METAPOST's superellipse with superness<0.5 does not give a superellipse

METAPOST's superellipse with superness<0.5 does not give a superellipse

[James Fisher]

[-- Attachment #1.1: Type: text/plain, Size: 2368 bytes --]

Hi again,


Another METAPOST problem.  For the sake of curiosity, I've been looking at
and playing with the superellipse() function in plain METAPOST.  This is all
fine and dandy until I try values of 'superness' less than 0.5, in which
case it generates shapes that are seemingly not superellipses.  At s=0.5,
the function generates a diamond shape -- which, AFAIK, is correct.
However, s<0.5, the points of the diamond immediately turn to curves.  (My
knowledge of superellipses here is just from
http://en.wikipedia.org/wiki/Superellipse -- try the image at
http://en.wikipedia.org/wiki/File:Lame_anima.gif to see how I expect the
shape to change with varying values of superness).

Some code follows -- perhaps someone could run it and tell me if, for
starters, they get the same as me.  (See
http://i49.tinypic.com/2ijqatl.jpgfor superellipse() with s=0.3).


Best,


James



% The following is a superellipse function at <
http://lists.foundry.supelec.fr/pipermail/metapost-commits/2008-June/000340.html
>;
% I think it's the superellipse function in my copy of METAPOST; it at least
has the same behaviour.
% It seems to calculate the vertices correctly, but not the way they join
(try changing all ... to --).
%
%def superellipse(expr r,t,l,b,s)=
%  r ... (s[xpart t,xpart r],s[ypart r,ypart t]){t-r} ...
%  t ... (s[xpart t,xpart l],s[ypart l,ypart t]){l-t} ...
%  l ... (s[xpart b,xpart l],s[ypart l,ypart b]){b-l} ...
%  b ... (s[xpart b,xpart r],s[ypart r,ypart b]){r-b} ... cycle
%enddef;

def supertest expr s =
  superellipse(
    ( 100, 50  ),
    ( 50,  100 ),
    ( 0,   50  ),
    ( 50,  0   ),
    s
    );
enddef;

% These >0 supernesses are fine, I think ...

beginfig(0);
  draw supertest 2;
endfig;

beginfig(1);
  draw supertest 1.01;
endfig;

% The following, 0.5>=superness<=1,
% are from visual reference definitely right

beginfig(2);
  draw supertest 1;
endfig;

beginfig(3);
  draw supertest 0.99;
endfig;

beginfig(4);
  draw supertest 0.7;
endfig;

beginfig(5);
  draw supertest 0.51;
endfig;

beginfig(6);
  draw supertest 0.5;
endfig;

% Now, for <0.5,
% things get problematic --
% the points in the shape generated by s=0.5
% should stay 'pointy'

beginfig(7);
  draw supertest 0.49;
endfig;

beginfig(8);
  draw supertest 0.3;
endfig;

beginfig(9);
  draw supertest 0.01;
endfig;

beginfig(10);
  draw supertest 0;
endfig;

end;

[-- Attachment #1.2: Type: text/html, Size: 2974 bytes --]

[-- Attachment #2: Type: text/plain, Size: 486 bytes --]

___________________________________________________________________________________
If your question is of interest to others as well, please add an entry to the Wiki!

maillist : ntg-context@ntg.nl / http://www.ntg.nl/mailman/listinfo/ntg-context
webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
archive  : http://foundry.supelec.fr/projects/contextrev/
wiki     : http://contextgarden.net
___________________________________________________________________________________

Re: METAPOST's superellipse with superness<0.5 does not give a superellipse

[James Fisher]

[-- Attachment #1.1: Type: text/plain, Size: 3427 bytes --]

I should say that the vertices of the superellipse are calculated
correctly.  The problem, it seems, is that for the vertices at right, top,
left, and bottom, the angles of entry and exit need to be explicitly
defined, rather than just relying on the '...' which coincidentally works
for s>=0.5.

I should say at this point that I am no maths whiz.  But, sticking with the
'right ... topright ... top' line, the angle calculation needs to satisfy,
for the exit angle of the first vertex:

For s=0, angle = 180 degrees (vector to the left)
For s = 0.5, angle = 135 degrees (45 degrees to the top left, producing a
straight line to create the diamond shape)
For s>0.5, angle = 90 degrees (vector vertically upwards)

Suffice to say that I don't know how to produce that elegantly.



James


On Mon, Mar 1, 2010 at 3:54 PM, James Fisher <jameshfisher@gmail.com> wrote:

> Hi again,
>
>
> Another METAPOST problem.  For the sake of curiosity, I've been looking at
> and playing with the superellipse() function in plain METAPOST.  This is all
> fine and dandy until I try values of 'superness' less than 0.5, in which
> case it generates shapes that are seemingly not superellipses.  At s=0.5,
> the function generates a diamond shape -- which, AFAIK, is correct.
> However, s<0.5, the points of the diamond immediately turn to curves.  (My
> knowledge of superellipses here is just from
> http://en.wikipedia.org/wiki/Superellipse -- try the image at
> http://en.wikipedia.org/wiki/File:Lame_anima.gif to see how I expect the
> shape to change with varying values of superness).
>
> Some code follows -- perhaps someone could run it and tell me if, for
> starters, they get the same as me.  (See
> http://i49.tinypic.com/2ijqatl.jpg for superellipse() with s=0.3).
>
>
> Best,
>
>
> James
>
>
>
> % The following is a superellipse function at <
> http://lists.foundry.supelec.fr/pipermail/metapost-commits/2008-June/000340.html
> >;
> % I think it's the superellipse function in my copy of METAPOST; it at
> least has the same behaviour.
> % It seems to calculate the vertices correctly, but not the way they join
> (try changing all ... to --).
> %
> %def superellipse(expr r,t,l,b,s)=
> %  r ... (s[xpart t,xpart r],s[ypart r,ypart t]){t-r} ...
> %  t ... (s[xpart t,xpart l],s[ypart l,ypart t]){l-t} ...
> %  l ... (s[xpart b,xpart l],s[ypart l,ypart b]){b-l} ...
> %  b ... (s[xpart b,xpart r],s[ypart r,ypart b]){r-b} ... cycle
> %enddef;
>
> def supertest expr s =
>   superellipse(
>     ( 100, 50  ),
>     ( 50,  100 ),
>     ( 0,   50  ),
>     ( 50,  0   ),
>     s
>     );
> enddef;
>
> % These >0 supernesses are fine, I think ...
>
> beginfig(0);
>   draw supertest 2;
> endfig;
>
> beginfig(1);
>   draw supertest 1.01;
> endfig;
>
> % The following, 0.5>=superness<=1,
> % are from visual reference definitely right
>
> beginfig(2);
>   draw supertest 1;
> endfig;
>
> beginfig(3);
>   draw supertest 0.99;
> endfig;
>
> beginfig(4);
>   draw supertest 0.7;
> endfig;
>
> beginfig(5);
>   draw supertest 0.51;
> endfig;
>
> beginfig(6);
>   draw supertest 0.5;
> endfig;
>
> % Now, for <0.5,
> % things get problematic --
> % the points in the shape generated by s=0.5
> % should stay 'pointy'
>
> beginfig(7);
>   draw supertest 0.49;
> endfig;
>
> beginfig(8);
>   draw supertest 0.3;
> endfig;
>
> beginfig(9);
>   draw supertest 0.01;
> endfig;
>
> beginfig(10);
>   draw supertest 0;
> endfig;
>
> end;
>
>

[-- Attachment #1.2: Type: text/html, Size: 4257 bytes --]

[-- Attachment #2: Type: text/plain, Size: 486 bytes --]

___________________________________________________________________________________
If your question is of interest to others as well, please add an entry to the Wiki!

maillist : ntg-context@ntg.nl / http://www.ntg.nl/mailman/listinfo/ntg-context
webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
archive  : http://foundry.supelec.fr/projects/contextrev/
wiki     : http://contextgarden.net
___________________________________________________________________________________

Re: METAPOST's superellipse with superness<0.5 does not give a superellipse

[Rory Molinari]

James's explanation appears to be right.

On p 126 of the METAFONTbook Knuth says that the "superness should be
between 0.5 (when you get a diamond) and 1.0 (when you get a square)".

Exercise 14.6 asks the reader to "Try superellimpse with superness
values less than 0.5 or greater than 1.0; explain why you get weird
shapes in such cases."  The answer is "There are inflection points,
because there are no bounding triangles for the '...' operations in
the superellipse macro ... unless 0.5 \leq s \leq 1."

Cheers,
Rory

On Mon, Mar 1, 2010 at 8:04 AM, James Fisher <jameshfisher@gmail.com> wrote:
> I should say that the vertices of the superellipse are calculated
> correctly.  The problem, it seems, is that for the vertices at right, top,
> left, and bottom, the angles of entry and exit need to be explicitly
> defined, rather than just relying on the '...' which coincidentally works
> for s>=0.5.
>
> I should say at this point that I am no maths whiz.  But, sticking with the
> 'right ... topright ... top' line, the angle calculation needs to satisfy,
> for the exit angle of the first vertex:
>
> For s=0, angle = 180 degrees (vector to the left)
> For s = 0.5, angle = 135 degrees (45 degrees to the top left, producing a
> straight line to create the diamond shape)
> For s>0.5, angle = 90 degrees (vector vertically upwards)
>
> Suffice to say that I don't know how to produce that elegantly.
>
>
>
> James
>
>
> On Mon, Mar 1, 2010 at 3:54 PM, James Fisher <jameshfisher@gmail.com> wrote:
>>
>> Hi again,
>>
>>
>> Another METAPOST problem.  For the sake of curiosity, I've been looking at
>> and playing with the superellipse() function in plain METAPOST.  This is all
>> fine and dandy until I try values of 'superness' less than 0.5, in which
>> case it generates shapes that are seemingly not superellipses.  At s=0.5,
>> the function generates a diamond shape -- which, AFAIK, is correct.
>> However, s<0.5, the points of the diamond immediately turn to curves.  (My
>> knowledge of superellipses here is just from
>> http://en.wikipedia.org/wiki/Superellipse -- try the image at
>> http://en.wikipedia.org/wiki/File:Lame_anima.gif to see how I expect the
>> shape to change with varying values of superness).
>>
>> Some code follows -- perhaps someone could run it and tell me if, for
>> starters, they get the same as me.  (See http://i49.tinypic.com/2ijqatl.jpg
>> for superellipse() with s=0.3).
>>
>>
>> Best,
>>
>>
>> James
>>
>>
>>
>> % The following is a superellipse function at
>> <http://lists.foundry.supelec.fr/pipermail/metapost-commits/2008-June/000340.html>;
>> % I think it's the superellipse function in my copy of METAPOST; it at
>> least has the same behaviour.
>> % It seems to calculate the vertices correctly, but not the way they join
>> (try changing all ... to --).
>> %
>> %def superellipse(expr r,t,l,b,s)=
>> %  r ... (s[xpart t,xpart r],s[ypart r,ypart t]){t-r} ...
>> %  t ... (s[xpart t,xpart l],s[ypart l,ypart t]){l-t} ...
>> %  l ... (s[xpart b,xpart l],s[ypart l,ypart b]){b-l} ...
>> %  b ... (s[xpart b,xpart r],s[ypart r,ypart b]){r-b} ... cycle
>> %enddef;
>>
>> def supertest expr s =
>>   superellipse(
>>     ( 100, 50  ),
>>     ( 50,  100 ),
>>     ( 0,   50  ),
>>     ( 50,  0   ),
>>     s
>>     );
>> enddef;
>>
>> % These >0 supernesses are fine, I think ...
>>
>> beginfig(0);
>>   draw supertest 2;
>> endfig;
>>
>> beginfig(1);
>>   draw supertest 1.01;
>> endfig;
>>
>> % The following, 0.5>=superness<=1,
>> % are from visual reference definitely right
>>
>> beginfig(2);
>>   draw supertest 1;
>> endfig;
>>
>> beginfig(3);
>>   draw supertest 0.99;
>> endfig;
>>
>> beginfig(4);
>>   draw supertest 0.7;
>> endfig;
>>
>> beginfig(5);
>>   draw supertest 0.51;
>> endfig;
>>
>> beginfig(6);
>>   draw supertest 0.5;
>> endfig;
>>
>> % Now, for <0.5,
>> % things get problematic --
>> % the points in the shape generated by s=0.5
>> % should stay 'pointy'
>>
>> beginfig(7);
>>   draw supertest 0.49;
>> endfig;
>>
>> beginfig(8);
>>   draw supertest 0.3;
>> endfig;
>>
>> beginfig(9);
>>   draw supertest 0.01;
>> endfig;
>>
>> beginfig(10);
>>   draw supertest 0;
>> endfig;
>>
>> end;
>>
>
>
> ___________________________________________________________________________________
> If your question is of interest to others as well, please add an entry to
> the Wiki!
>
> maillist : ntg-context@ntg.nl /
> http://www.ntg.nl/mailman/listinfo/ntg-context
> webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
> archive  : http://foundry.supelec.fr/projects/contextrev/
> wiki     : http://contextgarden.net
> ___________________________________________________________________________________
>
>
___________________________________________________________________________________
If your question is of interest to others as well, please add an entry to the Wiki!

maillist : ntg-context@ntg.nl / http://www.ntg.nl/mailman/listinfo/ntg-context
webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
archive  : http://foundry.supelec.fr/projects/contextrev/
wiki     : http://contextgarden.net
___________________________________________________________________________________

Re: METAPOST's superellipse with superness<0.5 does not give a superellipse

[James Fisher]

[-- Attachment #1.1: Type: text/plain, Size: 7265 bytes --]

I've come up with a crude function that's doing more like what I want.  I
have two problems with it:

1. The most important: I need to differentiate the equations that generate a
superellipse, in order to find the tangent at the defined vertices.  I have
failed to do this and so use a crude arbitrary power function.
2. Less important: what is the equivalent in METAFONT of the 'return'
keyword?  I just want superellipse() to return the shape, rather than draw
it.

James

def superellipse(expr r,t,l,b,s)=
  pair tr, tl, bl, br;
  tr = (s[xpart t,xpart r],s[ypart r,ypart t]);
  tl = (s[xpart t,xpart l],s[ypart l,ypart t]);
  bl = (s[xpart b,xpart l],s[ypart l,ypart b]);
  br = (s[xpart b,xpart r],s[ypart r,ypart b]);

  numeric theta;


  if s > 0.5:
    % Behave as in the normal superellipse function
    theta = 0;
  else:
    % This is a crude mockup of the kind of function that is required
    % to generate shapes with s<0.5 (apparently called astroids).
    % This satisfies:
    %
    %   s = 0.5,  theta = 0.5
    %   s = 0,    theta = 90
    %
    % But to find the actual function,
    % we need to differentiate, at an endpoint,
    % the equation that would produce one quadrant of the shape.

    theta = 90 - (s*s*s*7.11378661);
  fi

  fill  r{dir(90+theta)} ... tr{t-r} ... {dir(180-theta)}t &
        t{dir(180+theta)} ... tl{l-t} ... {dir(270-theta)}l &
        l{dir(270+theta)} ... bl{b-l} ... {dir(-theta)}b &
        b{dir(theta)} ... br{r-b} ... {dir(90-theta)}r &
        cycle;
enddef;

beginfig(0);
  superellipse(
    ( 100, 50  ),
    ( 50,  100 ),
    ( 0,   50  ),
    ( 50,  0   ),
    0.6
    );
endfig;

end;


On Mon, Mar 1, 2010 at 4:20 PM, Rory Molinari <quokka@gmail.com> wrote:

> James's explanation appears to be right.
>
> On p 126 of the METAFONTbook Knuth says that the "superness should be
> between 0.5 (when you get a diamond) and 1.0 (when you get a square)".
>
> Exercise 14.6 asks the reader to "Try superellimpse with superness
> values less than 0.5 or greater than 1.0; explain why you get weird
> shapes in such cases."  The answer is "There are inflection points,
> because there are no bounding triangles for the '...' operations in
> the superellipse macro ... unless 0.5 \leq s \leq 1."
>
> Cheers,
> Rory
>
> On Mon, Mar 1, 2010 at 8:04 AM, James Fisher <jameshfisher@gmail.com>
> wrote:
> > I should say that the vertices of the superellipse are calculated
> > correctly.  The problem, it seems, is that for the vertices at right,
> top,
> > left, and bottom, the angles of entry and exit need to be explicitly
> > defined, rather than just relying on the '...' which coincidentally works
> > for s>=0.5.
> >
> > I should say at this point that I am no maths whiz.  But, sticking with
> the
> > 'right ... topright ... top' line, the angle calculation needs to
> satisfy,
> > for the exit angle of the first vertex:
> >
> > For s=0, angle = 180 degrees (vector to the left)
> > For s = 0.5, angle = 135 degrees (45 degrees to the top left, producing a
> > straight line to create the diamond shape)
> > For s>0.5, angle = 90 degrees (vector vertically upwards)
> >
> > Suffice to say that I don't know how to produce that elegantly.
> >
> >
> >
> > James
> >
> >
> > On Mon, Mar 1, 2010 at 3:54 PM, James Fisher <jameshfisher@gmail.com>
> wrote:
> >>
> >> Hi again,
> >>
> >>
> >> Another METAPOST problem.  For the sake of curiosity, I've been looking
> at
> >> and playing with the superellipse() function in plain METAPOST.  This is
> all
> >> fine and dandy until I try values of 'superness' less than 0.5, in which
> >> case it generates shapes that are seemingly not superellipses.  At
> s=0.5,
> >> the function generates a diamond shape -- which, AFAIK, is correct.
> >> However, s<0.5, the points of the diamond immediately turn to curves.
> (My
> >> knowledge of superellipses here is just from
> >> http://en.wikipedia.org/wiki/Superellipse -- try the image at
> >> http://en.wikipedia.org/wiki/File:Lame_anima.gif to see how I expect
> the
> >> shape to change with varying values of superness).
> >>
> >> Some code follows -- perhaps someone could run it and tell me if, for
> >> starters, they get the same as me.  (See
> http://i49.tinypic.com/2ijqatl.jpg
> >> for superellipse() with s=0.3).
> >>
> >>
> >> Best,
> >>
> >>
> >> James
> >>
> >>
> >>
> >> % The following is a superellipse function at
> >> <
> http://lists.foundry.supelec.fr/pipermail/metapost-commits/2008-June/000340.html
> >;
> >> % I think it's the superellipse function in my copy of METAPOST; it at
> >> least has the same behaviour.
> >> % It seems to calculate the vertices correctly, but not the way they
> join
> >> (try changing all ... to --).
> >> %
> >> %def superellipse(expr r,t,l,b,s)=
> >> %  r ... (s[xpart t,xpart r],s[ypart r,ypart t]){t-r} ...
> >> %  t ... (s[xpart t,xpart l],s[ypart l,ypart t]){l-t} ...
> >> %  l ... (s[xpart b,xpart l],s[ypart l,ypart b]){b-l} ...
> >> %  b ... (s[xpart b,xpart r],s[ypart r,ypart b]){r-b} ... cycle
> >> %enddef;
> >>
> >> def supertest expr s =
> >>   superellipse(
> >>     ( 100, 50  ),
> >>     ( 50,  100 ),
> >>     ( 0,   50  ),
> >>     ( 50,  0   ),
> >>     s
> >>     );
> >> enddef;
> >>
> >> % These >0 supernesses are fine, I think ...
> >>
> >> beginfig(0);
> >>   draw supertest 2;
> >> endfig;
> >>
> >> beginfig(1);
> >>   draw supertest 1.01;
> >> endfig;
> >>
> >> % The following, 0.5>=superness<=1,
> >> % are from visual reference definitely right
> >>
> >> beginfig(2);
> >>   draw supertest 1;
> >> endfig;
> >>
> >> beginfig(3);
> >>   draw supertest 0.99;
> >> endfig;
> >>
> >> beginfig(4);
> >>   draw supertest 0.7;
> >> endfig;
> >>
> >> beginfig(5);
> >>   draw supertest 0.51;
> >> endfig;
> >>
> >> beginfig(6);
> >>   draw supertest 0.5;
> >> endfig;
> >>
> >> % Now, for <0.5,
> >> % things get problematic --
> >> % the points in the shape generated by s=0.5
> >> % should stay 'pointy'
> >>
> >> beginfig(7);
> >>   draw supertest 0.49;
> >> endfig;
> >>
> >> beginfig(8);
> >>   draw supertest 0.3;
> >> endfig;
> >>
> >> beginfig(9);
> >>   draw supertest 0.01;
> >> endfig;
> >>
> >> beginfig(10);
> >>   draw supertest 0;
> >> endfig;
> >>
> >> end;
> >>
> >
> >
> >
> ___________________________________________________________________________________
> > If your question is of interest to others as well, please add an entry to
> > the Wiki!
> >
> > maillist : ntg-context@ntg.nl /
> > http://www.ntg.nl/mailman/listinfo/ntg-context
> > webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
> > archive  : http://foundry.supelec.fr/projects/contextrev/
> > wiki     : http://contextgarden.net
> >
> ___________________________________________________________________________________
> >
> >
>
> ___________________________________________________________________________________
> If your question is of interest to others as well, please add an entry to
> the Wiki!
>
> maillist : ntg-context@ntg.nl /
> http://www.ntg.nl/mailman/listinfo/ntg-context
> webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
> archive  : http://foundry.supelec.fr/projects/contextrev/
> wiki     : http://contextgarden.net
>
> ___________________________________________________________________________________
>

[-- Attachment #1.2: Type: text/html, Size: 10391 bytes --]

[-- Attachment #2: Type: text/plain, Size: 486 bytes --]

___________________________________________________________________________________
If your question is of interest to others as well, please add an entry to the Wiki!

maillist : ntg-context@ntg.nl / http://www.ntg.nl/mailman/listinfo/ntg-context
webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
archive  : http://foundry.supelec.fr/projects/contextrev/
wiki     : http://contextgarden.net
___________________________________________________________________________________

Re: METAPOST's superellipse with superness<0.5 does not give a superellipse

[Rory Molinari]

For 2: I think Metafont makes more sense if you don't think of a macro
as a function that does some work in its own context and returns a
value, but as something that expands textually in place.  So there
isn't any concept of "return" as there aren't separate stack frames to
return "from" or "to".

So you could try something like this (completely untested, and I am a
Metafont beginner.  This is from memory, so check the book for the
right syntax, especially for the semicolons.)

def myPath(<args>) =
  begingroup
    save tr, tl, bl, br;
    pair tr, tl, bl, br;
    <some equations involving tr, tl, bl, br, and the args>;

    <a path expression>;
  endgroup;
enddef;

path aPath;
aPath := myPath(<args>);
fill aPath;

The construction

begingroup
  <statements>;
  <expression>;
endgroup;

lets you do some work in <statements>, and then give an <expression>.
The value of the group is the value of the <expression>.  The value
isn't really "returned"; it just appears wherever a "call" to myPath
appears.  Then the assignment to aPath is expanded by the interpreter
as

aPath := begingroup <etc> endgroup;

Cheers,
Rory

On Mon, Mar 1, 2010 at 9:57 AM, James Fisher <jameshfisher@gmail.com> wrote:
> I've come up with a crude function that's doing more like what I want.  I
> have two problems with it:
>
> 1. The most important: I need to differentiate the equations that generate a
> superellipse, in order to find the tangent at the defined vertices.  I have
> failed to do this and so use a crude arbitrary power function.
> 2. Less important: what is the equivalent in METAFONT of the 'return'
> keyword?  I just want superellipse() to return the shape, rather than draw
> it.
>
> James
>
> def superellipse(expr r,t,l,b,s)=
>   pair tr, tl, bl, br;
>   tr = (s[xpart t,xpart r],s[ypart r,ypart t]);
>   tl = (s[xpart t,xpart l],s[ypart l,ypart t]);
>   bl = (s[xpart b,xpart l],s[ypart l,ypart b]);
>   br = (s[xpart b,xpart r],s[ypart r,ypart b]);
>
>   numeric theta;
>
>
>   if s > 0.5:
>     % Behave as in the normal superellipse function
>     theta = 0;
>   else:
>     % This is a crude mockup of the kind of function that is required
>     % to generate shapes with s<0.5 (apparently called astroids).
>     % This satisfies:
>     %
>     %   s = 0.5,  theta = 0.5
>     %   s = 0,    theta = 90
>     %
>     % But to find the actual function,
>     % we need to differentiate, at an endpoint,
>     % the equation that would produce one quadrant of the shape.
>
>     theta = 90 - (s*s*s*7.11378661);
>   fi
>
>   fill  r{dir(90+theta)} ... tr{t-r} ... {dir(180-theta)}t &
>         t{dir(180+theta)} ... tl{l-t} ... {dir(270-theta)}l &
>         l{dir(270+theta)} ... bl{b-l} ... {dir(-theta)}b &
>         b{dir(theta)} ... br{r-b} ... {dir(90-theta)}r &
>         cycle;
> enddef;
>
> beginfig(0);
>   superellipse(
>     ( 100, 50  ),
>     ( 50,  100 ),
>     ( 0,   50  ),
>     ( 50,  0   ),
>     0.6
>     );
> endfig;
>
> end;
>
>
> On Mon, Mar 1, 2010 at 4:20 PM, Rory Molinari <quokka@gmail.com> wrote:
>>
>> James's explanation appears to be right.
>>
>> On p 126 of the METAFONTbook Knuth says that the "superness should be
>> between 0.5 (when you get a diamond) and 1.0 (when you get a square)".
>>
>> Exercise 14.6 asks the reader to "Try superellimpse with superness
>> values less than 0.5 or greater than 1.0; explain why you get weird
>> shapes in such cases."  The answer is "There are inflection points,
>> because there are no bounding triangles for the '...' operations in
>> the superellipse macro ... unless 0.5 \leq s \leq 1."
>>
>> Cheers,
>> Rory
>>
>> On Mon, Mar 1, 2010 at 8:04 AM, James Fisher <jameshfisher@gmail.com>
>> wrote:
>> > I should say that the vertices of the superellipse are calculated
>> > correctly.  The problem, it seems, is that for the vertices at right,
>> > top,
>> > left, and bottom, the angles of entry and exit need to be explicitly
>> > defined, rather than just relying on the '...' which coincidentally
>> > works
>> > for s>=0.5.
>> >
>> > I should say at this point that I am no maths whiz.  But, sticking with
>> > the
>> > 'right ... topright ... top' line, the angle calculation needs to
>> > satisfy,
>> > for the exit angle of the first vertex:
>> >
>> > For s=0, angle = 180 degrees (vector to the left)
>> > For s = 0.5, angle = 135 degrees (45 degrees to the top left, producing
>> > a
>> > straight line to create the diamond shape)
>> > For s>0.5, angle = 90 degrees (vector vertically upwards)
>> >
>> > Suffice to say that I don't know how to produce that elegantly.
>> >
>> >
>> >
>> > James
>> >
>> >
>> > On Mon, Mar 1, 2010 at 3:54 PM, James Fisher <jameshfisher@gmail.com>
>> > wrote:
>> >>
>> >> Hi again,
>> >>
>> >>
>> >> Another METAPOST problem.  For the sake of curiosity, I've been looking
>> >> at
>> >> and playing with the superellipse() function in plain METAPOST.  This
>> >> is all
>> >> fine and dandy until I try values of 'superness' less than 0.5, in
>> >> which
>> >> case it generates shapes that are seemingly not superellipses.  At
>> >> s=0.5,
>> >> the function generates a diamond shape -- which, AFAIK, is correct.
>> >> However, s<0.5, the points of the diamond immediately turn to curves.
>> >> (My
>> >> knowledge of superellipses here is just from
>> >> http://en.wikipedia.org/wiki/Superellipse -- try the image at
>> >> http://en.wikipedia.org/wiki/File:Lame_anima.gif to see how I expect
>> >> the
>> >> shape to change with varying values of superness).
>> >>
>> >> Some code follows -- perhaps someone could run it and tell me if, for
>> >> starters, they get the same as me.  (See
>> >> http://i49.tinypic.com/2ijqatl.jpg
>> >> for superellipse() with s=0.3).
>> >>
>> >>
>> >> Best,
>> >>
>> >>
>> >> James
>> >>
>> >>
>> >>
>> >> % The following is a superellipse function at
>> >>
>> >> <http://lists.foundry.supelec.fr/pipermail/metapost-commits/2008-June/000340.html>;
>> >> % I think it's the superellipse function in my copy of METAPOST; it at
>> >> least has the same behaviour.
>> >> % It seems to calculate the vertices correctly, but not the way they
>> >> join
>> >> (try changing all ... to --).
>> >> %
>> >> %def superellipse(expr r,t,l,b,s)=
>> >> %  r ... (s[xpart t,xpart r],s[ypart r,ypart t]){t-r} ...
>> >> %  t ... (s[xpart t,xpart l],s[ypart l,ypart t]){l-t} ...
>> >> %  l ... (s[xpart b,xpart l],s[ypart l,ypart b]){b-l} ...
>> >> %  b ... (s[xpart b,xpart r],s[ypart r,ypart b]){r-b} ... cycle
>> >> %enddef;
>> >>
>> >> def supertest expr s =
>> >>   superellipse(
>> >>     ( 100, 50  ),
>> >>     ( 50,  100 ),
>> >>     ( 0,   50  ),
>> >>     ( 50,  0   ),
>> >>     s
>> >>     );
>> >> enddef;
>> >>
>> >> % These >0 supernesses are fine, I think ...
>> >>
>> >> beginfig(0);
>> >>   draw supertest 2;
>> >> endfig;
>> >>
>> >> beginfig(1);
>> >>   draw supertest 1.01;
>> >> endfig;
>> >>
>> >> % The following, 0.5>=superness<=1,
>> >> % are from visual reference definitely right
>> >>
>> >> beginfig(2);
>> >>   draw supertest 1;
>> >> endfig;
>> >>
>> >> beginfig(3);
>> >>   draw supertest 0.99;
>> >> endfig;
>> >>
>> >> beginfig(4);
>> >>   draw supertest 0.7;
>> >> endfig;
>> >>
>> >> beginfig(5);
>> >>   draw supertest 0.51;
>> >> endfig;
>> >>
>> >> beginfig(6);
>> >>   draw supertest 0.5;
>> >> endfig;
>> >>
>> >> % Now, for <0.5,
>> >> % things get problematic --
>> >> % the points in the shape generated by s=0.5
>> >> % should stay 'pointy'
>> >>
>> >> beginfig(7);
>> >>   draw supertest 0.49;
>> >> endfig;
>> >>
>> >> beginfig(8);
>> >>   draw supertest 0.3;
>> >> endfig;
>> >>
>> >> beginfig(9);
>> >>   draw supertest 0.01;
>> >> endfig;
>> >>
>> >> beginfig(10);
>> >>   draw supertest 0;
>> >> endfig;
>> >>
>> >> end;
>> >>
>> >
>> >
>> >
>> > ___________________________________________________________________________________
>> > If your question is of interest to others as well, please add an entry
>> > to
>> > the Wiki!
>> >
>> > maillist : ntg-context@ntg.nl /
>> > http://www.ntg.nl/mailman/listinfo/ntg-context
>> > webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
>> > archive  : http://foundry.supelec.fr/projects/contextrev/
>> > wiki     : http://contextgarden.net
>> >
>> > ___________________________________________________________________________________
>> >
>> >
>>
>> ___________________________________________________________________________________
>> If your question is of interest to others as well, please add an entry to
>> the Wiki!
>>
>> maillist : ntg-context@ntg.nl /
>> http://www.ntg.nl/mailman/listinfo/ntg-context
>> webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
>> archive  : http://foundry.supelec.fr/projects/contextrev/
>> wiki     : http://contextgarden.net
>>
>> ___________________________________________________________________________________
>
>
> ___________________________________________________________________________________
> If your question is of interest to others as well, please add an entry to
> the Wiki!
>
> maillist : ntg-context@ntg.nl /
> http://www.ntg.nl/mailman/listinfo/ntg-context
> webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
> archive  : http://foundry.supelec.fr/projects/contextrev/
> wiki     : http://contextgarden.net
> ___________________________________________________________________________________
>
>
___________________________________________________________________________________
If your question is of interest to others as well, please add an entry to the Wiki!

maillist : ntg-context@ntg.nl / http://www.ntg.nl/mailman/listinfo/ntg-context
webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
archive  : http://foundry.supelec.fr/projects/contextrev/
wiki     : http://contextgarden.net
___________________________________________________________________________________

Re: METAPOST's superellipse with superness<0.5 does not give a superellipse

[James Fisher]

[-- Attachment #1.1: Type: text/plain, Size: 11263 bytes --]

Thanks, Rory.  In hind-sight I guess my presumptions look pretty silly.  The
reason I'm making mistakes like this is that I can't find any introductory
text along the lines of 'TeX for programmers'; i.e. people coming from your
typical modern imperative languages, who will (after looking at TeX code and
concluding that it *is* a programming language) expect things like
variables, flow control, classes, and the kind of and syntax of library
documentation that comes with these languages.  Perhaps the texts I've seen
should have included some content taken from function vs.
macro<http://www.google.co.uk/search?hl=en&source=hp&q=macro%20vs%20function&meta=&aq=f&oq=>discussions
elsewhere.

Best

James

On Mon, Mar 1, 2010 at 7:15 PM, Rory Molinari <quokka@gmail.com> wrote:

> For 2: I think Metafont makes more sense if you don't think of a macro
> as a function that does some work in its own context and returns a
> value, but as something that expands textually in place.  So there
> isn't any concept of "return" as there aren't separate stack frames to
> return "from" or "to".
>
> So you could try something like this (completely untested, and I am a
> Metafont beginner.  This is from memory, so check the book for the
> right syntax, especially for the semicolons.)
>
> def myPath(<args>) =
>  begingroup
>    save tr, tl, bl, br;
>     pair tr, tl, bl, br;
>     <some equations involving tr, tl, bl, br, and the args>;
>
>    <a path expression>;
>  endgroup;
> enddef;
>
> path aPath;
> aPath := myPath(<args>);
> fill aPath;
>
> The construction
>
> begingroup
>  <statements>;
>  <expression>;
> endgroup;
>
> lets you do some work in <statements>, and then give an <expression>.
> The value of the group is the value of the <expression>.  The value
> isn't really "returned"; it just appears wherever a "call" to myPath
> appears.  Then the assignment to aPath is expanded by the interpreter
> as
>
> aPath := begingroup <etc> endgroup;
>
> Cheers,
> Rory
>
> On Mon, Mar 1, 2010 at 9:57 AM, James Fisher <jameshfisher@gmail.com>
> wrote:
> > I've come up with a crude function that's doing more like what I want.  I
> > have two problems with it:
> >
> > 1. The most important: I need to differentiate the equations that
> generate a
> > superellipse, in order to find the tangent at the defined vertices.  I
> have
> > failed to do this and so use a crude arbitrary power function.
> > 2. Less important: what is the equivalent in METAFONT of the 'return'
> > keyword?  I just want superellipse() to return the shape, rather than
> draw
> > it.
> >
> > James
> >
> > def superellipse(expr r,t,l,b,s)=
> >   pair tr, tl, bl, br;
> >   tr = (s[xpart t,xpart r],s[ypart r,ypart t]);
> >   tl = (s[xpart t,xpart l],s[ypart l,ypart t]);
> >   bl = (s[xpart b,xpart l],s[ypart l,ypart b]);
> >   br = (s[xpart b,xpart r],s[ypart r,ypart b]);
> >
> >   numeric theta;
> >
> >
> >   if s > 0.5:
> >     % Behave as in the normal superellipse function
> >     theta = 0;
> >   else:
> >     % This is a crude mockup of the kind of function that is required
> >     % to generate shapes with s<0.5 (apparently called astroids).
> >     % This satisfies:
> >     %
> >     %   s = 0.5,  theta = 0.5
> >     %   s = 0,    theta = 90
> >     %
> >     % But to find the actual function,
> >     % we need to differentiate, at an endpoint,
> >     % the equation that would produce one quadrant of the shape.
> >
> >     theta = 90 - (s*s*s*7.11378661);
> >   fi
> >
> >   fill  r{dir(90+theta)} ... tr{t-r} ... {dir(180-theta)}t &
> >         t{dir(180+theta)} ... tl{l-t} ... {dir(270-theta)}l &
> >         l{dir(270+theta)} ... bl{b-l} ... {dir(-theta)}b &
> >         b{dir(theta)} ... br{r-b} ... {dir(90-theta)}r &
> >         cycle;
> > enddef;
> >
> > beginfig(0);
> >   superellipse(
> >     ( 100, 50  ),
> >     ( 50,  100 ),
> >     ( 0,   50  ),
> >     ( 50,  0   ),
> >     0.6
> >     );
> > endfig;
> >
> > end;
> >
> >
> > On Mon, Mar 1, 2010 at 4:20 PM, Rory Molinari <quokka@gmail.com> wrote:
> >>
> >> James's explanation appears to be right.
> >>
> >> On p 126 of the METAFONTbook Knuth says that the "superness should be
> >> between 0.5 (when you get a diamond) and 1.0 (when you get a square)".
> >>
> >> Exercise 14.6 asks the reader to "Try superellimpse with superness
> >> values less than 0.5 or greater than 1.0; explain why you get weird
> >> shapes in such cases."  The answer is "There are inflection points,
> >> because there are no bounding triangles for the '...' operations in
> >> the superellipse macro ... unless 0.5 \leq s \leq 1."
> >>
> >> Cheers,
> >> Rory
> >>
> >> On Mon, Mar 1, 2010 at 8:04 AM, James Fisher <jameshfisher@gmail.com>
> >> wrote:
> >> > I should say that the vertices of the superellipse are calculated
> >> > correctly.  The problem, it seems, is that for the vertices at right,
> >> > top,
> >> > left, and bottom, the angles of entry and exit need to be explicitly
> >> > defined, rather than just relying on the '...' which coincidentally
> >> > works
> >> > for s>=0.5.
> >> >
> >> > I should say at this point that I am no maths whiz.  But, sticking
> with
> >> > the
> >> > 'right ... topright ... top' line, the angle calculation needs to
> >> > satisfy,
> >> > for the exit angle of the first vertex:
> >> >
> >> > For s=0, angle = 180 degrees (vector to the left)
> >> > For s = 0.5, angle = 135 degrees (45 degrees to the top left,
> producing
> >> > a
> >> > straight line to create the diamond shape)
> >> > For s>0.5, angle = 90 degrees (vector vertically upwards)
> >> >
> >> > Suffice to say that I don't know how to produce that elegantly.
> >> >
> >> >
> >> >
> >> > James
> >> >
> >> >
> >> > On Mon, Mar 1, 2010 at 3:54 PM, James Fisher <jameshfisher@gmail.com>
> >> > wrote:
> >> >>
> >> >> Hi again,
> >> >>
> >> >>
> >> >> Another METAPOST problem.  For the sake of curiosity, I've been
> looking
> >> >> at
> >> >> and playing with the superellipse() function in plain METAPOST.  This
> >> >> is all
> >> >> fine and dandy until I try values of 'superness' less than 0.5, in
> >> >> which
> >> >> case it generates shapes that are seemingly not superellipses.  At
> >> >> s=0.5,
> >> >> the function generates a diamond shape -- which, AFAIK, is correct.
> >> >> However, s<0.5, the points of the diamond immediately turn to curves.
> >> >> (My
> >> >> knowledge of superellipses here is just from
> >> >> http://en.wikipedia.org/wiki/Superellipse -- try the image at
> >> >> http://en.wikipedia.org/wiki/File:Lame_anima.gif to see how I expect
> >> >> the
> >> >> shape to change with varying values of superness).
> >> >>
> >> >> Some code follows -- perhaps someone could run it and tell me if, for
> >> >> starters, they get the same as me.  (See
> >> >> http://i49.tinypic.com/2ijqatl.jpg
> >> >> for superellipse() with s=0.3).
> >> >>
> >> >>
> >> >> Best,
> >> >>
> >> >>
> >> >> James
> >> >>
> >> >>
> >> >>
> >> >> % The following is a superellipse function at
> >> >>
> >> >> <
> http://lists.foundry.supelec.fr/pipermail/metapost-commits/2008-June/000340.html
> >;
> >> >> % I think it's the superellipse function in my copy of METAPOST; it
> at
> >> >> least has the same behaviour.
> >> >> % It seems to calculate the vertices correctly, but not the way they
> >> >> join
> >> >> (try changing all ... to --).
> >> >> %
> >> >> %def superellipse(expr r,t,l,b,s)=
> >> >> %  r ... (s[xpart t,xpart r],s[ypart r,ypart t]){t-r} ...
> >> >> %  t ... (s[xpart t,xpart l],s[ypart l,ypart t]){l-t} ...
> >> >> %  l ... (s[xpart b,xpart l],s[ypart l,ypart b]){b-l} ...
> >> >> %  b ... (s[xpart b,xpart r],s[ypart r,ypart b]){r-b} ... cycle
> >> >> %enddef;
> >> >>
> >> >> def supertest expr s =
> >> >>   superellipse(
> >> >>     ( 100, 50  ),
> >> >>     ( 50,  100 ),
> >> >>     ( 0,   50  ),
> >> >>     ( 50,  0   ),
> >> >>     s
> >> >>     );
> >> >> enddef;
> >> >>
> >> >> % These >0 supernesses are fine, I think ...
> >> >>
> >> >> beginfig(0);
> >> >>   draw supertest 2;
> >> >> endfig;
> >> >>
> >> >> beginfig(1);
> >> >>   draw supertest 1.01;
> >> >> endfig;
> >> >>
> >> >> % The following, 0.5>=superness<=1,
> >> >> % are from visual reference definitely right
> >> >>
> >> >> beginfig(2);
> >> >>   draw supertest 1;
> >> >> endfig;
> >> >>
> >> >> beginfig(3);
> >> >>   draw supertest 0.99;
> >> >> endfig;
> >> >>
> >> >> beginfig(4);
> >> >>   draw supertest 0.7;
> >> >> endfig;
> >> >>
> >> >> beginfig(5);
> >> >>   draw supertest 0.51;
> >> >> endfig;
> >> >>
> >> >> beginfig(6);
> >> >>   draw supertest 0.5;
> >> >> endfig;
> >> >>
> >> >> % Now, for <0.5,
> >> >> % things get problematic --
> >> >> % the points in the shape generated by s=0.5
> >> >> % should stay 'pointy'
> >> >>
> >> >> beginfig(7);
> >> >>   draw supertest 0.49;
> >> >> endfig;
> >> >>
> >> >> beginfig(8);
> >> >>   draw supertest 0.3;
> >> >> endfig;
> >> >>
> >> >> beginfig(9);
> >> >>   draw supertest 0.01;
> >> >> endfig;
> >> >>
> >> >> beginfig(10);
> >> >>   draw supertest 0;
> >> >> endfig;
> >> >>
> >> >> end;
> >> >>
> >> >
> >> >
> >> >
> >> >
> ___________________________________________________________________________________
> >> > If your question is of interest to others as well, please add an entry
> >> > to
> >> > the Wiki!
> >> >
> >> > maillist : ntg-context@ntg.nl /
> >> > http://www.ntg.nl/mailman/listinfo/ntg-context
> >> > webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
> >> > archive  : http://foundry.supelec.fr/projects/contextrev/
> >> > wiki     : http://contextgarden.net
> >> >
> >> >
> ___________________________________________________________________________________
> >> >
> >> >
> >>
> >>
> ___________________________________________________________________________________
> >> If your question is of interest to others as well, please add an entry
> to
> >> the Wiki!
> >>
> >> maillist : ntg-context@ntg.nl /
> >> http://www.ntg.nl/mailman/listinfo/ntg-context
> >> webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
> >> archive  : http://foundry.supelec.fr/projects/contextrev/
> >> wiki     : http://contextgarden.net
> >>
> >>
> ___________________________________________________________________________________
> >
> >
> >
> ___________________________________________________________________________________
> > If your question is of interest to others as well, please add an entry to
> > the Wiki!
> >
> > maillist : ntg-context@ntg.nl /
> > http://www.ntg.nl/mailman/listinfo/ntg-context
> > webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
> > archive  : http://foundry.supelec.fr/projects/contextrev/
> > wiki     : http://contextgarden.net
> >
> ___________________________________________________________________________________
> >
> >
>
> ___________________________________________________________________________________
> If your question is of interest to others as well, please add an entry to
> the Wiki!
>
> maillist : ntg-context@ntg.nl /
> http://www.ntg.nl/mailman/listinfo/ntg-context
> webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
> archive  : http://foundry.supelec.fr/projects/contextrev/
> wiki     : http://contextgarden.net
>
> ___________________________________________________________________________________
>

[-- Attachment #1.2: Type: text/html, Size: 17033 bytes --]

[-- Attachment #2: Type: text/plain, Size: 486 bytes --]

___________________________________________________________________________________
If your question is of interest to others as well, please add an entry to the Wiki!

maillist : ntg-context@ntg.nl / http://www.ntg.nl/mailman/listinfo/ntg-context
webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
archive  : http://foundry.supelec.fr/projects/contextrev/
wiki     : http://contextgarden.net
___________________________________________________________________________________