Polar Coordinates
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2-D (Plane) Polar Coordinates
3-D (Spherical) Polar Coordinates

 

2-D (Plane) Polar Coordinates

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.

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3-D (Spherical) Polar Coordinates

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]

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Young-Hoo Kwon, 1998-