[Maxima-bugs] [ maxima-Bugs-2298099 ] atan2 & logarc

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Bugs item #2298099, was opened at 2008-11-16 12:50
Message generated for change (Comment added) made by crategus
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Category: Lisp Core - Trigonometry
Group: Includes proposed fix
Status: Open
Resolution: None
Priority: 5
Private: No
Submitted By: Barton Willis (willisbl)
Assigned to: Nobody/Anonymous (nobody)
Summary: atan2 & logarc

Initial Comment:
When logarc is true, atan2 returns incorrect values in
the left half plane:

 (%i5) logarc : true$

 (%i6) e : atan2(y,x);
 (%o6) -(%i*(log((%i*y)/x+1)-log(1-(%i*y)/x)))/2

Should be 3 %pi / 4, not - %pi / 4:

 (%i7) rectform(subst([y=1,x=-1], e));
 (%o7) -%pi/4

Should be -3 %pi / 4, not - %pi / 4:

 (%i8) rectform(subst([y=-1,x=-1],e));
 (%o8) %pi/4

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>Comment By: Dieter Kaiser (crategus)
Date: 2009-04-06 01:14

Message:
Remark: 

This topic was discussed already in the bug report SF [626721] "logarc of
atan2 wrong".
The function $logarc was fixed, but the simplifying function simpatan2 in
comm2.lisp has still the problem.
Therefore the flag logarc does not work.

Dieter Kaiser

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Comment By: Dieter Kaiser (crategus)
Date: 2009-04-05 21:27

Message:
A more general definition of atan2(y,x) is:

- %i * log((x+%i*y)/sqrt(x^2+y^2))

When we use this definition to transform atan2 for logarc:true, we will
get the correct results. The changed code is

;($logarc (logarc '%atan (div y x))) 

($logarc
  (mul -1 '$%i 
         (simplify (list '(%log) 
                      (div (add x (mul '$%i y))
                            (power (add (mul x x) (mul y y))
                                       (div 1 2)))))))

The examples of this bug report would give the correct results. The
testsuite has no problems:

(%i13) logarc:true$

(%i16) atan2(x,y);
(%o16) -%i*log((y+%i*x)/sqrt(y^2+x^2))

(%i17) rectform(subst([y=1,x=-1],atan2(y,x)));
(%o17) 3*%pi/4

(%i18) rectform(subst([y=-1,x=-1],atan2(y,x)));
(%o18) -3*%pi/4

Dieter Kaiser

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Comment By: Raymond Toy (rtoy)
Date: 2009-03-02 16:50

Message:
Perhaps this is really a bug in the logarc variable, or at least an
inconsistency.

atan2(y,x), logarc returns the expression you give above.  But

logarc(atan2(y,x)) -> -%i*log((x+%i*y)/sqrt(x^2+y^2)).

This latter expression gives 3*%pi/4  for x=-1, and y = 1.

This is an example where log(x/y) is not equal to log(x) - log(y), I
think.

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