const r3 = 1.7320508075688772;
const hr3 = 0.8660254037844386;
const ident = [1, 0, 0, 0, 1, 0];

function pt( x, y )
{
	return { x : x, y : y };
}

function hexPt( x, y )
{
	return pt( x + 0.5*y, hr3*y );
}

// Affine matrix inverse
function inv( T ) {
	const det = T[0]*T[4] - T[1]*T[3];
	return [T[4]/det, -T[1]/det, (T[1]*T[5]-T[2]*T[4])/det,
		-T[3]/det, T[0]/det, (T[2]*T[3]-T[0]*T[5])/det];
};

// Affine matrix multiply
function mul( A, B )
{
	return [A[0]*B[0] + A[1]*B[3], 
		A[0]*B[1] + A[1]*B[4],
		A[0]*B[2] + A[1]*B[5] + A[2],

		A[3]*B[0] + A[4]*B[3], 
		A[3]*B[1] + A[4]*B[4],
		A[3]*B[2] + A[4]*B[5] + A[5]];
}

function padd( p, q )
{
	return { x : p.x + q.x, y : p.y + q.y };
}

function psub( p, q )
{
	return { x : p.x - q.x, y : p.y - q.y };
}

// Rotation matrix
function trot( ang )
{
	const c = cos( ang );
	const s = sin( ang );
	return [c, -s, 0, s, c, 0];
}

// Translation matrix
function ttrans( tx, ty )
{
	return [1, 0, tx, 0, 1, ty];
}

function rotAbout( p, ang )
{
	return mul( ttrans( p.x, p.y ), 
		mul( trot( ang ), ttrans( -p.x, -p.y ) ) );
}

// Matrix * point
function transPt( M, P )
{
	return pt(M[0]*P.x + M[1]*P.y + M[2], M[3]*P.x + M[4]*P.y + M[5]);
}

// Match unit interval to line segment p->q
function matchSeg( p, q )
{
	return [q.x-p.x, p.y-q.y, p.x,  q.y-p.y, q.x-p.x, p.y];
};

// Match line segment p1->q1 to line segment p2->q2
function matchTwo( p1, q1, p2, q2 )
{
	return mul( matchSeg( p2, q2 ), inv( matchSeg( p1, q1 ) ) );
};

// Intersect two lines defined by segments p1->q1 and p2->q2
function intersect( p1, q1, p2, q2 )
{
    const d = (q2.y - p2.y) * (q1.x - p1.x) - (q2.x - p2.x) * (q1.y - p1.y);
    const uA = ((q2.x - p2.x) * (p1.y - p2.y) - (q2.y - p2.y) * (p1.x - p2.x)) / d;
    // const uB = ((q1.x - p1.x) * (p1.y - p2.y) - (q1.y - p1.y) * (p1.x - p2.x)) / d;

    return pt( p1.x + uA * (q1.x - p1.x), p1.y + uA * (q1.y - p1.y) );
}

const hat_outline = [
    hexPt(0, 0), hexPt(-1,-1), hexPt(0,-2), hexPt(2,-2),
    hexPt(2,-1), hexPt(4,-2), hexPt(5,-1), hexPt(4, 0),
    hexPt(3, 0), hexPt(2, 2), hexPt(0, 3), hexPt(0, 2),
    hexPt(-1, 2) ];