function [vx, vy] = make_region(convex)
%MAKE_REGION  Random 2D sampling region, returned as CCW vertices (VX, VY).
%   MAKE_REGION(true)  — a convex polygon (convex hull of random disk points).
%   MAKE_REGION(false) — a non-convex star polygon (star-shaped about the
%                        origin, so the origin is a valid interior start point).
if nargin < 1 || isempty(convex)
    convex = true;
end

if convex
    [vx, vy] = convex_region();
else
    [vx, vy] = star_region();
end

% CCW so the interior is to the left of each edge.
if signed_area(vx, vy) < 0
    vx = vx(end:-1:1);
    vy = vy(end:-1:1);
end
end

function [vx, vy] = convex_region()
% Convex hull of random points in a disk; anisotropic scaling for variety.
m = 12;
ang = 2 * pi * rand(m, 1);
r = sqrt(rand(m, 1));
px = 1.4 * r .* cos(ang);
py = 1.0 * r .* sin(ang);
k = convhull(px, py);
k = k(1:end-1);
vx = px(k);
vy = py(k);
end

function [vx, vy] = star_region()
% Spikes at sorted angles, alternating outer/inner radius. Small angle jitter
% keeps the angles ordered, so the polygon stays simple and star-shaped about
% the origin (lines through interior points can still leave and re-enter it).
spikes = 4 + randi(4);          % 5..8 spikes
k = 2 * spikes;
step = 2 * pi / k;
ang = (0:k - 1) * step + (rand(1, k) - 0.5) * step * 0.6;
r = zeros(1, k);
r(1:2:k) = 0.9 + 0.4 * rand(1, numel(1:2:k));    % outer
r(2:2:k) = 0.35 + 0.2 * rand(1, numel(2:2:k));   % inner
vx = (1.3 * r .* cos(ang)).';
vy = (1.0 * r .* sin(ang)).';
end

function A = signed_area(vx, vy)
%SIGNED_AREA  Shoelace area; positive when the vertices run counterclockwise.
n = numel(vx);
A = 0;
for i = 1:n
    j = mod(i, n) + 1;
    A = A + (vx(i) * vy(j) - vx(j) * vy(i));
end
A = A / 2;
end