Spherical To Cylindrical Coordinates Calculator

Conversion Formulas:

\[ \rho = r \sin(\phi) \] \[ z = r \cos(\phi) \] \[ \theta = \theta \]

Cylindrical Radius (ρ):

Azimuthal Angle (θ):

Height (z):

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1. What is Spherical to Cylindrical Conversion?

The spherical to cylindrical coordinates conversion transforms coordinates from the spherical system (r, θ, φ) to the cylindrical system (ρ, θ, z). This is useful in physics, engineering, and mathematics when different coordinate systems are better suited for different problems.

2. How Does the Calculator Work?

The calculator uses the following conversion formulas:

\[ \rho = r \sin(\phi) \] \[ z = r \cos(\phi) \] \[ \theta = \theta \]

Where:

\( r \) — Radial distance from origin (meters)
\( \theta \) — Azimuthal angle in xy-plane from x-axis (radians)
\( \phi \) — Polar angle from z-axis (radians)
\( \rho \) — Cylindrical radius (meters)
\( z \) — Height coordinate (meters)

Explanation: The conversion preserves the azimuthal angle θ while transforming the radial and vertical components.

3. Importance of Coordinate Conversion

Details: Different coordinate systems are better suited for different problems. Cylindrical coordinates are often used for problems with cylindrical symmetry, while spherical coordinates work well for spherical symmetry.

4. Using the Calculator

Tips: Enter the spherical coordinates (r, θ, φ) where r ≥ 0, θ is any real number, and φ is typically between 0 and π radians. All values must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between spherical and cylindrical coordinates?
A: Spherical uses (r, θ, φ) where r is distance from origin, θ is azimuthal angle, and φ is polar angle. Cylindrical uses (ρ, θ, z) where ρ is radial distance in xy-plane, θ is azimuthal angle, and z is height.

Q2: When should I use cylindrical coordinates?
A: Use cylindrical coordinates for problems with cylindrical symmetry, like calculating fields around a wire or fluid flow in a pipe.

Q3: What are typical angle ranges?
A: θ is usually between 0 and 2π radians, φ between 0 and π radians, but the calculator accepts any real values.

Q4: Can I input angles in degrees?
A: This calculator requires radians. Multiply degrees by π/180 to convert to radians.

Q5: What if my r value is negative?
A: The calculator requires r ≥ 0 as distance cannot be negative in standard coordinate systems.