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
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.
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