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 | – |
| – | – |
I | – |
| – | – |
III | + |
| + | – |
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.
Examples:
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.
Analytic
Geometry:
Cylindrical
Coordinates
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