---
id: 5900f53d1000cf542c510050
title: 'Problem 465: Polar polygons'
challengeType: 5
forumTopicId: 302140
dashedName: problem-465-polar-polygons
---

# --description--

The kernel of a polygon is defined by the set of points from which the entire polygon's boundary is visible. We define a polar polygon as a polygon for which the origin is strictly contained inside its kernel.

For this problem, a polygon can have collinear consecutive vertices. However, a polygon still cannot have self-intersection and cannot have zero area.

For example, only the first of the following is a polar polygon (the kernels of the second, third, and fourth do not strictly contain the origin, and the fifth does not have a kernel at all):

<img class="img-responsive center-block" alt="five example polygons" src="https://cdn.freecodecamp.org/curriculum/project-euler/polar-polygons.png" style="background-color: white; padding: 10px;">

Notice that the first polygon has three consecutive collinear vertices.

Let $P(n)$ be the number of polar polygons such that the vertices $(x, y)$ have integer coordinates whose absolute values are not greater than $n$.

Note that polygons should be counted as different if they have different set of edges, even if they enclose the same area. For example, the polygon with vertices [(0,0), (0,3), (1,1), (3,0)] is distinct from the polygon with vertices [(0,0), (0,3), (1,1), (3,0), (1,0)].

For example, $P(1) = 131$, $P(2) = 1\\,648\\,531$, $P(3) = 1\\,099\\,461\\,296\\,175$ and $P(343)\bmod 1\\,000\\,000\\,007 = 937\\,293\\,740$.

Find $P(7^{13})\bmod 1\\,000\\,000\\,007$.

# --hints--

`polarPolygons()` should return `585965659`.

```js
assert.strictEqual(polarPolygons(), 585965659);
```

# --seed--

## --seed-contents--

```js
function polarPolygons() {

  return true;
}

polarPolygons();
```

# --solutions--

```js
// solution required
```