function [vx, vy] = make_region()
%MAKE_REGION  Random convex polygon = convex hull of random points in a disk.
%   [VX, VY] = MAKE_REGION() returns the counterclockwise vertices of a random
%   convex region. Anisotropic scaling makes it a bit more interesting than a
%   circle. The vertices are ordered CCW so the interior is to the left of each
%   edge (which hit_and_run relies on for its inward half-plane normals).
m = 12;
ang = 2 * pi * rand(m, 1);
r = sqrt(rand(m, 1));          % sqrt -> uniform over the disk
px = 1.4 * r .* cos(ang);
py = 1.0 * r .* sin(ang);
k = convhull(px, py);          % boundary indices, closed (last == first)
k = k(1:end-1);                % drop the repeated closing vertex
vx = px(k);
vy = py(k);
% Ensure counterclockwise.
if signed_area(vx, vy) < 0
    vx = vx(end:-1:1);
    vy = vy(end:-1:1);
end
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