About

Coordinate system conversion is a routine operation in physics, engineering, and signal processing. Errors in quadrant assignment are the most common failure mode. A naive use of arctan(y ÷ x) collapses two quadrants into one and is undefined when x = 0. This tool uses the two-argument atan2(y, x) function, which resolves the angle correctly across all four quadrants and both axes. The radial distance r is computed as the Euclidean norm. Results are given in both radians and degrees to 8 decimal places.

The reverse path, polar to Cartesian, applies standard trigonometric projection. Note: this tool assumes a standard mathematical convention where θ = 0 lies along the positive x-axis and increases counterclockwise. Navigational or engineering conventions (clockwise from north) require manual offset. The interactive plot updates in real time to confirm the geometric interpretation of your input.

Formulas

Conversion from Cartesian coordinates (x, y) to Polar coordinates (r, θ) uses the Euclidean distance and two-argument arctangent:

r = √x² + y²

θ = atan2(y, x)

The reverse conversion from Polar to Cartesian applies trigonometric projection:

x = r ⋅ cos(θ)

y = r ⋅ sin(θ)

Angle unit conversion between radians and degrees:

θ_deg = θ_rad × 180π

Where x and y are the horizontal and vertical displacements from the origin. r is the radial distance (always ≥ 0). θ is the angle measured counterclockwise from the positive x-axis. atan2 returns values in the range (−π, π], correctly handling all four quadrants and axis-aligned points.

Reference Data

Quadrant	Sign of x	Sign of y	θ Range (Degrees)	θ Range (Radians)	Example Point	r	θ
I	+	+	0° to 90°	0 to π2	(3, 4)	5	53.13°
II	−	+	90° to 180°	π2 to π	(−3, 4)	5	126.87°
III	−	−	−180° to −90°	−π to −π2	(−3, −4)	5	−126.87°
IV	+	−	−90° to 0°	−π2 to 0	(3, −4)	5	−53.13°
+X Axis	+	0	0°	0	(5, 0)	5	0°
−X Axis	−	0	180°	π	(−5, 0)	5	180°
+Y Axis	0	+	90°	π2	(0, 5)	5	90°
−Y Axis	0	−	−90°	−π2	(0, −5)	5	−90°
Origin	0	0	Undefined	Undefined	(0, 0)	0	N/A
Common Reference Angles
-	1	0	0°	0 rad	(1, 0)	1	0°
-	√32	12	30°	π6	(0.866, 0.5)	1	30°
-	√22	√22	45°	π4	(0.707, 0.707)	1	45°
-	12	√32	60°	π3	(0.5, 0.866)	1	60°
-	0	1	90°	π2	(0, 1)	1	90°
-	−1	0	180°	π	(−1, 0)	1	180°
-	0	−1	−90°	−π2	(0, −1)	1	−90°

Frequently Asked Questions

The atan2 function returns angles in the range (−π, π], which corresponds to (−180°, 180°]. Negative angles indicate points below the x-axis (quadrants III and IV). For example, the point (1, −1) yields θ = −45°. If you need a 0° - 360° range, add 360° to any negative result.

At the origin, r = 0 and θ is mathematically undefined because no unique direction exists from a zero-length vector. The atan2(0, 0) function in JavaScript returns 0 by convention, but this tool flags the result as geometrically undefined to prevent misinterpretation.

The single-argument arctan(y÷x) only returns values in (−90°, 90°), making it impossible to distinguish between quadrants I and III (or II and IV) since both map to the same ratio. It also fails when x = 0 due to division by zero. The atan2 function accepts both arguments separately, inspects the signs of x and y independently, and returns the correct angle across the full (−180°, 180°] range.

Yes. JavaScript uses IEEE 754 double-precision floats, which support magnitudes from approximately 5 × 10⁻³²⁴ to 1.8 × 10³⁰⁸. The trigonometric functions and atan2 remain accurate for normal floating-point values. Precision loss only appears at the extremes of representable numbers. For everyday engineering and physics ranges (nanometers to astronomical units), precision to 15 - 16 significant digits is maintained.

Standard math convention measures θ counterclockwise from the east (positive x-axis). Navigation bearings measure clockwise from north (positive y-axis). To convert: bearing = 90° − θ_deg. If the result is negative, add 360°. This tool outputs the mathematical convention; apply this offset manually for navigational use.

Cylindrical coordinates (r, θ, z) use the same planar polar conversion for the x-y plane. Spherical coordinates (ρ, θ, φ) add a zenith angle. The azimuthal angle θ from this converter is directly usable in both systems. Ensure consistent sign and range conventions match your target system before reusing the output.