# Rotate a point around a circle

Given a point `(x, y)`

and an angle `θ`

(theta) measured in radians, the coordinates of the point `(x', y')`

found by rotating the original point counterclockwise around a circle centered on `(0, 0)`

are:

```
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
```

To rotate clockwise, flip the sign of `θ`

— if it’s positive then make it negative, and vice versa.

If the circle isn’t centered on `(0, 0)`

, find the delta between the center of the circle and the origin, subtract the delta from the point before rotating, then add that delta back to the new point.

Keep in mind that the initial coordinates must be used for both calculations. In particular, avoid this issue where one of the coordinates is modified, and then that modified coordinate is used to calculate the other coordinate:

```
const point = { x: 10, y: 10 };
const theta = Math.PI / 8;
// ❌ incorrect
point.x = point.x * Math.cos(theta) - point.y * Math.sin(theta);
point.y = point.x * Math.sin(theta) + point.y * Math.cos(theta); // point.x is different now!
```

Instead, either assign the result to a new variable, or — if the original object must be mutated — store the previous `x`

and `y`

and use them when calculating the result.

```
const point = { x: 10, y: 10 };
const theta = Math.PI / 8;
// ✅ correct if not mutating original object
const result = {
x: point.x * Math.cos(theta) - point.y * Math.sin(theta),
y: point.x * Math.sin(theta) + point.y * Math.cos(theta),
};
// ✅ correct if mutating original object
const { x, y } = point;
point.x = x * Math.cos(theta) - y * Math.sin(theta);
point.y = x * Math.sin(theta) + y * Math.cos(theta);
```