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 x^{2} + y^{2} = c^{2} 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>
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