Spherical Coordinates Calculator
Convert between Cartesian (rectangular) and spherical coordinate systems
Coordinate Converter
Cartesian Coordinates
Spherical Coordinates
Conversion Formulas:
r = √(x² + y² + z²)
θ = arccos(z/r)
φ = arctan(y/x)
Example Calculation
Cartesian to Spherical
Given: Point (3, 4, 5) in Cartesian coordinates
Calculate:
r = √(3² + 4² + 5²) = √50 ≈ 7.071
θ = arccos(5/7.071) ≈ 45.0°
φ = arctan(4/3) ≈ 53.13°
Result: (7.071, 45.0°, 53.13°)
Spherical to Cartesian
Given: (r=10, θ=30°, φ=45°)
Calculate:
x = 10 × sin(30°) × cos(45°) ≈ 3.536
y = 10 × sin(30°) × sin(45°) ≈ 3.536
z = 10 × cos(30°) ≈ 8.660
Result: (3.536, 3.536, 8.660)
Coordinate Systems
Cartesian (Rectangular)
Uses three perpendicular axes (x, y, z) intersecting at origin
Spherical
Uses radius (r), polar angle (θ), and azimuth angle (φ)
Parameter Ranges
Quick Tips
r represents distance from origin
θ is angle from positive z-axis
φ is angle in xy-plane from x-axis
Common in physics and engineering
Understanding Spherical Coordinates
What are Spherical Coordinates?
Spherical coordinates are a three-dimensional coordinate system that specifies points using a radial distance and two angles. This system is particularly useful for problems with spherical symmetry, such as in physics, astronomy, and engineering.
The Three Components
- •r (radius): Distance from origin to point
- •θ (theta): Polar angle from positive z-axis
- •φ (phi): Azimuthal angle in xy-plane from x-axis
Conversion Formulas
Cartesian to Spherical
r = √(x² + y² + z²)
θ = arccos(z/r)
φ = arctan(y/x)
Spherical to Cartesian
x = r × sin(θ) × cos(φ)
y = r × sin(θ) × sin(φ)
z = r × cos(θ)
Applications of Spherical Coordinates
🌌 Physics
- • Gravitational fields
- • Electric fields of spheres
- • Quantum mechanics (orbitals)
- • Wave propagation
🔭 Astronomy
- • Celestial navigation
- • Satellite positioning
- • Star coordinates
- • Planetary motion
🛰️ Engineering
- • Antenna design
- • Radar systems
- • 3D modeling
- • Geographic coordinates
🗺️ Geography
- • GPS positioning
- • Earth mapping
- • Navigation systems
- • Distance calculations
💻 Computer Graphics
- • 3D rendering
- • Camera positioning
- • Lighting calculations
- • Virtual reality
🔬 Mathematics
- • Triple integrals
- • Surface area calculations
- • Volume computations
- • Vector calculus
Step-by-Step Conversion Guide
Converting Cartesian to Spherical
Step 1: Calculate the Radius (r)
The radius is the 3D distance from the origin to the point. Use the Pythagorean theorem in three dimensions:
r = √(x² + y² + z²)
Step 2: Calculate the Polar Angle (θ)
The polar angle is measured from the positive z-axis. Use the inverse cosine function:
θ = arccos(z/r)
Range: 0 ≤ θ ≤ π (0° to 180°)
Step 3: Calculate the Azimuthal Angle (φ)
The azimuthal angle is measured in the xy-plane from the positive x-axis:
φ = arctan(y/x)
Use atan2(y, x) for correct quadrant handling
Converting Spherical to Cartesian
Step 1: Calculate x coordinate
x = r × sin(θ) × cos(φ)
Step 2: Calculate y coordinate
y = r × sin(θ) × sin(φ)
Step 3: Calculate z coordinate
z = r × cos(θ)
Common Use Cases and Examples
Example 1: Point on Sphere
Find spherical coordinates of point (1, 1, √2) on unit sphere
Given: (x, y, z) = (1, 1, √2)
Calculate r: √(1² + 1² + 2) = 2
Calculate θ: arccos(√2/2) = 45°
Calculate φ: arctan(1/1) = 45°
Result: (2, 45°, 45°)
Example 2: Antenna Position
Convert antenna location (r=100m, θ=60°, φ=30°) to Cartesian
Given: r=100m, θ=60°, φ=30°
Calculate x: 100×sin(60°)×cos(30°) ≈ 75.0m
Calculate y: 100×sin(60°)×sin(30°) ≈ 43.3m
Calculate z: 100×cos(60°) = 50.0m
Result: (75.0, 43.3, 50.0)m
Frequently Asked Questions
What is the difference between spherical and cylindrical coordinates?
Spherical coordinates use a radius and two angles (r, θ, φ), while cylindrical coordinates use a radius, an angle, and a height (ρ, φ, z). Spherical coordinates are better for problems with spherical symmetry (like planets), while cylindrical coordinates work well for problems with cylindrical symmetry (like pipes or columns).
Why use spherical coordinates instead of Cartesian?
Spherical coordinates simplify many problems involving spherical symmetry. For example, calculating the gravitational field of a planet or the electric field of a charged sphere becomes much easier in spherical coordinates. They also naturally describe positions on spheres, making them ideal for astronomy and geography.
What are the valid ranges for each coordinate?
The radius r must be non-negative (r ≥ 0). The polar angle θ ranges from 0 to π (0° to 180°), measured from the positive z-axis. The azimuthal angle φ ranges from 0 to 2π (0° to 360°), measured in the xy-plane from the positive x-axis counterclockwise.
How do I handle the arctangent for φ correctly?
When calculating φ = arctan(y/x), use the atan2(y, x) function available in most programming languages and calculators. This function correctly handles all four quadrants and avoids division by zero when x = 0. Simple arctan only returns values between -π/2 and π/2.
Can spherical coordinates represent any point in 3D space?
Yes, any point in 3D space can be uniquely represented in spherical coordinates, except for the origin (0, 0, 0) where the angles are undefined. For the origin, r = 0 and the angles θ and φ can be any value. Points on the z-axis have undefined φ because there's no projection onto the xy-plane.