1. What Are Cylindrical Coordinates?

Cylindrical coordinates (ρ, φ, z) are a three-dimensional coordinate system that extends polar coordinates by adding a height coordinate (z). They are useful for problems with cylindrical symmetry.

2. How Does the Conversion Work?

The conversion from Cartesian to cylindrical coordinates uses these formulas:

Where:

• \( \rho \) — Radial distance from z-axis
• \( \phi \) — Azimuthal angle (in radians)
• \( z \) — Same as Cartesian z-coordinate

Note: The calculator uses atan2(y,x) which properly handles all quadrants and special cases (like x=0).

3. Applications of Cylindrical Coordinates

Common Uses: Electromagnetics, fluid dynamics, heat transfer, and any physics problem with cylindrical symmetry. Particularly useful for describing objects like pipes, wires, or cylindrical tanks.

4. Using the Calculator

Instructions: Enter Cartesian coordinates (x,y,z) and the calculator will compute the equivalent cylindrical coordinates (ρ, φ, z). φ is shown in both radians and degrees.

5. Frequently Asked Questions (FAQ)

Q1: What's the range of φ?
A: Typically -π to π radians (-180° to 180°) when using atan2, which ensures the correct quadrant.

Q2: How is this different from spherical coordinates?
A: Spherical coordinates use two angles and a radial distance from origin, while cylindrical keeps one Cartesian coordinate (z).

Q3: What happens when x=0?
A: The calculator handles this case correctly (φ=π/2 or -π/2 depending on y's sign), unlike simple arctan(y/x).

Q4: Can I convert back to Cartesian?
A: Yes, using x = ρcosφ, y = ρsinφ, z = z.

Q5: When should I use cylindrical vs Cartesian?
A: Use cylindrical when the problem has cylindrical symmetry - it often simplifies equations and boundary conditions.