordinates.
72     class cCoordGeo
73     {
74     public:
75     cCoordGeo();
76     cCoordGeo(double lat, double lon, double alt) :
77     m_Lat(lat), m_Lon(lon), m_Alt(alt) {}
78     virtual ~cCoordGeo() {};
79    
80     double m_Lat; // Latitude, radians (negative south)
81     double m_Lon; // Longitude, radians (negative west)
82     double m_Alt; // Altitude, km (above mean sea level)
83     };
84    
85     //////////////////////////////////////////////////////////////////////
86     // Topocentric-Horizon coordinates.
87     class cCoordTopo
88     {
89     public:
90     cCoordTopo();
91     cCoordTopo(double az, double el, double rng, double rate) :
92     m_Az(az), m_El(el), m_Range(rng), m_RangeRate(rate) {}
93     virtual ~cCoordTopo() {};
94    
95     double m_Az; // Azimuth, radians
96     double m_El; // Elevation, radians
97     double m_Range; // Range, kilometers
98     double m_RangeRate; // Range rate of change, km/sec
99     // Negative value means "towards observer"
100     };
101    
102     // cVector.h: interface for the cVector class.
103     //
104     // Copyright 2003 (c) Michael F. Henry
105     //
106     //////////////////////////////////////////////////////////////////////
107    
108     class cVector
109     {
110     public:
111     cVector(double x = 0.0, double y = 0.0, double z = 0.0, double w = 0.0) :
112     m_x(x), m_y(y), m_z(z), m_w(w) {}
113     virtual ~cVector() {};
114    
115     void Sub(const cVector&); // subtraction
116     void Mul(double factor); // multiply each component by 'factor'
117    
118     double Angle(const cVector&) const; // angle between two vectors
119     double Magnitude() const; // vector magnitude
120     double Dot(const cVector& vec) const; // dot product
121    
122     // protected:
123     double m_x;
124     double m_y;
125     double m_z;
126     double m_w;
127     };
128     //
129     // cTle.h
130     //
131     // This class will accept a single set of two-line elements and then allow
132     // a client to request specific fields, such as epoch, mean motion,
133     // etc., from the set.
134     //
135     // Copyright 1996-2003 Michael F. Henry
136     //
137     /////////////////////////////////////////////////////////////////////////////
138     class cTle
139     {
140     public:
141     cTle(string&, string&, string&);
142     cTle(const cTle &tle);
143     ~cTle();
144    
145     enum eTleLine
146     {
147     LINE_ZERO,
148     LINE_ONE,
149     LINE_TWO
150     };
151    
152     enum eField
153     {
154     FLD_FIRST,
155     FLD_NORADNUM = FLD_FIRST,
156     FLD_INTLDESC,
157     FLD_SET, // TLE set number
158     FLD_EPOCHYEAR, // Epoch: Last two digits of year
159     FLD_EPOCHDAY, // Epoch: Fractional Julian Day of year
160     FLD_ORBITNUM, // Orbit at epoch
161     FLD_I, // Inclination
162     FLD_RAAN, // R.A. ascending node
163     FLD_E, // Eccentricity
164     FLD_ARGPER, // Argument of perigee
165     FLD_M, // Mean anomaly
166     FLD_MMOTION, // Mean motion
167     FLD_MMOTIONDT, // First time derivative of mean motion
168     FLD_MMOTIONDT2,// Second time derivative of mean motion
169     FLD_BSTAR, // BSTAR Drag
170     FLD_LAST // MUST be last
171     };
172    
173     enum eUnits
174     {
175     U_FIRST,
176     U_RAD = U_FIRST, // radians
177     U_DEG, // degrees
178     U_NATIVE, // TLE format native units (no conversion)
179     U_LAST // MUST be last
180     };
181    
182     void Initialize();
183    
184     static int CheckSum(const string&);
185     static bool IsValidLine(string&, eTleLine);
186     static string ExpToDecimal(const string&);
187    
188     static void TrimLeft(string&);
189     static void TrimRight(string&);
190    
191     double getField(eField fld, // which field to retrieve
192     eUnits unit = U_NATIVE, // return units in rad, deg etc.
193     string *pstr = NULL, // return ptr for str value
194     bool bStrUnits = false) // 'true': append units to str val
195     const;
196     string getName() const { return m_strName; }
197     string getLine1() const { return m_strLine1;}
198     string getLine2() const { return m_strLine2;}
199    
200     protected:
201     static double ConvertUnits(double val, eField fld, eUnits units);
202    
203     private:
204     string getUnits(eField) const;
205     double getFieldNumeric(eField) const;
206    
207     // Satellite name and two data lines
208     string m_strName;
209     string m_strLine1;
210     string m_strLine2;
211    
212     // Converted fields, in atof()-readable form
213     string m_Field[FLD_LAST];
214    
215     // Cache of field values in "double" format
216     typedef int FldKey;
217     FldKey Key(eUnits u, eField f) const { return (u * 100) + f; }
218     mutable map<FldKey, double> m_mapCache;
219     };
220    
221     ///////////////////////////////////////////////////////////////////////////
222     //
223     // TLE data format
224     //
225     // [Reference: T.S. Kelso]
226     //
227     // Two line element data consists of three lines in the following format:
228     //
229     // AAAAAAAAAAAAAAAAAAAAAA
230     // 1 NNNNNU NNNNNAAA NNNNN.NNNNNNNN +.NNNNNNNN +NNNNN-N +NNNNN-N N NNNNN
231     // 2 NNNNN NNN.NNNN NNN.NNNN NNNNNNN NNN.NNNN NNN.NNNN NN.NNNNNNNNNNNNNN
232     //
233     // Line 0 is a twenty-two-character name.
234     //
235     // Lines 1 and 2 are the standard Two-Line Orbital Element Set Format identical
236     // to that used by NORAD and NASA. The format description is:
237     //
238     // Line 1
239     // Column Description
240     // 01-01 Line Number of Element Data
241     // 03-07 Satellite Number
242     // 10-11 International Designator (Last two digits of launch year)
243     // 12-14 International Designator (Launch number of the year)
244     // 15-17 International Designator (Piece of launch)
245     // 19-20 Epoch Year (Last two digits of year)
246     // 21-32 Epoch (Julian Day and fractional portion of the day)
247     // 34-43 First Time Derivative of the Mean Motion
248     // or Ballistic Coefficient (Depending on ephemeris type)
249     // 45-52 Second Time Derivative of Mean Motion (decimal point assumed;
250     // blank if N/A)
251     // 54-61 BSTAR drag term if GP4 general perturbation theory was used.
252     // Otherwise, radiation pressure coefficient. (Decimal point assumed)
253     // 63-63 Ephemeris type
254     // 65-68 Element number
255     // 69-69 Check Sum (Modulo 10)
256     // (Letters, blanks, periods, plus signs = 0; minus signs = 1)
257     //
258     // Line 2
259     // Column Description
260     // 01-01 Line Number of Element Data
261     // 03-07 Satellite Number
262     // 09-16 Inclination [Degrees]
263     // 18-25 Right Ascension of the Ascending Node [Degrees]
264     // 27-33 Eccentricity (decimal point assumed)
265     // 35-42 Argument of Perigee [Degrees]
266     // 44-51 Mean Anomaly [Degrees]
267     // 53-63 Mean Motion [Revs per day]
268     // 64-68 Revolution number at epoch [Revs]
269     // 69-69 Check Sum (Modulo 10)
270     //
271     // All other columns are blank or fixed.
272     //
273     // Example:
274     //
275     // NOAA 6
276     // 1 11416U 86 50.28438588 0.00000140 67960-4 0 5293
277     // 2 11416 98.5105 69.3305 0012788 63.2828 296.9658 14.24899292346978
278    
279     //
280     // cJulian.h
281     //
282     // Copyright (c) 2003 Michael F. Henry
283     //
284     //
285     // See note in cJulian.cpp for information on this class and the epoch dates
286     //
287     const double EPOCH_JAN1_00H_1900 = 2415019.5; // Jan 1.0 1900 = Jan 1 1900 00h UTC
288     const double EPOCH_JAN1_12H_1900 = 2415020.0; // Jan 1.5 1900 = Jan 1 1900 12h UTC
289     const double EPOCH_JAN1_12H_2000 = 2451545.0; // Jan 1.5 2000 = Jan 1 2000 12h UTC
290    
291     //////////////////////////////////////////////////////////////////////////////
292     class cJulian
293     {
294     public:
295     cJulian() { Initialize(2000, 1); }
296     explicit cJulian(time_t t); // Create from time_t
297     explicit cJulian(int year, double day); // Create from year, day of year
298     explicit cJulian(int year, // i.e., 2004
299     int mon, // 1..12
300     int day, // 1..31
301     int hour, // 0..23
302     int min, // 0..59
303     double sec = 0.0); // 0..(59.999999...)
304     ~cJulian() {};
305    
306     double toGMST() const; // Greenwich Mean Sidereal Time
307     double toLMST(double lon) const; // Local Mean Sideral Time
308     time_t toTime() const; // To time_t type - avoid using
309    
310     double FromJan1_00h_1900() const { return m_Date - EPOCH_JAN1_00H_1900; }
311     double FromJan1_12h_1900() const { return m_Date - EPOCH_JAN1_12H_1900; }
312     double FromJan1_12h_2000() const { return m_Date - EPOCH_JAN1_12H_2000; }
313    
314     void getComponent(int *pYear, int *pMon = NULL, double *pDOM = NULL) const;
315     double getDate() const { return m_Date; }
316    
317     void addDay (double day) { m_Date += day; }
318     void addHour(double hr ) { m_Date += (hr / HR_PER_DAY ); }
319     void addMin (double min) { m_Date += (min / MIN_PER_DAY); }
320     void addSec (double sec) { m_Date += (sec / SEC_PER_DAY); }
321    
322     double spanDay (const cJulian& b) const { return m_Date - b.m_Date; }
323     double spanHour(const cJulian& b) const { return spanDay(b) * HR_PER_DAY; }
324     double spanMin (const cJulian& b) const { return spanDay(b) * MIN_PER_DAY; }
325     double spanSec (const cJulian& b) const { return spanDay(b) * SEC_PER_DAY; }
326    
327     static bool IsLeapYear(int y)
328     { return (y % 4 == 0 && y % 100 != 0) || (y % 400 == 0); }
329    
330     protected:
331     void Initialize(int year, double day);
332    
333     double m_Date; // Julian date
334     };
335     //
336     // cEci.h
337     //
338     // Copyright (c) 2003 Michael F. Henry
339     //
340     //////////////////////////////////////////////////////////////////////
341     // class cEci
342     // Encapsulates an Earth-Centered Inertial position, velocity, and time.
343     class cEci
344     {
345     public:
346     cEci() { m_VecUnits = UNITS_NONE; }
347     cEci(const cCoordGeo &geo, const cJulian &cJulian);
348     cEci(const cVector &pos, const cVector &vel,
349     const cJulian &date, bool IsAeUnits = true);
350     virtual ~cEci() {};
351    
352     cCoordGeo toGeo();
353    
354     cVector getPos() const { return m_pos; }
355     cVector getVel() const { return m_vel; }
356     cJulian getDate() const { return m_date; }
357    
358     void setUnitsAe() { m_VecUnits = UNITS_AE; }
359     void setUnitsKm() { m_VecUnits = UNITS_KM; }
360     bool UnitsAreAe() const { return m_VecUnits == UNITS_AE; }
361     bool UnitsAreKm() const { return m_VecUnits == UNITS_KM; }
362     void ae2km(); // Convert position, velocity vector units from AE to km
363    
364     protected:
365     void MulPos(double factor) { m_pos.Mul(factor); }
366     void MulVel(double factor) { m_vel.Mul(factor); }
367    
368     enum VecUnits
369     {
370     UNITS_NONE, // not initialized
371     UNITS_AE,
372     UNITS_KM,
373     };
374    
375     cVector m_pos;
376     cVector m_vel;
377     cJulian m_date;
378     VecUnits m_VecUnits;
379     };
380     //
381     // cNoradBase.h
382     //
383     // This class provides a base class for the NORAD SGP4/SDP4
384     // orbit models.
385     //
386     // Copyright (c) 2003 Michael F. Henry
387     //
388     #pragma once
389    
390     //////////////////////////////////////////////////////////////////////////////
391    
392     class cEci;
393     class cOrbit;
394    
395     //////////////////////////////////////////////////////////////////////////////
396    
397     class cNoradBase
398     {
399     public:
400     cNoradBase(const cOrbit&);
401     virtual ~cNoradBase() {};
402    
403     virtual bool getPosition(double tsince, cEci &eci) = 0;
404    
405     protected:
406     cNoradBase& operator=(const cNoradBase&);
407    
408     void Initialize();
409     bool FinalPosition(double incl, double omega, double e,
410     double a, double xl, double xnode,
411     double xn, double tsince, cEci &eci);
412    
413     const cOrbit &m_Orbit;
414    
415     // Orbital parameter variables which need only be calculated one
416     // time for a given orbit (ECI position time-independent).
417     double m_satInc; // inclination
418     double m_satEcc; // eccentricity
419    
420     double m_cosio; double m_theta2; double m_x3thm1; double m_eosq;
421     double m_betao2; double m_betao; double m_aodp; double m_xnodp;
422     double m_s4; double m_qoms24; double m_perigee; double m_tsi;
423     double m_eta; double m_etasq; double m_eeta; double m_coef;
424     double m_coef1; double m_c1; double m_c2; double m_c3;
425     double m_c4; double m_sinio; double m_a3ovk2; double m_x1mth2;
426     double m_xmdot; double m_omgdot; double m_xhdot1; double m_xnodot;
427     double m_xnodcf; double m_t2cof; double m_xlcof; double m_aycof;
428     double m_x7thm1;
429     };
430     //
431     // cOrbit.h
432     //
433     // This is the header file for the class cOrbit. This class accepts a
434     // single satellite's NORAD two-line element set and provides information
435     // regarding the satellite's orbit such as period, axis length,
436     // ECI coordinates/velocity, etc., using the SGP4/SDP4 orbital models.
437     //
438     // Copyright (c) 2002-2003 Michael F. Henry
439     //
440     #pragma once
441    
442     #include "math.h"
443    
444     using namespace std;
445     //////////////////////////////////////////////////////////////////////////////
446    
447     class cVector;
448     class cGeoCoord;
449     class cEci;
450    
451     //////////////////////////////////////////////////////////////////////////////
452     class cOrbit
453     {
454     public:
455     cOrbit(const cTle &tle);
456     virtual ~cOrbit();
457    
458     // Return satellite ECI data at given minutes since element's epoch.
459     bool getPosition(double tsince, cEci *pEci) const;
460    
461     double Inclination() const { return radGet(cTle::FLD_I); }
462     double Eccentricity() const { return m_tle.getField(cTle::FLD_E); }
463     double RAAN() const { return radGet(cTle::FLD_RAAN); }
464     double ArgPerigee() const { return radGet(cTle::FLD_ARGPER); }
465     double BStar() const { return m_tle.getField(cTle::FLD_BSTAR) / AE;}
466     double Drag() const { return m_tle.getField(cTle::FLD_MMOTIONDT); }
467     double mnMotion() const { return m_tle.getField(cTle::FLD_MMOTION); }
468     double mnAnomaly() const { return radGet(cTle::FLD_M); }
469     double mnAnomaly(cJulian t) const; // mean anomaly (in radians) at time t
470    
471     cJulian Epoch() const { return m_jdEpoch; }
472    
473     double TPlusEpoch(const cJulian &t) const; // time span [t - epoch] in secs
474    
475     string SatName(bool fAppendId = false) const;
476    
477     // "Recovered" from the input elements
478     double SemiMajor() const { return m_aeAxisSemiMajorRec; }
479     double SemiMinor() const { return m_aeAxisSemiMinorRec; }
480     double mnMotionRec() const { return m_mnMotionRec; } // mn motion, rads/min
481     double Major() const { return 2.0 * SemiMajor(); } // major axis in AE
482     double Minor() const { return 2.0 * SemiMinor(); } // minor axis in AE
483     double Perigee() const { return m_kmPerigeeRec; } // perigee in km
484     double Apogee() const { return m_kmApogeeRec; } // apogee in km
485     double Period() const; // period in seconds
486    
487     protected:
488     double radGet(cTle::eField fld) const
489     { return m_tle.getField(fld, cTle::U_RAD); }
490    
491     double degGet(cTle::eField fld) const
492     { return m_tle.getField(fld, cTle::U_DEG); }
493    
494     private:
495     cTle m_tle;
496     cJulian m_jdEpoch;
497     cNoradBase *m_pNoradModel;
498    
499     // Caching variables; note units are not necessarily the same as tle units
500     mutable double m_secPeriod;
501    
502     // Caching variables recovered from the input TLE elements
503     double m_aeAxisSemiMinorRec; // semi-minor axis, in AE units
504     double m_aeAxisSemiMajorRec; // semi-major axis, in AE units
505     double m_mnMotionRec; // radians per minute
506     double m_kmPerigeeRec; // perigee, in km
507     double m_kmApogeeRec; // apogee, in km
508     };
509    
510     //
511     // cNoradSGP4.h
512     //
513     // This class implements the NORAD Simple General Perturbation 4 orbit
514     // model. This model provides the ECI coordiantes/velocity of satellites
515     // with orbit periods less than 225 minutes.
516     //
517     // Copyright (c) 2003 Michael F. Henry
518     //
519     #pragma once
520    
521     class cOrbit;
522    
523     //////////////////////////////////////////////////////////////////////////////
524     class cNoradSGP4 : public cNoradBase
525     {
526     public:
527     cNoradSGP4(const cOrbit &orbit);
528     virtual ~cNoradSGP4() {};
529    
530     virtual bool getPosition(double tsince, cEci &eci);
531    
532     protected:
533     double m_c5;
534     double m_omgcof;
535     double m_xmcof;
536     double m_delmo;
537     double m_sinmo;
538     };
539    
540     //
541     // cNoradSDP4.h
542     //
543     // This class implements the NORAD Simple Deep Perturbation 4 orbit
544     // model. This model provides the ECI coordinates/velocity of satellites
545     // with periods >= 225 minutes.
546     //
547     // Copyright (c) 2003 Michael F. Henry
548     //
549     #pragma once
550    
551     class cOrbit;
552    
553     //////////////////////////////////////////////////////////////////////////////
554     class cNoradSDP4 : public cNoradBase
555     {
556     public:
557     cNoradSDP4(const cOrbit &orbit);
558     virtual ~cNoradSDP4() {};
559    
560     virtual bool getPosition(double tsince, cEci &eci);
561    
562     protected:
563     bool DeepInit(double *eosq, double *sinio, double *cosio, double *m_betao,
564     double *m_aodp, double *m_theta2, double *m_sing, double *m_cosg,
565     double *m_betao2,double *xmdot, double *omgdot, double *xnodott);
566    
567     bool DeepSecular(double *xmdf, double *omgadf,double *xnode, double *emm,
568     double *xincc, double *xnn, double *tsince);
569     bool DeepCalcDotTerms (double *pxndot, double *pxnddt, double *pxldot);
570     void DeepCalcIntegrator(double *pxndot, double *pxnddt, double *pxldot,
571     const double &delt);
572     bool DeepPeriodics(double *e, double *xincc, double *omgadf,
573     double *xnode, double *xmam);
574     double m_sing;
575     double m_cosg;
576    
577     // Deep Initialization
578     double eqsq; double siniq; double cosiq; double rteqsq; double ao;
579     double cosq2; double sinomo; double cosomo; double bsq; double xlldot;
580     double omgdt; double xnodot;
581    
582     // Deep Secular, Periodic
583     double xll; double omgasm; double xnodes; double _em;
584     double xinc; double xn; double t;
585    
586     // Variables shared by "Deep" routines
587     double dp_e3; double dp_ee2; double dp_savtsn; double dp_se2;
588     double dp_se3; double dp_sgh2; double dp_sgh3; double dp_sgh4;
589     double dp_sghs; double dp_sh2; double dp_sh3; double dp_si2;
590     double dp_si3; double dp_sl2; double dp_sl3; double dp_sl4;
591     double dp_xgh2; double dp_xgh3; double dp_xgh4; double dp_xh2;
592     double dp_xh3; double dp_xi2; double dp_xi3; double dp_xl2;
593     double dp_xl3; double dp_xl4; double dp_xqncl; double dp_zmol;
594     double dp_zmos;
595    
596     double dp_atime; double dp_d2201; double dp_d2211; double dp_d3210;
597     double dp_d3222; double dp_d4410; double dp_d4422; double dp_d5220;
598     double dp_d5232; double dp_d5421; double dp_d5433; double dp_del1;
599     double dp_del2; double dp_del3; double dp_fasx2; double dp_fasx4;
600     double dp_fasx6; double dp_omegaq; double dp_sse; double dp_ssg;
601     double dp_ssh; double dp_ssi; double dp_ssl; double dp_step2;
602     double dp_stepn; double dp_stepp; double dp_thgr; double dp_xfact;
603     double dp_xlamo; double dp_xli; double dp_xni;
604    
605     bool dp_iresfl;
606     bool dp_isynfl;
607    
608     // DeepInit vars that change with epoch
609     double dpi_c; double dpi_ctem; double dpi_day; double dpi_gam;
610     double dpi_stem; double dpi_xnodce; double dpi_zcosgl; double dpi_zcoshl;
611     double dpi_zcosil; double dpi_zsingl; double dpi_zsinhl; double dpi_zsinil;
612     double dpi_zx; double dpi_zy;
613    
614     };
615    
616     #endif

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