Cartesian to Spherical Coordinates Calculator
Spherical Coordinates Result
Understanding Cartesian to Spherical Transformation
In three-dimensional space, we often need to switch between different coordinate systems. While the Cartesian system (x, y, z) is intuitive for linear measurements, the Spherical coordinate system (r, θ, φ) is essential in fields like physics, engineering, and astronomy, where radial symmetry or angular positions are prioritized.
The Mathematics Behind the Conversion
This calculator uses the ISO convention (commonly used in physics) to perform the transformation. Here are the fundamental formulas used:
2. Inclination/Polar Angle (θ) = arccos(z / r)
3. Azimuthal Angle (φ) = atan2(y, x)
Parameter Definitions
- Radial Distance (r): The distance from the origin (0,0,0) to the point. It is always non-negative.
- Inclination (θ): Also known as the polar angle or zenith angle. It is the angle measured down from the positive z-axis. It typically ranges from 0 to π (180°).
- Azimuth (φ): The angle measured in the xy-plane from the positive x-axis. It typically ranges from -π to π (or 0 to 2π).
Practical Example
Suppose you have a point located at Cartesian coordinates x = 3, y = 4, z = 5.
- Calculate r: √(3² + 4² + 5²) = √(9 + 16 + 25) = √50 ≈ 7.071
- Calculate θ: arccos(5 / 7.071) ≈ 0.785 radians (or 45°)
- Calculate φ: atan2(4, 3) ≈ 0.927 radians (or 53.13°)
The resulting spherical coordinates are approximately (r=7.071, θ=45°, φ=53.13°).
Why use Spherical Coordinates?
Spherical coordinates are particularly useful when dealing with phenomena related to spheres, such as light distribution from a point source, the gravitational pull of planets, or calculating signal paths in antenna theory. Converting Cartesian coordinates allows researchers to simplify complex differential equations into more manageable forms.