%% Polynomial level curve of constant width
% Nick Trefethen, May 2022
%%
% (Chebfun example geom/ConstantWidth.m)
%%
% A _Mathematics Today_ column by Alan Champneys [2] describes
% a fascinating example that can be found in Wikipedia [4] and
% goes back to a paper by Rabinowitz [3]; see also [1].
% Here is a bivariate polynomial $p(x,y)$:
tic
r2 = @(x,y) x.^2 + y.^2; xy = @(x,y) x.^2 - 3*y.^2;
p = @(x,y) r2(x,y).^4 - 45*r2(x,y).^3 - 41283*r2(x,y).^2 + ...
7950960*r2(x,y) + 16*xy(x,y).^3 + 48*r2(x,y).*xy(x,y).^2 + ...
x.*xy(x,y).*(16*r2(x,y).^2 - 5544*r2(x,y) + 266382) - 373248000;
%%
% The result is that the zero set of $p$ has constant width in
% the complex plane! -- like the British 50p coin.
% Let's verify this in Chebfun. Here's the domain, computed
% with the Chebfun2 |roots| command:
pc = chebfun2(p,[-11 11 -11 11]);
r = roots(pc);
copper = [.722 .451 .20];
fill(real(r),imag(r),copper)
axis(12*[-1 1 -1 1]), axis square, grid on
%%
% We compute its width measured in 5 different directions, and
% they agree to 5 digits, which is not bad considering the size of
% the coefficients.
disp('theta/pi width')
for theta = pi*(0:4)/5;
a = exp(1i*theta);
width = max(real(a*r)) - min(real(a*r));
fprintf('%8.5f %12.8f\n',theta/pi,width)
end
%%
% The exact result should be 18, as can be verified by
% setting $y=0$, in which case the polynomial reduces to
p = @(x) x^8 + 16*x^7 + 19*x^6 - 5544*x^5 - 41283*x^4 + 266382*x^3 + 7950960*x^2 - 373248000;
%%
% We confirm that this is zero at $x=-8$ and $x=10$:
p(-8)
p(10)
%%
% Just for fun let's compute the perimeter of the coin, presumably
% also accurate to about 5 digits:
perimeter = norm(diff(r),1)
%%
time_for_this_example = toc
%%
% [1] M. Bardet and T. Bayen, On the degree of the polynomial
% defining a planar algebraic curves of constant width,
% arXiv:1312.4358v1, 2013.
%
% [2] A. Champneys, Westward Ho! Musing on mathematics and
% mechanics, _Mathematics Today_, April 2022, 56--59.
%
% [3] S. Rabinowitz, A polynomial curve of constant width,
% _Missouri Journal of Mathematical Sciences_ 9 (1997), 23--27.
%
% [4] Wikipedia, "Curve of constant width".