Math.atan2()

The Math.atan2() function returns the angle in the plane (in radians) between the positive x-axis and the ray from (0, 0) to the point (x, y), for Math.atan2(y, x).

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Syntax

Math.atan2(y, x)

Parameters

y

The y coordinate of the point.

x

The x coordinate of the point.

Return value

The angle in radians (between -π and π, inclusive) between the positive x-axis and the ray from (0, 0) to the point (x, y).

Description

The Math.atan2() method measures the counterclockwise angle θ, in radians, between the positive x-axis and the point (x, y). Note that the arguments to this function pass the y-coordinate first and the x-coordinate second.

A simple diagram showing the angle returned by atan2(y, x)

Math.atan2() is passed separate x and y arguments, while Math.atan() is passed the ratio of those two arguments. Math.atan2(y, x) differs from Math.atan(y / x) in the following cases:

x y Math.atan2(y, x) Math.atan(y / x)
Infinity Infinity π / 4 NaN
Infinity -Infinity -π / 4 NaN
-Infinity Infinity 3π / 4 NaN
-Infinity -Infinity -3π / 4 NaN
0 0 0 NaN
0 -0 -0 NaN
< 0 (including -0) 0 π 0
< 0 (including -0) -0 0
-Infinity > 0 π -0
-0 > 0 π / 2 -π / 2
-Infinity < 0 0
-0 < 0 -π / 2 π / 2

In addition, for points in the second and third quadrants (x < 0), Math.atan2() would output an angle less than - π 2 -\frac{\pi}{2} or greater than π 2 \frac{\pi}{2} .

Because atan2() is a static method of Math, you always use it as Math.atan2(), rather than as a method of a Math object you created (Math is not a constructor).

Examples

Using Math.atan2()

Math.atan2(90, 15); // 1.4056476493802699
Math.atan2(15, 90); // 0.16514867741462683

Difference between Math.atan2(y, x) and Math.atan(y / x)

The following script prints all inputs that produce a difference between Math.atan2(y, x) and Math.atan(y / x).

const formattedNumbers = new Map([
  [-Math.PI, "-π"],
  [(-3 * Math.PI) / 4, "-3π/4"],
  [-Math.PI / 2, "-π/2"],
  [-Math.PI / 4, "-π/4"],
  [Math.PI / 4, "π/4"],
  [Math.PI / 2, "π/2"],
  [(3 * Math.PI) / 4, "3π/4"],
  [Math.PI, "π"],
  [-Infinity, "-∞"],
  [Infinity, "∞"],
]);

function format(template, ...args) {
  return String.raw(
    { raw: template },
    ...args.map((num) =>
      (Object.is(num, -0)
        ? "-0"
        : formattedNumbers.get(num) ?? String(num)
      ).padEnd(5)
    )
  );
}

console.log(`| x     | y     | atan2 | atan  |
|-------|-------|-------|-------|`);

for (const x of [-Infinity, -1, -0, 0, 1, Infinity]) {
  for (const y of [-Infinity, -1, -0, 0, 1, Infinity]) {
    const atan2 = Math.atan2(y, x);
    const atan = Math.atan(y / x);
    if (!Object.is(atan2, atan)) {
      console.log(format`| ${x} | ${y} | ${atan2} | ${atan} |`);
    }
  }
}

The output is:

| x     | y     | atan2 | atan  |
|-------|-------|-------|-------|
| -∞    | -∞    | -3π/4 | NaN   |
| -∞    | -1    | -π    | 0     |
| -∞    | -0    | -π    | 0     |
| -∞    | 0     | π     | -0    |
| -∞    | 1     | π     | -0    |
| -∞    | ∞     | 3π/4  | NaN   |
| -1    | -∞    | -π/2  | π/2   |
| -1    | -1    | -3π/4 | π/4   |
| -1    | -0    | -π    | 0     |
| -1    | 0     | π     | -0    |
| -1    | 1     | 3π/4  | -π/4  |
| -1    | ∞     | π/2   | -π/2  |
| -0    | -∞    | -π/2  | π/2   |
| -0    | -1    | -π/2  | π/2   |
| -0    | -0    | -π    | NaN   |
| -0    | 0     | π     | NaN   |
| -0    | 1     | π/2   | -π/2  |
| -0    | ∞     | π/2   | -π/2  |
| 0     | -0    | -0    | NaN   |
| 0     | 0     | 0     | NaN   |
| ∞     | -∞    | -π/4  | NaN   |
| ∞     | ∞     | π/4   | NaN   |

Specifications

Specification
ECMAScript Language Specification
# sec-math.atan2

Browser compatibility

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See also