Cylindrical To Spherical Calculator

Cylindrical to Spherical Conversion:

\[ \rho = \sqrt{r^2 + z^2} \] \[ \theta = \phi \] \[ \phi = \arctan(r/z) \]

Spherical Radius (ρ):

Azimuth Angle (θ):

Polar Angle (φ):

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1. What is Cylindrical to Spherical Conversion?

Cylindrical to spherical conversion transforms coordinates from the cylindrical coordinate system (r, φ, z) to the spherical coordinate system (ρ, θ, φ). This conversion is essential in mathematics, physics, and engineering applications involving three-dimensional space.

2. How Does the Calculator Work?

The calculator uses the following conversion formulas:

\[ \rho = \sqrt{r^2 + z^2} \] \[ \theta = \phi \] \[ \phi = \arctan(r/z) \]

Where:

\( r \) — Cylindrical radius (distance from z-axis)
\( \phi \) — Azimuth angle (same in both systems)
\( z \) — Height coordinate
\( \rho \) — Spherical radius (distance from origin)
\( \theta \) — Polar angle (angle from positive z-axis)

Explanation: The conversion preserves the azimuth angle while calculating the radial distance and polar angle based on the cylindrical coordinates.

3. Coordinate System Fundamentals

Details: Cylindrical coordinates are useful for problems with cylindrical symmetry, while spherical coordinates are ideal for spherical symmetry. Understanding both systems is crucial for solving complex 3D problems in physics and engineering.

4. Using the Calculator

Tips: Enter cylindrical radius (r) and height (z) in consistent units. The azimuth angle (φ) should be in radians. All values must be valid (r ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: What are the typical units for these coordinates?
A: All units must be consistent. Common units include meters, centimeters, or any length unit for r, z, and ρ. Angles are typically in radians.

Q2: How do I handle negative z values?
A: Negative z values are acceptable and will affect the polar angle calculation. The calculator handles all real z values correctly.

Q3: What is the range of the polar angle φ?
A: The polar angle typically ranges from 0 to π radians (0° to 180°), representing the angle from the positive z-axis.

Q4: Can I use degrees instead of radians?
A: The calculator expects angles in radians. Convert degrees to radians by multiplying by π/180 before input.

Q5: When is this conversion most useful?
A: This conversion is particularly useful in electromagnetism, fluid dynamics, and quantum mechanics problems where different symmetries are present.