ATan2 - Two-Argument Arctangent (y, x)
Calculator for calculating the angle in all four quadrants
ATan2 Function Calculator
Instructions
Enter the Cartesian coordinates (y, x). The function calculates the polar angle φ in the correct quadrant with a range from -180° to +180° or -π to +π.
Visualization
Angle φ = 45°
ATan2 - Overview
Special Feature
Unlike the normal arctangent, ATan2 expects two arguments (y, x) and can output the result in a range of 360°, covering all four quadrants.
Definition
The ATan2 function calculates the polar angle φ from Cartesian coordinates, considering the signs of y and x to determine the correct quadrant.
\(\displaystyle \varphi = \text{atan2}(y, x) = \arctan\left(\frac{y}{x}\right) \)
with range: \( \varphi \in [-\pi, \pi] \) or [-180°, 180°]
Quadrant Assignment
| I. Quadrant: | x > 0, y > 0 | 0° to 90° |
| II. Quadrant: | x < 0, y > 0 | 90° to 180° |
| III. Quadrant: | x < 0, y < 0 | -180° to -90° |
| IV. Quadrant: | x > 0, y < 0 | -90° to 0° |
Advantages over ATan
- All quadrants: Full 360° range
- Sign consideration: Considers y and x separately
- No division by zero: Safe calculation when x = 0
- Unique assignment: Each point has a unique angle
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Description of the ATan2 Function
Fundamentals
The ATan2 function is an extended form of the arctangent that receives two Cartesian coordinates (y, x) as arguments and returns the polar angle φ in the correct quadrant.
Formula:
\(\displaystyle \varphi = \text{atan2}(y, x) \)
equivalent to
\(\displaystyle \varphi = \arctan\left(\frac{y}{x}\right) \text{ with quadrant correction} \)
Cartesian to Polar Coordinates
When the function atan2(y, x) is passed the two Cartesian coordinates, you get the polar angle φ located in the correct quadrant:
\(\displaystyle (x, y) \rightarrow (r, \varphi) \)
\(\displaystyle r = \sqrt{x^2 + y^2}, \quad \varphi = \text{atan2}(y, x) \)
Quadrant Determination
The function considers the signs of both coordinates to determine the correct quadrant:
x > 0: Quadrant I or IV
x < 0: Quadrant II or III
y > 0: Quadrant I or II
y < 0: Quadrant III or IV
Detailed Examples
Example 1: First Quadrant
Given: y = 3, x = 4
Calculation:
\(\displaystyle \varphi = \text{atan2}(3, 4) = \arctan\left(\frac{3}{4}\right) \)
\(\displaystyle \varphi \approx 36.87° \)
Example 2: Second Quadrant
Given: y = 3, x = -4
Calculation:
\(\displaystyle \varphi = \text{atan2}(3, -4) \)
\(\displaystyle = 180° - \arctan\left(\frac{3}{4}\right) \approx 143.13° \)
Note: ATan would give the wrong quadrant here!
Example 3: Special Cases
- \( \text{atan2}(0, 1) = 0° \) (positive x-axis)
- \( \text{atan2}(1, 0) = 90° \) (positive y-axis)
- \( \text{atan2}(0, -1) = 180° \) (negative x-axis)
- \( \text{atan2}(-1, 0) = -90° \) (negative y-axis)
Comparison ATan vs ATan2
ATan(y/x): Range -90° to +90° (only 2 quadrants)
ATan2(y, x): Range -180° to +180° (all 4 quadrants)
Properties
- Domain: \( (x, y) \in \mathbb{R}^2 \setminus \{(0,0)\} \)
- Range: \( \varphi \in [-\pi, \pi] \) or [-180°, 180°]
- Quadrants: All four quadrants are correctly covered
- Continuity: Continuous except on the negative x-axis
- Symmetry: atan2(-y, -x) = atan2(y, x) ± π
- Special cases:
- atan2(0, 0) is undefined
- atan2(y, 0) = ±90° for y ≠ 0
Practical Applications
- Robotics: Joint angle calculation and inverse kinematics
- Computer graphics: 2D rotations and sprite orientation
- Navigation: Bearing angles and course calculations
- Game development: Object orientation to target points
- Signal processing: Phase angle determination
- Radar/Sonar: Target direction determination
- CAD/CAM: Angle calculations in all quadrants
Important Note
The ATan2 function is indispensable when the correct quadrant is important. Unlike the simple ATan function, which only covers the range from -90° to +90°, ATan2 provides values in the full range from -180° to +180°. This is particularly important in robotics, navigation, and computer graphics, where the exact direction of a vector must be determined. The function also avoids division by zero, as x and y are passed separately.