Cylindrical coordinate system

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

A point P is given as <math>(r, \theta, h)</math>. In terms of the Cartesian coordinate system:

• <math>r</math> 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.
• <math>\theta</math> is the angle between the positive x-axis and the line OP', measured anti-clockwise.
• <math>h</math> is the same as <math>z</math>.

Some mathematicians indeed use <math>(r, \theta, z)</math>.

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 = r cosθ

y = r sinθ

z = h

Conversion from Cartesian to cylindrical coordinates

<math>r = \sqrt{x^2 + y^2}</math>

<math>\theta = \arctan\frac{y}{x}</math>

<math>h = z\,</math>