Understanding the ATAN2() Function in SQLite

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The ATAN2() function in SQLite calculates the arc tangent (inverse tangent) of the ratio of two given numeric values, typically representing the y-coordinate and x-coordinate of a point.

Unlike ATAN(), which calculates the angle based only on a single tangent value, ATAN2() considers both the x and y values to determine the correct quadrant for the resulting angle.

Syntax

ATAN2(Y, X)
  • Y: The y-coordinate (or opposite side length in trigonometric terms).
  • X: The x-coordinate (or adjacent side length in trigonometric terms).

The result will be an angle in radians between -π and π, allowing for a complete angle measurement around a circle (from -180° to 180°). This accounts for the direction and quadrant, as both coordinates are considered.

Difference Between ATAN() and ATAN2()

  • ATAN(X) calculates the arc tangent of a single value, providing an angle in radians between -π/2 and π/2 (i.e., from -90° to 90°). It does not distinguish between points in different quadrants, so it only gives the angle based on the tangent’s absolute value.
  • ATAN2(Y, X), however, uses both Y and X to identify the correct quadrant of the angle, returning a more complete range of angles (-π to π) based on the coordinates. This is particularly useful for 2D coordinate systems, where knowing the quadrant is essential.

Example 1

Here’s a quick example to demonstrate how it works:

SELECT ATAN2(5, 7);

Output:

0.620249485982821

Example 2

Suppose we have a table points with columns x and y that represent coordinates on a 2D plane. We want to calculate the angle between each point and the positive x-axis.

CREATE TABLE points (x REAL, y REAL);

INSERT INTO points (x, y) VALUES (1, 1), (-1, 1), (-1, -1), (1, -1), (0, 1);

Now, we can use ATAN2() to calculate the angle for each point:

SELECT x, y, ATAN2(y, x) AS angle_in_radians
FROM points;

Output:

x     y     angle_in_radians  
----  ----  ------------------
1.0   1.0   0.785398163397448 
-1.0  1.0   2.35619449019234  
-1.0  -1.0  -2.35619449019234 
1.0   -1.0  -0.785398163397448
0.0   1.0   1.5707963267949

In this example:

  • For the point (1, 1), ATAN2(1, 1) returns 0.785398163397448 radians (π/4), which is in the first quadrant.
  • For (-1, 1), ATAN2(1, -1) returns approximately 2.35619449019234 radians (3π/4), indicating the second quadrant.
  • For (-1, -1), ATAN2(-1, -1) returns approximately -2.35619449019234 radians, indicating the third quadrant.
  • For (1, -1), ATAN2(-1, 1) returns approximately -0.785398163397448 radians, indicating the fourth quadrant.
  • For (0, 1), ATAN2(1, 0) returns 1.5707963267949 radians (π/2), which is directly on the positive y-axis.

The ATAN2() function is essential in applications like computer graphics, physics, and navigation, where calculating the angle of a vector in a 2D space requires precise quadrant-based angles.