Cartesian to Spherical 3D Conversion Calculator

Convert Cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ) in 3D space.

Input Parameters

Calculation Results

Calculation Formula

The spherical coordinates (r, θ, φ) are calculated from Cartesian coordinates (x, y, z) as follows:

Where:
r = √(x² + y² + z²)
θ = atan2(y, x)
φ = acos(z/r)

Cartesian to Spherical 3D Conversion Calculator Calculator Usage Guide

Learn how to use the Cartesian to Spherical 3D Conversion Calculator and its working principles

Overview

This calculator converts Cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ) in 3D space. Spherical coordinates are useful in various fields such as physics, engineering, and computer graphics.

Input Parameters

  • X: The x-coordinate of the Cartesian point.
  • Y: The y-coordinate of the Cartesian point.
  • Z: The z-coordinate of the Cartesian point.

Output Parameters

  • Radius (r): The distance from the origin to the point.
  • Azimuthal Angle (θ): The angle in the xy-plane from the positive x-axis.
  • Polar Angle (φ): The angle from the positive z-axis.

Formulae

The conversion from Cartesian to spherical coordinates is based on the following formulae:

r = √(x² + y² + z²)

θ = atan2(y, x)

φ = acos(z/r)

Example

For a Cartesian point (1, 1, 1):

r = √(1² + 1² + 1²) = √3 ≈ 1.732

θ = atan2(1, 1) ≈ 0.785 radians (or 45°)

φ = acos(1/√3) ≈ 0.955 radians (or 55.3°)

Applications

Spherical coordinates are commonly used in:

  • 3D modeling and computer graphics
  • Physics problems involving spherical symmetry
  • Navigation systems
  • Geographic coordinates (latitude and longitude can be converted to spherical coordinates)