![](https://secure.gravatar.com/avatar/0efa34edd38fb2fccbcc3909ddc2aaed.jpg?s=120&d=mm&r=g)
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
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
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....; % 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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