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 UrnOct 4, 2018 at 12:59
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2 Answers
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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 $\pi$ .
answered Oct 11, 2018 at 5:52
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$\begingroup$ Could you explain what is done in line 214? The case arctan = 0 for x = 0 and y = 0? $\endgroup$– UweOct 11, 2018 at 11:52
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$\begingroup$ @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? $\endgroup$ Oct 11, 2018 at 13:14
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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. $\endgroup$– UweOct 11, 2018 at 15:44
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$\begingroup$ @DrSheldon Do you want to take a stab at space.stackexchange.com/q/31196/195 ? $\endgroup$ Oct 11, 2018 at 16:06
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1$\begingroup$ @RussellBorogove: I've been working on it and hope to have an answer later today! $\endgroup$ Oct 11, 2018 at 16:27
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Here you can see source code of the Apollo 11 Guidance computer.
Inside, see Subroutine for ARCTAN.
Answers your question?
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5$\begingroup$ That's a link's only answer, and only understandable for people who can read assembly. The question asks which mathematical approach was used, not so much the source code. $\endgroup$– HobbesOct 4, 2018 at 12:06
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$\begingroup$ @Hobbes There are different assembly languages, at least one per type of processor. I'm not sure what language the code is using, but it's not one I'm familiar with, and I've been around for a while. I can't tell what algorithm the code is using. $\endgroup$ Oct 4, 2018 at 17:51
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2$\begingroup$ @DavidThornley see answers to How did the Apollo computers evaluate transcendental functions like sine, arctangent, log? for more on the particular flavor $\endgroup$– uhohOct 4, 2018 at 20:27
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$\begingroup$ RobinC welcome to Stack Exchange! It's a big different than other sites you may have used. You can get a feeling for what kinds of answers are well-received by looking around at other answers and have a look at How to Answer as well. The four-quadrant correct atan2 is really important in engineering applications and so it is certainly implemented somehow If ARCTAN doesn't do that (you have not yet said if it does or doesn't) then this will be handled by whatever calls ARCTAN and so your answer should expand on that. $\endgroup$– uhohOct 4, 2018 at 20:33
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$\begingroup$ Right now it's called a "link-only" answer and this is strongly discouraged (and down-voted) because links break and then the answer becomes useless and the question becomes unanswered again. The idea is to write answers that are helpful to future readers as well as the OP. $\endgroup$– uhohOct 4, 2018 at 20:34