A hypocycloid is the curve drawn by a point on a small circle
rolling inside a larger circle. The parametric equations of a
hypocycloid centered at the origin, and starting at the right most
point is given by:
$$x(t) = (R - r)\cos(t) + r\cos\left( \frac{R-r}{r}t \right)$$
$$y(t) = (R - r)\sin(t) - r\sin\left( \frac{R - r}{r}t \right)$$
Where \(R\) is the radius of the large circle and \(r\) the radius
of the small circle.
Let \(C(R, r)\) be the set of distinct points with integer
coordinates on the hypocycloid with radius \(R\) and \(r\) and for
which there is a corresponding value of \(t\) such that
\(\sin(t)\) and \(\cos(t)\) are rational numbers.
Let \(\sum_{(x, y)\in C(R, r)} |x|+|y|\) be the sum of the
absolute values of the \(x\) and \(y\) coordinates of the points
in \(C(R, r)\).
Let \(T(N) = \sum_{R=3}^{N}\sum_{r=1}^{\left
\lfloor{\frac{R-1}{2}}\right \rfloor } S(R, r)\) be the sum of
\(S(R, r)\) for \(R\) and \(r\) positive integers, \(R\leq N\) and
\(2r < R.\)
You are given:
$$C(3, 1) = \{(3, 0), (-1, 2), (-1,0), (-1,-2)\}$$
$$C(2500, 1000) = \{(2500, 0), (772, 2376), (772, -2376),$$
$$(516, 1792), (516, -1792), (500, 0), (68, 504),$$
$$(68, -504), (-1356, 1088), (-1356, -1088),$$
$$(-1500, 1000), (-1500, -1000)\}$$
Note: \((-625, 0)\) is not an element of \(C(2500, 1000)\)
because \(\sin(t)\) is not a rational number for the
corresponding values of \(t\).
$$S(3, 1) = (|3|+|0|) + (|-1| + |2|) + (|-1| + |0|) + $$
$$(|-1| + |-2|) = 10$$
$$T(3) = 10; T(10) = 524 ;T(100) = 580442;$$
$$T\left(10^{3}\right) = 583108600.$$
Find \(T\left(10^{6}\right).\)
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