The 2-D polar coordinate system involves the distance from the
origin and an azimuth angle. Figure 1 shows the 2-D polar
coordinate system, where *r* is the distance from the origin to point *P*,
and is the azimuth angle measured from the horizontal (*X*) axis in the
counterclockwise direction. Thus, the position of point *P* is described as (*r*,
). *r* &
are the 2-D polar coordinates.

**Figure
1**

The polar coordinates are useful in describing the human body motion
since the essence of the human body motion is the joint motions. The segments undergo
rotations about the joint centers and using the azimuth angle (while *r* = const)
in describing the body motion is more efficient than using the Cartesian coordinates. One
thing to note here is that coordinates *r* &
are not the same kind.

The relationship between the 2-D Cartesian coordinates and the 2-D
polar coordinates can be summarized as

**[1]**

where .
Or

**[2]**

In some cases, one may use the Cartesian coordinates in the sense of
the polar coordinates, as shown on Figure 2.

**Figure
2**

The
axis is the radial axis while the
axis is the tangential axis.
Using the *P-* system (using the
Cartesian system in the sense of the polar system) is
sometimes more convenient than using the polar coordinate system since it combines the
advantages of both systems. Motion in the direction of the
axis is
the radial motion while that in the direction of the
axis reflects the motion due to rotation. One problem here is that the
orientation of the
& * *axes changes as point *P* moves.

The 3-D polar coordinate system or the spherical coordinate system
involves the distance from the origin and 2 angles (Figure 3).

**Figure 3**

The position of point P is described as (*r*, ,
), where *r* = the distance from the origin (*O*),
= the horizontal azimuth angle measured
on the *XY* plane from the *X* axis in the counterclockwise direction, and
= the azimuth angle measured from the *Z*
axis. Again, the coordinates are not the same kind.

The relationship between the spherical coordinates and the Cartesian
coordinates can be summarized as

**[3]**

where
and . Or

**[4]**

Similar to the 2-D moving Cartesian coordinate system shown in Figure 2, one can also define a moving 3-D Cartesian system. See Figure 4. Axis
is the radial axis while axes
&
shows the normal axes to .

**Figure 4**

The moving 3-D Cartesian coordinate system shown in Figure 4 is useful in describing the motion of point *P* in
terms of the radial motion and the rotations about the origin. Note here that
the Cartesian coordinate system shown in Figure 4
is a right-hand system. On the contrary, the Cartesian system normally used in
biomechanics or motion analysis is a left-hand system.

The unit vectors (see Vector
for the details of the unit vector) of the axes can be described in those of
the global reference frame. Let the three unit vectors of system *O-XYZ*
be **i**, **j**, and **k**. The unit vectors of system *P-*
(, ,
and ) are then

.
**[5]**