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Category Math & Algebra

r Radial distance

θ Azimuthal angle

z Height

Angle unit:

Presets:

Cylindrical (r, θ, z)

r—

θ—

z—

Cartesian (x, y, z)

x—

y—

z—

Spherical (ρ, θ, φ)

ρ—

θ—

φ—

Additional Properties

Distance from origin—

Distance from z-axis—

Quadrant (xy)—

Jacobian |J|—

3D Visualization Drag to rotate

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About

Coordinate system errors propagate silently through integration bounds, vector field computations, and boundary condition definitions. A sign mistake in the θ component or a forgotten r factor in the volume element dV = r ⋅ dr ⋅ dθ ⋅ dz invalidates the entire result. This calculator performs exact conversions between cylindrical (r, θ, z), Cartesian (x, y, z), and spherical (ρ, θ, φ) coordinate systems using standard ISO 31-11 conventions. It handles quadrant-aware arctangent via atan2 and supports both degree and radian input.

The tool assumes the physics convention for spherical coordinates where θ is the azimuthal angle and φ is the polar angle measured from the positive z-axis. Note: the mathematics convention swaps θ and φ. Verify which convention your textbook uses before applying results. The interactive 3D visualization renders the point and its projections onto coordinate planes to build geometric intuition.

Formulas

The fundamental conversion from cylindrical to Cartesian coordinates uses the polar projection onto the xy-plane.

x = r cos θ

y = r sin θ

z = z

The inverse transformation from Cartesian to cylindrical recovers the radial distance and azimuthal angle.

r = √x² + y²

θ = atan2(y, x)

Conversion between cylindrical and spherical coordinates (physics convention where φ is the polar angle from the z-axis).

ρ = √r² + z²

φ = atan2(r, z)

The gradient operator in cylindrical coordinates applies scale factors to partial derivatives.

∇f = ∂f∂r e_r + 1r ∂f∂θ e_θ + ∂f∂z e_z

The Laplacian in cylindrical coordinates includes the critical 1r factor.

∇²f = 1r ∂∂r (r ∂f∂r) + 1r² ∂²f∂θ² + ∂²f∂z²

Where r = radial distance from z-axis, θ = azimuthal angle in xy-plane measured from positive x-axis, z = height along the vertical axis, ρ = radial distance from origin (spherical), φ = polar angle from positive z-axis (physics convention).

Reference Data

Property	Cylindrical	Cartesian	Spherical
Coordinates	(r, θ, z)	(x, y, z)	(ρ, θ, φ)
Range of r / ρ	r ≥ 0	x ∈ R	ρ ≥ 0
Range of θ	[0, 2π)	-	[0, 2π)
Range of φ	-	-	[0, π]
Volume Element	r dr dθ dz	dx dy dz	ρ² sinφ dρ dθ dφ
Scale Factor h₁	1	1	1
Scale Factor h₂	r	1	ρ
Scale Factor h₃	1	1	ρ sinφ
Unit Vectors	e_r, e_θ, e_z	e_x, e_y, e_z	e_ρ, e_θ, e_φ
Orthogonal	Yes	Yes	Yes
Best for Symmetry	Axial / Cylindrical	None (general)	Radial / Spherical
Common Applications	Pipes, solenoids, rotating shafts	General mechanics, CAD	Gravity, EM radiation, antennas
Laplacian Form	1r ∂∂r(r ∂f∂r) + ...	∂²f∂x² + ∂²f∂y² + ∂²f∂z²	Complex (see formulas)
x from Cylindrical	r cosθ	-	ρ sinφ cosθ
y from Cylindrical	r sinθ	-	ρ sinφ sinθ
z from Cylindrical	z	-	ρ cosφ
r from Cartesian	√x² + y²	-	-
θ from Cartesian	atan2(y, x)	-	atan2(y, x)
ρ from Cartesian	-	-	√x² + y² + z²
Jacobian Determinant	r	1	ρ² sinφ
ISO Convention	ISO 31-11	ISO 31-11	ISO 31-11 (physics)

Frequently Asked Questions

When r = 0, the point lies on the z-axis. The azimuthal angle θ becomes undefined (any value yields the same point). The calculator returns θ = 0 by convention, matching the behavior of atan2(0, 0). The Laplacian and gradient operators contain 1r terms that become singular at r = 0. Apply L'Hôpital's rule or use Cartesian coordinates near the axis for differential operations.

ISO 31-11 defines two conventions. In physics, θ is the azimuthal angle (in the xy-plane, range [0, 2π)) and φ is the polar angle from the z-axis (range [0, π]). In mathematics, these symbols are swapped. This calculator uses the physics convention. Mixing conventions inverts the Jacobian determinant and corrupts volume integrals. Always check your source material before substituting values.

The standard arctan(yx) returns values in (−π2, π2) and cannot distinguish quadrants II and III from I and IV. The atan2(y, x) function uses the signs of both arguments to return the correct angle in (−π, π]. This calculator normalizes negative results to [0, 2π) by adding 2π when the raw result is negative.

The Jacobian matrix of the transformation maps differential elements between coordinate systems. For cylindrical coordinates, ∂(x, y, z)∂(r, θ, z) evaluates to r. This means a differential volume element dx dy dz transforms to r dr dθ dz. Omitting the r factor is one of the most common integration errors in multivariable calculus. It arises because arclength in the θ direction scales linearly with r.

By standard convention, r ≥ 0. A negative r value is geometrically equivalent to the point at (|r|, θ + π, z). Some polar plotting software allows negative r for convenience. This calculator accepts negative r input and automatically normalizes it: it takes the absolute value and adds π to θ, then reduces θ modulo 2π.

The coordinate system itself is geometry-dependent, not medium-dependent. Choose cylindrical coordinates when the problem has axial symmetry (pipes, coaxial cables, rotating machinery). The material properties (thermal conductivity, permittivity) appear as coefficients inside the differential operators, not in the coordinate transformation. However, anisotropic materials where conductivity differs along e_r vs e_θ require tensor notation in that coordinate system, making the Laplacian more complex than the scalar form shown here.