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# Cylindrical coordinate system

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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 $(r, \theta, h)$. 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.
• $\theta$ 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 $(r, \theta, z)$.

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.

x = r cosθ
y = r sinθ
z = h

$r = \sqrt{x^2 + y^2}$
$\theta = \arctan\frac{y}{x}$
$h = z\,$

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