function G = triangle_geometry(P)
% TRIANGLE_GEOMETRY Node polyline of the open triangle centerline.
%
% An equilateral triangle of side side_length_m in the x-y plane, apex up:
% base from V2 (bottom left) to V3 (bottom right) along y = 0, apex V1 above
% the origin. The rod is open at the bottom-left corner: gap_m of arclength
% is removed there, half from the base and half from the left leg, so the
% centerline runs A -> V3 -> V1 -> B counterclockwise, where A and B sit
% gap_m / 2 from V2 along their legs. Total rod length is 3 L - gap.
%
% The polyline is discretized into about n_elements equal-length elements,
% with the two closed corners landing exactly on nodes. Arclength fraction 0
% is the free end on the base, 1 the free end on the left leg; strike_pos and
% roll_pos2 are expressed in this coordinate.
%
% Returns G with:
%   nodes    2 x nn node coordinates (m)
%   sfrac    1 x nn arclength fraction of each node
%   totalLen rod arclength (m)
%   nn, ne   node count and per-leg element counts (1 x 3)

L = P.side_length_m;
g = min(P.gap_m, 0.8 * L);
V1 = [0; L * sqrt(3) / 2];
V2 = [-L / 2; 0];
V3 = [L / 2; 0];
e1 = (V3 - V2) / L;
e3 = (V2 - V1) / L;
A = V2 + e1 * (g / 2);
B = V2 - e3 * (g / 2);

starts = [A, V3, V1];
stops = [V3, V1, B];
lens = [L - g / 2, L, L - g / 2];
total = lens(1) + lens(2) + lens(3);

ne = zeros(1, 3);
for k = 1:3
  ne(k) = max(2, round(P.n_elements * lens(k) / total));
end
nn = ne(1) + ne(2) + ne(3) + 1;

nodes = zeros(2, nn);
sarc = zeros(1, nn);
nodes(:, 1) = A;
idx = 1;
acc = 0;
for k = 1:3
  for j = 1:ne(k)
    idx = idx + 1;
    frac = j / ne(k);
    nodes(:, idx) = starts(:, k) + (stops(:, k) - starts(:, k)) * frac;
    sarc(idx) = acc + lens(k) * frac;
  end
  acc = acc + lens(k);
end

G.nodes = nodes;
G.sfrac = sarc / total;
G.totalLen = total;
G.nn = nn;
G.ne = ne;
end