% ex_hodge.m — Hodge decomposition of a tangential vector field.
%
% Any tangential vector field f on a surface splits into curl-free,
% divergence-free, and harmonic parts:
%
%     f = grad(u) + n x grad(v) + w.
%
% hodge() computes this by solving two Laplace-Beltrami problems. We use a
% modest torus here so the demo stays quick (the docs use a finer mesh).

mip load --install flatironinstitute/flatironinstitute/surfacefun

p = 8;
nu = 8;
nv = 16;
dom = surfacemesh.torus(p + 1, nu, nv);

% A smooth random tangential vector field.
rng(0)
bb = boundingbox(dom);
gx = randnfun3(10, bb);
gy = randnfun3(10, bb);
gz = randnfun3(10, bb);
g = cross([0 1 1], surfacefunv(@(x, y, z) gx(x, y, z), ...
                               @(x, y, z) gy(x, y, z), ...
                               @(x, y, z) gz(x, y, z), dom));
vn = normal(dom);
f = -cross(vn, vn, g);   % project g onto the tangent plane

% Compute the decomposition.
[u, v, w] = hodge(f);

figure(1);
quiver(f, 0.3, 4)
title('Tangential field f');

figure(2);
quiver(grad(u), 0.3, 4)
title('Curl-free part  grad(u)');

figure(3);
quiver(cross(vn, grad(v)), 0.3, 4)
title('Divergence-free part  n x grad(v)');

figure(4);
quiver(w, 0.3, 4)
title('Harmonic part  w');

% The harmonic remainder should be nearly divergence-free.
fprintf('|| div(w) || = %.3e\n', norm(div(w)));