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 <quo...@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 <jameshfis...@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 <jameshfis...@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;
> >>
> >
> >
> >
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> >
> ___________________________________________________________________________________
> >
> >
>
> ___________________________________________________________________________________
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>
> maillist : ntg-context@ntg.nl /
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>
> ___________________________________________________________________________________
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___________________________________________________________________________________
If your question is of interest to others as well, please add an entry to the
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maillist : ntg-context@ntg.nl / http://www.ntg.nl/mailman/listinfo/ntg-context
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