Here on wolfram, I found the polynomial form and parametric form of the famous Klein bottle.
But when I put the parametric form in Eq.(2)~(4) into the polynomial form in Eq.(1):
klein = @(x, y, z) (x .^ 2 + y .^ 2 + z .^ 2 + 2 * y - 1) .* ((x .^ 2 + y .^ 2 + z .^ 2 - 2 * y - 1) .^ 2 - 8 * z .^ 2) + 16 * x .* z .* (x .^ 2 + y .^ 2 + z .^ 2 - 2 * y - 1);
fx = @(u, v) cos(u) .* (cos(u / 2) .* (sqrt(2) + cos(v)) + (sin(u / 2) .* sin(v) .* cos(v)));
fy = @(u, v) sin(u) .* (cos(u / 2) .* (sqrt(2) + cos(v)) + (sin(u / 2) .* sin(v) .* cos(v)));
fz = @(u, v) -1 * sin(u / 2) .* (sqrt(2) + cos(v)) + cos(u / 2) .* sin(v) .* cos(v);
u = 0;
v = 0;
X = fx(u, v);
Y = fy(u, v);
Z = fz(u, v);
result = klein(X, Y, Z);
disp(num2str(result));
Something went wrong! The result is far from zero (112.5685 instead!)
Why? Is there anything wrong with the equations on the website?