Cartesian to Spherical Calculator

Cartesian to Spherical Coordinate Converter

Conversion Results

Radial Distance (r):

Polar Angle (θ – Inclination)

Azimuthal Angle (φ)

Understanding Cartesian to Spherical Conversion

This calculator converts three-dimensional Cartesian coordinates (x, y, z) into Spherical coordinates (r, θ, φ). This system is widely used in physics, engineering, and computer graphics to describe positions in 3D space using distances and angles.

Conversion Formulas

  • Radius (r): r = √(x² + y² + z²)
  • Polar Angle (θ – Inclination): θ = arccos(z / r)
  • Azimuthal Angle (φ): φ = atan2(y, x)

Coordinate Definitions

The Radius (r) is the distance from the origin. The Polar Angle (θ) represents the angle from the positive z-axis (ranging from 0 to π radians or 180°). The Azimuthal Angle (φ) represents the angle from the positive x-axis in the xy-plane (ranging from -π to π radians, or -180° to 180°).

Example Calculation

If you have a point at (x=1, y=1, z=1):

  1. r = √(1² + 1² + 1²) = √3 ≈ 1.732
  2. θ = arccos(1 / √3) ≈ 54.74°
  3. φ = atan2(1, 1) = 45°
function calculateSpherical() { var x = parseFloat(document.getElementById('coordX').value); var y = parseFloat(document.getElementById('coordY').value); var z = parseFloat(document.getElementById('coordZ').value); if (isNaN(x) || isNaN(y) || isNaN(z)) { alert("Please enter valid numeric values for all coordinates."); return; } // Calculate Radial Distance (r) var r = Math.sqrt(x * x + y * y + z * z); var thetaRad = 0; var phiRad = 0; if (r !== 0) { // Polar Angle (theta) – inclination from z-axis thetaRad = Math.acos(z / r); // Azimuthal Angle (phi) – angle in xy-plane phiRad = Math.atan2(y, x); } else { // At origin, angles are technically undefined, but usually set to 0 thetaRad = 0; phiRad = 0; } // Convert to degrees var thetaDeg = thetaRad * (180 / Math.PI); var phiDeg = phiRad * (180 / Math.PI); // Display Results document.getElementById('results').style.display = 'block'; document.getElementById('resR').innerText = r.toFixed(5); document.getElementById('resThetaDeg').innerHTML = "Degrees: " + thetaDeg.toFixed(4) + "°"; document.getElementById('resThetaRad').innerHTML = "Radians: " + thetaRad.toFixed(5) + " rad"; document.getElementById('resPhiDeg').innerHTML = "Degrees: " + phiDeg.toFixed(4) + "°"; document.getElementById('resPhiRad').innerHTML = "Radians: " + phiRad.toFixed(5) + " rad"; }

How to Convert Cartesian to Spherical Coordinates

In mathematics and physics, switching between coordinate systems is essential for simplifying complex problems. While Cartesian coordinates (x, y, z) are excellent for measuring distances along straight axes, Spherical coordinates (r, θ, φ) are superior for systems involving rotation, symmetry, or central forces, such as planetary orbits or electromagnetic fields.

The Geometry of the Conversion

To visualize the conversion, imagine a point in 3D space. The r value is a straight line from the origin to that point. The θ (theta) angle tells you how far down to "tilt" from the vertical Z-axis. Finally, the φ (phi) angle tells you how far to rotate around the vertical axis starting from the X-axis.

Common Use Cases

  • Astronomy: Mapping stars on the celestial sphere.
  • Engineering: Designing 3D components like domes or circular antennas.
  • Robotics: Calculating the reach and rotation of robotic arms.
  • Computer Graphics: Implementing camera controls that orbit around an object.

Standard Convention Warning

Note that conventions can vary between mathematics and physics. In physics, θ is often the polar angle and φ is the azimuthal angle. Some mathematical textbooks swap these labels. This calculator follows the standard physics convention where θ is the inclination from the z-axis.

Table: Key Conversion Points

Cartesian (x, y, z) Spherical (r, θ, φ)
(1, 0, 0) (1, 90°, 0°)
(0, 1, 0) (1, 90°, 90°)
(0, 0, 1) (1, 0°, 0°)

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