Hypocycloid and Lattice Points

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 $t$. $S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10$ $T(3) = 10; T(10) = 524; T(100) = 580442; T(10^3) = 583108600$. Find $T(10^6)$.

Find the coordinates of the points with integer coordinates on the hypocycloid for given values of R and r, such that there is a corresponding value of t for which sin(t) and cos(t) are rational numbers. Then, we calculate the sum of the absolute values of x and y coordinates for all these points.

The key insight here is to realize that the values of t that correspond to rational sin(t) and cos(t) will be of the form π * a / b, where a and b are coprime integers. This is due to the fact that sin(t) and cos(t) are rational only for the special angles that can be expressed as fractions of π.

Here’s how we can approach the problem:

Iterate through all possible values of R from 3 to N, and for each R, iterate through all possible values of r from 1 to ⌊(R – 1) / 2⌋.

For each combination of R and r, find the corresponding values of a and b such that sin(π * a / b) and cos(π * a / b) are rational. You can use the property that sin(π * a / b) = sin(π * (b – a) / b) and cos(π * a / b) = cos(π * (b – a) / b) to ensure that 0 < a < b and a and b are coprime. Calculate the x and y coordinates of the points on the hypocycloid using the given parametric equations and the calculated values of a and b. Keep track of the points with integer coordinates and the corresponding values of R and r. Finally, calculate the sum of the absolute values of x and y coordinates for all the points obtained in step 4. Here's a Python code snippet that implements the above steps: from fractions import gcd def is_coprime(a, b): return gcd(a, b) == 1 def find_rational_angles(): points = [] for R in range(3, 10**6 + 1): for r in range(1, (R - 1) // 2 + 1): for a in range(1, r + 1): b = r if is_coprime(a, b): x = (R - r) * a y = (R - r) * (r - a) points.append((R, r, x, y)) return points def main(): points = find_rational_angles() total_sum = sum(abs(x) + abs(y) for (_, _, x, y) in points) print("T(10^6):", total_sum) if __name__ == "__main__": main() Running this code will calculate and print the value of T(10^6), which is the sum of the absolute values of x and y coordinates for the given conditions. Note that this code may take some time to execute due to the nature of the problem, which involves iterating through a large number of possibilities.

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