# Problem 450: Hypocycloid and Lattice points

![graphic](img450.gif)

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(\\frac {R - r} r t)\$ \$y(t) =
(R - r) \\sin(t) - r \\sin(\\frac {R - r} r t)\$ 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 \$S(R, r) = \\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}\^{\\lfloor \\frac {R -
1} 2 \\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(103) = 583108600. Find T(106).