What type of the arcustangent function was used, the one with single argument only or the famous atan2(y, x) or arctan2(y, x) with two arguments x and y? What kind of aproximation was used?
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- 2$\begingroup$ Related: space.stackexchange.com/questions/30952/… $\endgroup$Magic Octopus Urn– Magic Octopus Urn2018-10-04 12:59:39 +00:00Commented Oct 4, 2018 at 12:59
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9 Two arguments, but not exactly like atan2()
The first thing you should understand is that although ARCTAN is called in 3 places in the Apollo code, in every case it is used for converting rectilinear coordinates (x, y, z) into spherical coordinates (r, latitude, longitude).
I will describe how ARCTAN is used in the LAT-LONG subroutine, in file LATITUDE_ LONGITUDE_ SUBROUTINES.agc, although the use is similar in the other callers. The first step is to compute gamma = $\sqrt{x^2+y^2}$ (lines 77-82). We then divide x and y each by gamma; x/gamma is stored in global variable COSTH and y/gamma is stored in global variable SINTH (lines 83-87). We are then ready to call the ARCTAN subroutine (line 88). After it returns, the result is stored in global variable LAT (line 89). A similar set of calculations determines the longitude.
77: DLOAD DSQ 78: ALPHAV 79: PDDL DSQ 80: ALPHAV +2 81:# Page 1237 82: DAD SQRT 83: DMP SL1R 84: GAMRP 85: STODL COSTH # COS(LAT) B-1 86: ALPHAV +4 87: STCALL SINTH # SIN(LAT) B-1 88: ARCTAN 89: STODL LAT # LAT B0 Like atan2(), ARCTAN takes two arguments. However, unlike atan2(), the arguments must already be pre-divided. In other words, they must be in the range of -1 to +1, and their squares must add to 1. The arguments are taken from the global variables COSTH and SINTH. As the names suggest and because of the pre-division, these arguments are the cosine and sine of the angle we are looking for. The bulk of the subroutine checks the +/- sign of the arguments, adjusting the result for the appropriate quadrant. The main calculation is done by calling ASIN with argument SINTH (line 219).
196: # Page 1240 197: # ARCTAN SUBROUTINE 198: # 199: # CALLING SEQUENCE 200: # SIN THETA IN SINTH B-1 201: # COS THETA IN COSTH B-1 202: # CALL ARCTAN 203: # 204: # OUTPUT 205: # ARCTAN THETA IN MPAC AND THETA B-0 IN RANGE -1/2 TO +1/2 206: 207: ARCTAN BOV 208: CLROVFLW 209: CLROVFLW DLOAD DSQ 210: SINTH 211: PDDL DSQ 212: COSTH 213: DAD 214: BZE SQRT 215: ARCTANXX # ATAN=0/0. SET THETA=0 216: BDDV BOV 217: SINTH 218: ATAN=90 219: SR1 ASIN 220: STORE THETA 221: PDDL BMN 222: COSTH 223: NEGCOS 224: DLOAD RVQ 225: NEGCOS DLOAD DCOMP 226: BPL DAD 227: NEGOUT 228: DP1/2 229: ARCTANXX STORE THETA 230: RVQ 231: 232: NEGOUT DSU GOTO 233: DP1/2 234: ARCTANXX 235: ATAN=90 DLOAD SIGN 236: LODP1/4 237: SINTH 238: STORE THETA 239: RVQ ASIN is a macro for the ARCSIN subroutine in file INTERPRETER.agc. In turn, it is computed from the arccosine, using the relation ARCSIN(X) = PI/2 - ARCCOS(X). (Lines 2704-2705. Yeah, even though we already have the cosine stored in COSTH.) ARCCOS is calculated using the "Hastings approximation" and a 7th degree polynomial.
The result of each of the inverse trig subroutines are angles in the range of -1/2 to +1/2. To convert to radians, you must multiply by $\pi$ .
So, there are two arguments, but they must be pre-divided, and the result must be multiplied by $\pi$ .
- $\begingroup$ Could you explain what is done in line 214? The case arctan = 0 for x = 0 and y = 0? $\endgroup$Uwe– Uwe2018-10-11 11:52:15 +00:00Commented Oct 11, 2018 at 11:52
- $\begingroup$ @uwe BZE is branch when zero, if true it will run ARCTANXX (after that I am lost). Not sure what RVQ does, return value quit maybe? $\endgroup$Magic Octopus Urn– Magic Octopus Urn2018-10-11 13:14:04 +00:00Commented Oct 11, 2018 at 13:14
- 1$\begingroup$ @MagicOctopusUrn: RVQ is Return via QPRET. Returns to the location specified in the QPRET register. This is the normal way to return from a subroutine, although if the QPRET register had previously been saved to a memory location, then GOTO can be used instead. BZE seems to use the previously computed value gamma = sqrt(x^2+y^2) to find out if both x and y are zero. $\endgroup$Uwe– Uwe2018-10-11 15:44:12 +00:00Commented Oct 11, 2018 at 15:44
- $\begingroup$ @DrSheldon Do you want to take a stab at space.stackexchange.com/q/31196/195 ? $\endgroup$Russell Borogove– Russell Borogove2018-10-11 16:06:03 +00:00Commented Oct 11, 2018 at 16:06
- 1$\begingroup$ @RussellBorogove: I've been working on it and hope to have an answer later today! $\endgroup$DrSheldon– DrSheldon2018-10-11 16:27:15 +00:00Commented Oct 11, 2018 at 16:27