atan2 function

atan2 function

[Thangalin via ntg-context]

Hi there,

How is atan2 called? I rolled my own as follows:

% SOT
\startuseMPgraphic{HexGridBaseGraphic}
  vardef atantwo( expr ax, ay, bx, by ) =
    save theta;

    numeric dx;
    numeric dy;
    numeric theta;

    dx := bx - ax;
    dy := by - ay;
    theta := 0;

    if (dx > 0):
      theta := atan( dy / dx );
    elseif (dx < 0) and (dy >= 0):
      theta := atan( dy / dx ) + pi;
    elseif (dx < 0) and (dy < 0):
      theta := atan( dy / dx ) - pi;
    elseif (dx == 0) and (dy > 0):
      theta := pi / 2;
    elseif (dx == 0) and (dy < 0):
      theta := -pi / 2;
    fi;

    %theta := atan( dy, dx );

    theta
  enddef;
% EOT

When this runs, the expected result is generated:

https://pdfhost.io/v/xdNxANU18_scaled

When uncommenting "atan( dy, dx )", an unexpected result is generated:

https://pdfhost.io/v/QdfU89IL._scaled

Is atan with two parameters supposed to behave like atan2?

Cheers!
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___________________________________________________________________________________

Re: atan2 function

[Max Chernoff via ntg-context]

Hi,

> How is atan2 called? I rolled my own as follows:

> Is atan with two parameters supposed to behave like atan2?

At mp-math.mpxl:167 there is:

       vardef atan   primary x = angle(1,x)       enddef ;
   
The MetaPost manual says:

   The angle operator takes a pair and computes the two-argument
   arctangent; i.e., angle is the inverse of the dir operator
   
So it looks like "angle" is the function that you want for "atan2". 

Thanks,
-- Max
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Re: atan2 function

[Thangalin via ntg-context]

Thank you, Max.

The angle function doesn't appear to provide the same calculation as
my atantwo in all cases.

https://pdfhost.io/v/Oqj7XmibJ_scaled

The shorter line segment should be directed towards the vertex closest
to the longer line segment. I tried using both:

    theta := angle( dx, dy );
    theta := angle( dy, dx );

Neither made a difference, in some cases the angle differs from what I'd expect.

My implementation is based on the first version given here:

https://en.wikipedia.org/wiki/Atan2#Definition_and_computation

I have something that works, so this is more of a curiosity as to
there being a difference between my implementation of atantwo and the
angle function.

Here's an example with many lines and the angle function:
https://pdfhost.io/v/1T4jgBnxh_scaled

On Wed, Oct 12, 2022 at 11:42 PM Max Chernoff <mseven@telus.net> wrote:
>
> Hi,
>
> > How is atan2 called? I rolled my own as follows:
>
> > Is atan with two parameters supposed to behave like atan2?
>
> At mp-math.mpxl:167 there is:
>
>        vardef atan   primary x = angle(1,x)       enddef ;
>
> The MetaPost manual says:
>
>    The angle operator takes a pair and computes the two-argument
>    arctangent; i.e., angle is the inverse of the dir operator
>
> So it looks like "angle" is the function that you want for "atan2".
>
> Thanks,
> -- Max
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___________________________________________________________________________________

Re: atan2 function

[Max Chernoff via ntg-context]

Hi,

> The angle function doesn't appear to provide the same calculation as
> my atantwo in all cases.

They both give the same results, but "angle" gives a result in degrees
while "atantwo" gives a result in radians. This demo:

   \startMPpage
       vardef atantwo( expr dy, dx ) =
           save theta;
   
           numeric theta;
   
           theta := 0;
   
           if (dx > 0):
               theta := atan( dy / dx );
           elseif (dx < 0) and (dy >= 0):
               theta := atan( dy / dx ) + pi;
           elseif (dx < 0) and (dy < 0):
               theta := atan( dy / dx ) - pi;
           elseif (dx == 0) and (dy > 0):
               theta := pi / 2;
           elseif (dx == 0) and (dy < 0):
               theta := -pi / 2;
           fi;
   
           theta
       enddef;
   
       def showangles(expr dx, dy) =
           message "---------------"
           message "atantwo " & decimal atantwo(dy, dx)
           message "angle " & decimal (angle(dx, dy) * pi / 180)
           message "angle " & decimal angle(dx, dy)
           message "(" & decimal dx & ", " & decimal dy & ")"
       enddef;
   
       showangles(1, 0);
       showangles(1, 1);
       showangles(0, 1);
       showangles(-1, 1);
       showangles(-1, 0);
       showangles(-1, -1);
       showangles(0, -1);
       showangles(1, -1);
   \stopMPpage

gives:

   (1, 0)
   angle 0
   angle 0
   atantwo 0
   ---------------
   (1, 1)
   angle 45
   angle 0.78539816339744828
   atantwo 0.78539816339744828
   ---------------
   (0, 1)
   angle 90
   angle 1.5707963267948966
   atantwo 1.5707963267948966
   ---------------
   (-1, 1)
   angle 135
   angle 2.3561944901923448
   atantwo 2.3561944901923448
   ---------------
   (-1, 0)
   angle 180
   angle 3.1415926535897931
   atantwo 3.1415926535897931
   ---------------
   (-1, -1)
   angle -135
   angle -2.3561944901923448
   atantwo -2.3561944901923448
   ---------------
   (0, -1)
   angle -90
   angle -1.5707963267948966
   atantwo -1.5707963267948966
   ---------------
   (1, -1)
   angle -45
   angle -0.78539816339744828
   atantwo -0.78539816339744828
   ---------------
   
Thanks,
-- Max
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Re: atan2 function

[Hans Hagen via ntg-context]

On 10/14/2022 2:35 AM, Max Chernoff via ntg-context wrote:
> 
> Hi,
> 
>> The angle function doesn't appear to provide the same calculation as
>> my atantwo in all cases.
> 
> They both give the same results, but "angle" gives a result in degrees
> while "atantwo" gives a result in radians. This demo:
> 
>     \startMPpage
>         vardef atantwo( expr dy, dx ) =
>             save theta;
>     
>             numeric theta;
>     
>             theta := 0;
>     
>             if (dx > 0):
>                 theta := atan( dy / dx );
>             elseif (dx < 0) and (dy >= 0):
>                 theta := atan( dy / dx ) + pi;
>             elseif (dx < 0) and (dy < 0):
>                 theta := atan( dy / dx ) - pi;
>             elseif (dx == 0) and (dy > 0):
>                 theta := pi / 2;
>             elseif (dx == 0) and (dy < 0):
>                 theta := -pi / 2;
>             fi;
>     
>             theta
>         enddef;
>     
>         def showangles(expr dx, dy) =
>             message "---------------"
>             message "atantwo " & decimal atantwo(dy, dx)
>             message "angle " & decimal (angle(dx, dy) * pi / 180)
>             message "angle " & decimal angle(dx, dy)
>             message "(" & decimal dx & ", " & decimal dy & ")"
>         enddef;
>     
>         showangles(1, 0);
>         showangles(1, 1);
>         showangles(0, 1);
>         showangles(-1, 1);
>         showangles(-1, 0);
>         showangles(-1, -1);
>         showangles(0, -1);
>         showangles(1, -1);
>     \stopMPpage
So you suggest to add atantwo? As side note, you can redure your 
definition to:

          vardef atantwo( expr dy, dx ) =
              if (dx > 0):
                  atan( dy / dx )
              elseif (dx < 0) and (dy >= 0):
                  atan( dy / dx ) + pi
              elseif (dx < 0) and (dy < 0):
                  atan( dy / dx ) - pi
              elseif (dx == 0) and (dy > 0):
                  pi / 2
              elseif (dx == 0) and (dy < 0):
                  -pi / 2
              else :
                  0
              fi
          enddef ;

and then

     vardef atantwo(expr dy, dx) =
         if dx == 0 :
             if dy < 0 : - fi pi / 2
         else :
             atan(dy/dx) if dx > 0 : if dy < 0 : - else : + fi pi fi
         fi
     enddef ;

going further makes us end up in an one-line obscurity

Hans

-----------------------------------------------------------------
                                           Hans Hagen | PRAGMA ADE
               Ridderstraat 27 | 8061 GH Hasselt | The Netherlands
        tel: 038 477 53 69 | www.pragma-ade.nl | www.pragma-pod.nl
-----------------------------------------------------------------

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Re: atan2 function

[Alan Braslau via ntg-context]

Are you using MP in scaled integer or in doubleprecision mode?

I have not looked into the MP angle operator code, but my suspicion is 
that it is based on some super-clever John Hobby scheme optimized for 
scaled integer calculations.

Hans has integrated the entire math library (and more) into lmtx.

On 13/10/22 13/10/22, 13:51, Thangalin via ntg-context wrote:
> Thank you, Max.
> 
> The angle function doesn't appear to provide the same calculation as
> my atantwo in all cases.
> 
> https://pdfhost.io/v/Oqj7XmibJ_scaled
> 
> The shorter line segment should be directed towards the vertex closest
> to the longer line segment. I tried using both:
> 
>      theta := angle( dx, dy );
>      theta := angle( dy, dx );
> 
> Neither made a difference, in some cases the angle differs from what I'd expect.
> 
> My implementation is based on the first version given here:
> 
> https://en.wikipedia.org/wiki/Atan2#Definition_and_computation
> 
> I have something that works, so this is more of a curiosity as to
> there being a difference between my implementation of atantwo and the
> angle function.
> 
> Here's an example with many lines and the angle function:
> https://pdfhost.io/v/1T4jgBnxh_scaled
> 
> On Wed, Oct 12, 2022 at 11:42 PM Max Chernoff <mseven@telus.net> wrote:
>>
>> Hi,
>>
>>> How is atan2 called? I rolled my own as follows:
>>
>>> Is atan with two parameters supposed to behave like atan2?
>>
>> At mp-math.mpxl:167 there is:
>>
>>         vardef atan   primary x = angle(1,x)       enddef ;
>>
>> The MetaPost manual says:
>>
>>     The angle operator takes a pair and computes the two-argument
>>     arctangent; i.e., angle is the inverse of the dir operator
>>
>> So it looks like "angle" is the function that you want for "atan2".
>>
>> Thanks,
>> -- Max
> ___________________________________________________________________________________
> If your question is of interest to others as well, please add an entry to the Wiki!
> 
> maillist : ntg-context@ntg.nl / https://www.ntg.nl/mailman/listinfo/ntg-context
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> archive  : https://bitbucket.org/phg/context-mirror/commits/
> wiki     : https://contextgarden.net
> ___________________________________________________________________________________

\

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Re: atan2 function

[Taco Hoekwater via ntg-context]

> On 14 Oct 2022, at 14:26, Alan Braslau via ntg-context <ntg-context@ntg.nl> wrote:
> 
> Are you using MP in scaled integer or in doubleprecision mode?
> 
> I have not looked into the MP angle operator code, but my suspicion is that it is based on some super-clever John Hobby scheme optimized for scaled integer calculations.

The scaled version of “angle” is a handwritten approximation in web, not using
any math library. I really doubt that that code is faster than calling atan2()
on a modern machine, but I’ve kept it in because using a lib might give slightly 
different results.

The double version of “angle" uses atan2() internally already, so that the 
atantwo macro is just duplicating the internals for that case.

The decimal version of “angle’ uses a dedicated decNumberAtan2() library call.

> 
> Hans has integrated the entire math library (and more) into lmtx.
> 
> On 13/10/22 13/10/22, 13:51, Thangalin via ntg-context wrote:
>> Thank you, Max.
>> The angle function doesn't appear to provide the same calculation as
>> my atantwo in all cases.
>> https://pdfhost.io/v/Oqj7XmibJ_scaled
>> The shorter line segment should be directed towards the vertex closest
>> to the longer line segment. I tried using both:
>>     theta := angle( dx, dy );
>>     theta := angle( dy, dx );
>> Neither made a difference, in some cases the angle differs from what I'd expect.
>> My implementation is based on the first version given here:
>> https://en.wikipedia.org/wiki/Atan2#Definition_and_computation
>> I have something that works, so this is more of a curiosity as to
>> there being a difference between my implementation of atantwo and the
>> angle function.
>> Here's an example with many lines and the angle function:
>> https://pdfhost.io/v/1T4jgBnxh_scaled
>> On Wed, Oct 12, 2022 at 11:42 PM Max Chernoff <mseven@telus.net> wrote:
>>> 
>>> Hi,
>>> 
>>>> How is atan2 called? I rolled my own as follows:
>>> 
>>>> Is atan with two parameters supposed to behave like atan2?
>>> 
>>> At mp-math.mpxl:167 there is:
>>> 
>>>        vardef atan   primary x = angle(1,x)       enddef ;
>>> 
>>> The MetaPost manual says:
>>> 
>>>    The angle operator takes a pair and computes the two-argument
>>>    arctangent; i.e., angle is the inverse of the dir operator
>>> 
>>> So it looks like "angle" is the function that you want for "atan2".
>>> 
>>> Thanks,
>>> -- Max
>> ___________________________________________________________________________________
>> If your question is of interest to others as well, please add an entry to the Wiki!
>> maillist : ntg-context@ntg.nl / https://www.ntg.nl/mailman/listinfo/ntg-context
>> webpage  : https://www.pragma-ade.nl / http://context.aanhet.net
>> archive  : https://bitbucket.org/phg/context-mirror/commits/
>> wiki     : https://contextgarden.net
>> ___________________________________________________________________________________
> 
> \
> 
> ___________________________________________________________________________________
> If your question is of interest to others as well, please add an entry to the Wiki!
> 
> maillist : ntg-context@ntg.nl / https://www.ntg.nl/mailman/listinfo/ntg-context
> webpage  : https://www.pragma-ade.nl / http://context.aanhet.net
> archive  : https://bitbucket.org/phg/context-mirror/commits/
> wiki     : https://contextgarden.net
> ___________________________________________________________________________________

— 
Taco Hoekwater              E: taco@bittext.nl
genderfluid (all pronouns)



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Re: atan2 function

[luigi scarso via ntg-context]

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On Fri, 14 Oct 2022 at 14:58, Taco Hoekwater via ntg-context <
ntg-context@ntg.nl> wrote:

>
>
> > On 14 Oct 2022, at 14:26, Alan Braslau via ntg-context <
> ntg-context@ntg.nl> wrote:
> >
> > Are you using MP in scaled integer or in doubleprecision mode?
> >
> > I have not looked into the MP angle operator code, but my suspicion is
> that it is based on some super-clever John Hobby scheme optimized for
> scaled integer calculations.
>
> The scaled version of “angle” is a handwritten approximation in web, not
> using
> any math library. I really doubt that that code is faster than calling
> atan2()
> on a modern machine, but I’ve kept it in because using a lib might give
> slightly
> different results.
>

iirc, in metapost  the whole scaled number system has no dependency  from
any external math library
(I have not checked the lmtx source yet).

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___________________________________________________________________________________

Re: atan2 function

[Alan Braslau via ntg-context]

On Fri, 14 Oct 2022 10:07:19 +0200
Hans Hagen via ntg-context <ntg-context@ntg.nl> wrote:

> So you suggest to add atantwo? As side note, you can redure your 
> definition to:
> 
>      vardef atantwo(expr dy, dx) =
>          if dx == 0 :
>              if dy < 0 : - fi pi / 2
>          else :
>              atan(dy/dx) if dx > 0 : if dy < 0 : - else : + fi pi fi
>          fi
>      enddef ;
> 
> going further makes us end up in an one-line obscurity

An absurdity, as further discussion on this thread has shown:
atan is derived from angle, which is essentially atan2.
There is no need to circle around.

As a side point, I now use MP doubleprecision mode by default, although
I do believe that it is still preferable to use classic scaled integer
mode (and minifun) for text graphical embellishments. The only
practical point to consider when using doubleprecision mode for me is
the need to explicitly use format with "decimal i" to round the text
rendering of numbers.

Alan
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Re: atan2 function

[Thangalin via ntg-context]

I don't think an atantwo is needed. I *thought* I had read somewhere
that atan( y, x ) was equivalent to calling atan2 in Lua. Ensuring
there's no breakage when x == y would be nice, though. It was a little
surprising to see angle return degrees rather than radians, but it
does simplify my code:

      dc := vbc - vac;
      dr := vbr - var;
      vi := 0;

      if not( dc == dr ):
        vi := round( angle( dc, dr ) / 60 );
      fi;

      % Compute the direction towards the first segment (to vertex of an edge).
      vangle := vi * 60 * pi / 180;

Even simpler would be:

      dc := vbc - vac;
      dr := vbr - var;
      vi := round( angle( dc, dr ) / 60 ); % returns 0 when dc == dr

      % Compute the direction towards the first segment (to vertex of an edge).
      vangle := vi * 60 * pi / 180;

Or accepting a third argument as the return value in the special case:

      vi := round( angle( dc, dr, 0 ) / 60 ); % returns 0 when dc == dr

Cheers!
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Re: atan2 function

[Alan Braslau via ntg-context]

On Fri, 14 Oct 2022 11:59:49 -0700
Thangalin <thangalin@gmail.com> wrote:

> I don't think an atantwo is needed. I *thought* I had read somewhere
> that atan( y, x ) was equivalent to calling atan2 in Lua. Ensuring
> there's no breakage when x == y would be nice, though. It was a little
> surprising to see angle return degrees rather than radians, but it
> does simplify my code:
> 
>       dc := vbc - vac;
>       dr := vbr - var;
>       vi := 0;
> 
>       if not( dc == dr ):
>         vi := round( angle( dc, dr ) / 60 );
>       fi;
> 
>       % Compute the direction towards the first segment (to vertex of
> an edge). vangle := vi * 60 * pi / 180;
> 
> Even simpler would be:
> 
>       dc := vbc - vac;
>       dr := vbr - var;
>       vi := round( angle( dc, dr ) / 60 ); % returns 0 when dc == dr
> 
>       % Compute the direction towards the first segment (to vertex of
> an edge). vangle := vi * 60 * pi / 180;
> 
> Or accepting a third argument as the return value in the special case:
> 
>       vi := round( angle( dc, dr, 0 ) / 60 ); % returns 0 when dc ==
> dr
> 
> Cheers!

vi := if (dc = dr) : 0 else : round (angle(dc,dr)/60) fi ;

Alan
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Re: atan2 function

[Thangalin via ntg-context]

That's certainly tighter, thank you, Alan.

Would making the conditional part of the API be useful to others?

Cheers!

On Fri, Oct 14, 2022 at 3:42 PM Alan Braslau <alan.braslau@icloud.com> wrote:
>
> On Fri, 14 Oct 2022 11:59:49 -0700
> Thangalin <thangalin@gmail.com> wrote:
>
> > I don't think an atantwo is needed. I *thought* I had read somewhere
> > that atan( y, x ) was equivalent to calling atan2 in Lua. Ensuring
> > there's no breakage when x == y would be nice, though. It was a little
> > surprising to see angle return degrees rather than radians, but it
> > does simplify my code:
> >
> >       dc := vbc - vac;
> >       dr := vbr - var;
> >       vi := 0;
> >
> >       if not( dc == dr ):
> >         vi := round( angle( dc, dr ) / 60 );
> >       fi;
> >
> >       % Compute the direction towards the first segment (to vertex of
> > an edge). vangle := vi * 60 * pi / 180;
> >
> > Even simpler would be:
> >
> >       dc := vbc - vac;
> >       dr := vbr - var;
> >       vi := round( angle( dc, dr ) / 60 ); % returns 0 when dc == dr
> >
> >       % Compute the direction towards the first segment (to vertex of
> > an edge). vangle := vi * 60 * pi / 180;
> >
> > Or accepting a third argument as the return value in the special case:
> >
> >       vi := round( angle( dc, dr, 0 ) / 60 ); % returns 0 when dc ==
> > dr
> >
> > Cheers!
>
> vi := if (dc = dr) : 0 else : round (angle(dc,dr)/60) fi ;
>
> Alan
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Re: atan2 function

[Alan Braslau via ntg-context]

On Fri, 14 Oct 2022 16:09:41 -0700
Thangalin <thangalin@gmail.com> wrote:

> Would making the conditional part of the API be useful to others?

Of course, $\tan^{-1}(\pm 1)$ is undefined (mathematically).

Alan
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