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 h) which measures the height of a point above the plane.

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

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• $r$ 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.
• $heta$ 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, heta, z).

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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.

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Cylindrical coordinate system: Conversion from cylindrical to Cartesian coordinates ►