The cylindrical coordinate system is a three-dimensional system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted h) which measures the height of a point above the plane.

A point P is given as (r,θ,h). In terms of the Cartesian coordinate system:

• r is the distance from O to P', the orthogonal projection of the point P onto the XY plane. This is the same as the distance of P to the z-axis.
• θ is the angle between the positive x-axis and the line OP', measured anti-clockwise.
• h is the same as z.

Some mathematicians indeed use (r,θ,z).

Cylindrical coordinates are useful in analyzing surfaces that are symmetrical about an axis, with the z-axis chosen as the axis of symmetry. For example, the infinitely long circular cylinder that has the Cartesian equation x² + y² = c² has the very simple equation r = c in cylindrical coordinates. Hence the name "cylindrical" coordinates.

Conversion from cylindrical to Cartesian coordinates

x = rcosθ y = rsinθ z = h

Conversion from Cartesian to cylindrical coordinates

Conversion from cylindrical to spherical coordinates

where φ is the azimuth and θ' is the latitude.

Conversion from spherical to cylindrical coordinates

r = ρcosθ θ' = φ h = ρsinθ

where φ is azimuth and θ is latitude.

See also

• Cartesian coordinate system
• Spherical coordinate system
• Parabolic coordinate system