difference between atan2(x,y) and atan(x/y)

I'm a bit stumped here, maybe someone can help me out?

y = atan (sin(2*pi*x) / cos(2*pi*x)); 

s = atan2 (sin(2*pi*x) , cos(2*pi*x));

I would expect y and s to be equal, but they're not.

Most surprising to me is that the maximum value of s appears to be pi,
despite the fact that the limit of atan2(a,b) as a/b->infinity is pi/2
(right?)

Can anyone clue me in as to why my expectations are being defied here?  I
think the root of my misunderstanding lies in my not understanding the
differences between atan(a/b) and atan2(a,b).

Thanks, any help is much appreciated

atan2(y,x) is used to find the angle from the point (0,0) to the point
(x,y), using the x-axis as reference; e.g.,

atan2 (.707, .707) = pi/4 => 45 degrees
atan2 (.707, -.707) = 3*pi/4 => 135 degrees
atan2 (-.707, .707) = -pi/4 = -45 degrees
atan2 (-.707, -.707) = -3*pi/4 => -135 degrees

(the signs of x and y determine the quadrant of the result)

On Jan 8, 6:13 pm, "maxplanck" <erik.bo...@comcast.net> wrote:
> I'm a bit stumped here, maybe someone can help me out?
>
> y = atan (sin(2*pi*x) / cos(2*pi*x));
>
> s = atan2 (sin(2*pi*x) , cos(2*pi*x));
>
> I would expect y and s to be equal, but they're not.
>
> Most surprising to me is that the maximum value of s appears to be pi,
> despite the fact that the limit of atan2(a,b) as a/b->infinity is pi/2
> (right?)
>
> Can anyone clue me in as to why my expectations are being defied here?  I
> think the root of my misunderstanding lies in my not understanding the
> differences between atan(a/b) and atan2(a,b).
>
> Thanks, any help is much appreciated

On Jan 8, 6:13 pm, "maxplanck" <erik.bo...@comcast.net> wrote:
> I'm a bit stumped here, maybe someone can help me out?
>
> y = atan (sin(2*pi*x) / cos(2*pi*x));
>
> s = atan2 (sin(2*pi*x) , cos(2*pi*x));
>
> I would expect y and s to be equal, but they're not.
>
> Most surprising to me is that the maximum value of s appears to be pi,
> despite the fact that the limit of atan2(a,b) as a/b->infinity is pi/2
> (right?)
>
> Can anyone clue me in as to why my expectations are being defied here?  I
> think the root of my misunderstanding lies in my not understanding the
> differences between atan(a/b) and atan2(a,b).

atan2(Im{z}, Re{z}) is meant to be the angle of a complex number z.

    arg{z} = atan2(Im{z}, Re{z})

when using the atan2(Im{z}, Re{z}) for the angle of a complex number,
it is correct only if Re(z)>0. for the case Re(z)<0, your atan2(Im{z},
Re{z}) expression for the angle is off by a constant value of pi.


sometimes we say:

             {arctan(Im{z}/Re{z})            for Re{z}>0
    arg{z} = {
             {arctan(Im{z}/Re{z}) +/- pi     for Re{z}<0

or better:


             {arctan(Im{z}/Re{z})         Re{z}>0, Im{z}<|Re{z}|
    arg{z} = {
             {sgn(Im{z})*pi/2 - arctan(Re{z}/Im{z})  otherwize

atan2() deals with getting the angle right, no matter which quadrant z
ends up in.

r b-j

maxplanck wrote:
> I'm a bit stumped here, maybe someone can help me out?
> 
> y = atan (sin(2*pi*x) / cos(2*pi*x)); 
> 
> s = atan2 (sin(2*pi*x) , cos(2*pi*x));
> 
> I would expect y and s to be equal, but they're not.
> 
> Most surprising to me is that the maximum value of s appears to be pi,
> despite the fact that the limit of atan2(a,b) as a/b->infinity is pi/2
> (right?)
> 
> Can anyone clue me in as to why my expectations are being defied here?  I
> think the root of my misunderstanding lies in my not understanding the
> differences between atan(a/b) and atan2(a,b).
> 
> Thanks, any help is much appreciated

How much trig do you remember? Can you simplify sin(x)/cos(x)? Are you 
familiar with the concept of "principal value"?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:
> maxplanck wrote:
>> I'm a bit stumped here, maybe someone can help me out?
>>
>> y = atan (sin(2*pi*x) / cos(2*pi*x));
>> s = atan2 (sin(2*pi*x) , cos(2*pi*x));
>>
>> I would expect y and s to be equal, but they're not.
>>
>> Most surprising to me is that the maximum value of s appears to be pi,
>> despite the fact that the limit of atan2(a,b) as a/b->infinity is pi/2
>> (right?)
>>
>> Can anyone clue me in as to why my expectations are being defied here?  I
>> think the root of my misunderstanding lies in my not understanding the
>> differences between atan(a/b) and atan2(a,b).
>>
>> Thanks, any help is much appreciated
> 
> How much trig do you remember? Can you simplify sin(x)/cos(x)? Are you 
> familiar with the concept of "principal value"?

This is a DSP group, so why not express it in terms of information....

Division is lossy, as maps the answers to -ve/-ve to the same space as 
+ve/+ve ones. Same for -ve/+ve and +ve/-ve. atan2() lets the functiom 
have the complete raw information, so it can handle all four quadrants 
properly.

Steve

"maxplanck" <erik.bowen@comcast.net> writes:

> I'm a bit stumped here, maybe someone can help me out?
> 
> y = atan (sin(2*pi*x) / cos(2*pi*x)); 
> 
> s = atan2 (sin(2*pi*x) , cos(2*pi*x));
> 
> I would expect y and s to be equal, but they're not.
> 
> Most surprising to me is that the maximum value of s appears to be pi,
> despite the fact that the limit of atan2(a,b) as a/b->infinity is pi/2
> (right?)
> 
> Can anyone clue me in as to why my expectations are being defied here?  I
> think the root of my misunderstanding lies in my not understanding the
> differences between atan(a/b) and atan2(a,b).
> 
> Thanks, any help is much appreciated

y is just the principle value, which will only be -pi/2 and pi/2.

s is the four quandrant version which can be between -pi and pi.

http://en.wikipedia.org/wiki/Atan

(see the section entitled "Two argument variant of arctangent").

Ciao,

Peter K.

-- 
"And he sees the vision splendid 
of the sunlit plains extended
And at night the wondrous glory of the everlasting stars."

 

What I meant to say was the angle between the x-axis and the line
segment connecting the points (0,0) and (x,y)

On Jan 8, 6:50 pm, predict...@gmail.com wrote:
> atan2(y,x) is used to find the angle from the point (0,0) to the point
> (x,y), using the x-axis as reference; e.g.,
>
> atan2 (.707, .707) = pi/4 => 45 degrees
> atan2 (.707, -.707) = 3*pi/4 => 135 degrees
> atan2 (-.707, .707) = -pi/4 = -45 degrees
> atan2 (-.707, -.707) = -3*pi/4 => -135 degrees
>
> (the signs of x and y determine the quadrant of the result)
>
> On Jan 8, 6:13 pm, "maxplanck" <erik.bo...@comcast.net> wrote:
>
> > I'm a bit stumped here, maybe someone can help me out?
>
> > y = atan (sin(2*pi*x) / cos(2*pi*x));
>
> > s = atan2 (sin(2*pi*x) , cos(2*pi*x));
>
> > I would expect y and s to be equal, but they're not.
>
> > Most surprising to me is that the maximum value of s appears to be pi,
> > despite the fact that the limit of atan2(a,b) as a/b->infinity is pi/2
> > (right?)
>
> > Can anyone clue me in as to why my expectations are being defied here?  I
> > think the root of my misunderstanding lies in my not understanding the
> > differences between atan(a/b) and atan2(a,b).
>
> > Thanks, any help is much appreciated

In case if you can get hold of the book 
"understanding dsp" by richard lyons a
nice section is written,

13.21 an efficient arctangent approximation.
which tells about how to approximate

theta = atan(Q/I) where Q and I can be both
positive as well as negative numbers. 

actually the tan(theta) plot can go both 
positive infinity and negative infinity.
but atan plot output is always bounded
between -pi and pi, though the input argument
can go both +/- infinity.

so when you consider Q and I seperately
(meaning signs of it), we get the sense
of quadrant. Else the sense of quadrant is
somewhat lost when we have a single quantity 
Q/I as an argument.

Bharat Pathak
Arithos Designs

i just noticed a mistake:

On Jan 8, 6:52 pm, robert bristow-johnson <r...@audioimagination.com>
wrote:
>
> atan2(Im{z}, Re{z}) is meant to be the angle of a complex number z.
>
>     arg{z} = atan2(Im{z}, Re{z})
>
> when using the atan2(Im{z}, Re{z}) for the angle of a complex number,
> it is correct only if Re(z)>0.  for the case Re(z)<0, your atan2(Im{z},
> Re{z}) expression for the angle is off by a constant value of pi.

i meant to say:

"when using the arctan(Im{z}/Re{z}) for the angle of a complex number,
it is correct only if Re(z)>0.  for the case Re(z)<0, your
arctan(Im{z}/Re{z}) expression for the angle is off by a constant
value of pi."

atan2(Im{z}, Re{z}) is correct (for the principal angle) in anycase
where z is not 0.

also

>            {arctan(Im{z}/Re{z})         Re{z}>0, Im{z}<|Re{z}|
>   arg{z} = {
>            {sgn(Im{z})*pi/2 - arctan(Re{z}/Im{z})    otherwize

is slightly wrong.  but i think the rest of it was accurate.

the 4-quadrant expressions for arg{z} are useful for the case where
you might not have an efficient implementation of atan2() available to
you in your DSP context, e.g. phase-vocoder or sinusoidal modeling.
then you have to combine this

            { arctan(Im{z}/Re{z})         Re{z}>0, |Im{z}|<=Re{z}
            {
            { pi/2 - arctan(Re{z}/Im{z})  Im{z}>0, |Re{z}|<=Im{z}
            {
  Arg{z} =  {-pi/2 - arctan(Re{z}/Im{z})  Im{z}<0, |Re{z}|<=-Im{z}
            {
            { arctan(Im{z}/Re{z}) + pi    Re{z}<0, 0<=Im{z}<=-Re{z}
            {
            { arctan(Im{z}/Re{z}) - pi    Re{z}<0, Re{z}<=Im{z}<0


with a means of implementing arctan(), such as

    arctan(x) ~= x/f(x^2)

where

    f(u)  =     1.0
             +  0.33288950512027 * u
             + -0.08467922817644 * u^2
             +  0.03252232640125 * u^3
             + -0.00749305860992 * u^4

    for |x| <= 1 .

r b-j

On Jan 8, 7:56�pm, p.kootsoo...@remove.ieee.org (Peter K.) wrote:
> "maxplanck" <erik.bo...@comcast.net> writes:
> > I'm a bit stumped here, maybe someone can help me out?
>
> > y = atan (sin(2*pi*x) / cos(2*pi*x));
>
> > s = atan2 (sin(2*pi*x) , cos(2*pi*x));
>
> > I would expect y and s to be equal, but they're not.
>
> > Most surprising to me is that the maximum value of s appears to be pi,
> > despite the fact that the limit of atan2(a,b) as a/b->infinity is pi/2
> > (right?)
>
> > Can anyone clue me in as to why my expectations are being defied here? &#4294967295;I
> > think the root of my misunderstanding lies in my not understanding the
> > differences between atan(a/b) and atan2(a,b).
>
> > Thanks, any help is much appreciated
>
> y is just the principle value, which will only be -pi/2 and pi/2.
>
> s is the four quandrant version which can be between -pi and pi.
>
> http://en.wikipedia.org/wiki/Atan
>
> (see the section entitled "Two argument variant of arctangent").
>
> Ciao,
>
> Peter K.
>
> --
> "And he sees the vision splendid
> of the sunlit plains extended
> And at night the wondrous glory of the everlasting stars."- Hide quoted text -
>
> - Show quoted text -

Peter,

Word is "principal' not your word "principle".  Correct wordings most
value if you are too be Dsp Guru!

Ciao,

Kamar Ruptan