---
id: 5900f52e1000cf542c510041
title: 'Problem 450: Hypocycloid and Lattice points'
challengeType: 5
forumTopicId: 302123
dashedName: problem-450-hypocycloid-and-lattice-points
---

# --description--

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).

# --hints--

`euler450()` should return 583333163984220900.

```js
assert.strictEqual(euler450(), 583333163984220900);
```

# --seed--

## --seed-contents--

```js
function euler450() {

  return true;
}

euler450();
```

# --solutions--

```js
// solution required
```