I will find an xy dataset satisfying an implicit equation.

(x^2 + y^2)^3 + (15*x + 3.3*y)*(x^2 + y^2)^2 + (62.5*x^2 + 20*x*y - 8.37*y^2)*(x^2 + y^2) - 17*x^3 + 11*x^2*y + 17*x*y^2 - 11*y^3 + 432*x^2 - 24*x*y + 67*y^2 - 82*x + 400*y - 1037=0

I have the above equation and I want to obtain 200 xy points satisfying the equation. How can I find?

Answer 1

[Bruno Luong]

f = @(x,y)(x.^2 + y.^2).^3 + (15.*x + 3.3.*y).*(x.^2 + y.^2).^2 + (62.5.*x.^2 + 20.*x.*y - 8.37.*y.^2).*(x.^2 + y.^2) - 17.*x.^3 + 11.*x.^2.*y + 17.*x.*y.^2 - 11.*y.^3 + 432.*x.^2 - 24.*x.*y + 67.*y.^2 - 82.*x + 400.*y - 1037;

n = 8;
while true
    xg = linspace(-8,8,n+1);
    yg = linspace(-8,8,n+1);
    [Xg,Yg] = meshgrid(xg,yg);
    z=f(Xg,Yg);
    close all
    a = contour(Xg,Yg,z,[0 0]);
    if a(2,1) >= 200
        break
    end
    n = 2*n;
end

xy = a(:,2:end);
x = xy(1,:);
y = xy(2,:);

hold on
axis equal
h1=plot(x, y, '.b');

for k=1:size(xy,2)
    if ismember(x(k),xg)
        y(k) = fzero(@(y) f(x(k),y), y(k));
    else
        x(k) = fzero(@(x) f(x,y(k)), x(k));
    end
end
xy = [x(:), y(:)]

xy = 253×2

-2.4309 -6.7500 -2.5000 -6.7679 -2.6250 -6.7968 -2.7500 -6.8211 -2.8750 -6.8409 -3.0000 -6.8561 -3.1250 -6.8667 -3.2500 -6.8727 -3.3750 -6.8739 -3.5000 -6.8704

h2=plot(x, y, '.r');
legend([h1 h2],'approximation', 'accurate')

figure
plot(f(x,y)) % should be close to 0

[kadir can erbas]

This is exactly what I want. Thanks. :)