In a 3D Cartesian coordinate system, a point P is referred to by three real
numbers (coordinates), indicating the positions of the perpendicular projections from the
point to three fixed, perpendicular, graduated lines, called the **axes**
which intersect
at the **origin**.
Often the x-axis is imagined to be horizontal and pointing roughly toward the
viewer (out of the page), the y-axis is also horizontal and pointing to the
right, and the z-axis is vertical, pointing up. The system is called **right-handed**
if it can be rotated so that the three axes are in the position as shown in the
figure above. The x-coordinate of of the point P in the figure is a,
the y-coordinate is b, and the z-coordinate is c.

Learn more by exploring the tutorial on **three-dimensional
rectangular (Cartesian) coordinate systems**.

(Note: To explore the tutorial on you need to to download the free **
LiveMath plug-in**.)

To define a **cylindrical coordinate system**, we take an axis (usually called the
**z-axis**) and a ** perpendicular
plane**, on which we choose a ray (the initial ray)
originating at the intersection of the plane and the axis (the **origin**).
The coordinates of a point P are the ** polar coordinates (r, f)**
of the projection of P on the plane, and the coordinate z of the projection of P
on the z-axis. The coordinate r is always positive and the range of f
is from 0 to 2p
(360^{o}).

To
transform from Cartesian to cylindrical coordinates and vice versa, we use
the transformation equations

x
= r cosf, y = r sinf,
z = z,

r
= (x^{2 }+ y^{2})^{1/2}, f
= tan^{-1}(y/x), z = z.

To define ** spherical coordinates**, we take an axis (the ** polar
axis**) and a
perpendicular plane (the ** equatorial plane**), on which we choose a ray (the
initial ray) originating at the intersection of the plane and the axis (the **
origin** O). The coordinates of a point P are the distance ** r** from P to the origin;
the angle ** q** (zenith) between the line OP and the
positive polar axis; and the angle **f**
(azimuth) between the initial ray and the projection of OP onto the equatorial
plane. The range of f
is from 0 to 2p
(360^{o})., and the range of q is from 0 to p
(180^{o}).

To transform from Cartesian to
spherical coordinates and vice versa, we use
the transformation equations

x
= r sinq cosf, y = r
sinq
sinf, z = r cosq,

r
= (x^{2 }+ y^{2 }+ z^{2})^{1/2},
q =tan^{-1}(z/(x^{2}+y^{2})^{1/2}),
f
= tan^{-1}(y/x).