Spherical Cylindrical Coordinates

Reference data and engineering information about spherical cylindrical coordinates for standard organizations applications.

sphericalcylindricalcoordinates

Overview

Engineering reference data for Spherical Cylindrical Coordinates in standard organizations.

$ISO \; 9001:2015$

Quality management systems.

$ASTM \; E8$

Standard test methods for tension testing.

$ANSI/ASME \; B16.5$

Pipe flanges and flanged fittings.

Symbol	Description	Unit
$ISO$	International Organization for Standardization	—
$ASTM$	American Society for Testing and Materials	—
$ANSI$	American National Standards Institute	—

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Comparison of 3D coordinate systems
Property	Cartesian	Cylindrical	Spherical
Coordinates	$(x, y, z)$	$( ho, \Phi, z)$	$(r, \Phi, \theta)$
Radial Distance	$r = \sqrt{x^2 + y^2 + z^2}$	$\rho = \sqrt{x^2 + y^2}$	$r = \sqrt{x^2 + y^2 + z^2}$
Azimuthal Angle	$\Phi = \arctan(y/x)$	$\Phi = \arctan(y/x)$	$\Phi = \arctan(y/x)$
Polar Angle	N/A	N/A	$\theta = \arccos(z/r)$
Best Geometry	Rectangular	Cylinders, pipes	Spheres, cones

Source: engineeringtoolbox.com

$r$ — radial distance from origin to point $P$
$\Phi$ — azimuthal angle in the $xy$ -plane from the positive $x$ -axis (range: $0 \leq \Phi < 2\pi$ )
$\theta$ — polar angle from the positive $z$ -axis (range: $0 \leq \theta \leq \pi$ )

Common applications: Problems with spherical symmetry such as gravitational fields, electromagnetic waves, and quantum mechanical orbitals.

$\rho$ — perpendicular distance from the $z$ -axis to point $P$
$\Phi$ — azimuthal angle in the $xy$ -plane from the positive $x$ -axis (range: $0 \leq \Phi < 2\pi$ )
$z$ — height along the $z$ -axis (identical to Cartesian $z$ )

Common applications: Problems with cylindrical symmetry such as pipe flow, coaxial cables, and cylindrical tanks.

In both systems, $\Phi$ represents the same azimuthal angle measured from the $x$ -axis
The relationship between spherical and cylindrical radial distances: $r = \sqrt{\rho^2 + z^2}$
Coordinate transformations are essential for simplifying integrals over domains with natural symmetry