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  Analytic Geometry: Cylindrical Coordinates  

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Cylindrical Coordinates

Cylindrical coordinates are obtained by using polar coordinates in a plane, and then adding a z-axis perpendicular to the plane passing through the pole. This gives three coordinates (r,theta,z) for any point.

To transform from rectangular Cartesian coordinates (x,y,z) to cylindrical ones and back, use the following formulas:

x = r cos(theta),
y = r sin(theta),
z = z,

r = ±sqrt(x2+y2),
theta = arctan(y/x),
z = z.

The sign of r is determined by which of the values of the arctangent function is chosen:

  Sign of x     Sign of y     Quadrant of theta     Sign of r  
++ I+
++ III
+ II+
+ IV
+ II
+ IV+

The quadrant of theta can always be chosen to make r positive, if it is so desired.

Common Uses

The most common use of cylindrical coordinates is to give the equation of a surface of revolution. If the z-axis is taken as the axis of revolution, then the equation will not involve theta at all.


  • A paraboloid of revolution might have equation

    z = r2.

    This is the surface you would get by rotating the parabola z = x2 in the xz-plane about the z-axis. The cartesian coordinate equation of the paraboloid of revolution would be z = x2 + y2.

  • A right circular cylinder of radius a whose axis is the z-axis has equation

    r = R.

  • A a sphere with center at the origin and radius R will have equation

    r2 + z2 = R2.

  • A right circular cone with vertex at the origin and axis the z-axis has equation

    z = m r.

As another kind of example, a helix has the following equations:

r = R,
z = a theta.

Sometimes a change from rectangular to cylindrical coordinates makes computing difficult multiple integrals simpler.

Compiled by Robert L. Ward.

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