## Hypocycloid and Lattice points

### Problem 450

Published on Sunday, 15th December 2013, 07:00 am; Solved by 123; Difficulty rating: 100%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 $2r < R$.

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`(10^{3}) = 583108600.

Find `T`(10^{6}).