Distance vs. Position
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Angular Distance vs. Relative Angular Position
Computation of Angular Distance
Computation of Relative Angular Position

Angular Distance vs. Relative Angular Position

One elementary concept in mechanics is the difference between distance and displacement (or relative position). Distance has the magnitude information only (scalar) while both displacement and relative position have the directional information in addition to the magnitude (vector). Displacement and relative position are essentially the same except the fact that displacement deals with the same object's motion in time.

This conceptual difference causes some problem in the user-angle computation. Depending on the situation, certain user-angles fall into the angular distance category while the others belong to the relative angular position category. Their angle computation methods are different and, therefore, the angles must be computed using the right calculation methods.

There are two main criteria that determine the type of the user-angle in question:

Dimension of the analysis: In 2-D analysis, all user-angles can be regarded as relative angular positions. In 3-D analysis, it depends on the second criterion.
Projection: In 3-D analysis, all projected angles can be regarded as relative angular position. All non-projected angles are angular distances.

Figure 1 illustrates angular distances (a) and relative angular position (b). As shown in the figure, angular distances only give the magnitude of the angle between two lines regardless of the perspective. On the other hand, relative angular position requires lines 1 and 2 to be specifically identified: the relative position of line 2 to line 1. It also requires a particular reference frame that describes the plane in which the angle is defined. Any 2-D angles suffice these requirements.

    Figure 1

It is generally more advantageous to use relative angular positions rather than angular distances since they provide one additional piece of information: direction in conjunction with a particular perspective.

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Computation of Angular Distance

Angular distance can be computed from the scalar product of the two line vectors that form the angle:

.    [1]

The return value of the inverse cosine function ranges

.    [2]

The unit is radian (rad).

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Computation of Relative Angular Position

Any 2-D or projected 3-D angles provide the relative position of line 2 to line 1. Let P and p be the reference frame for projection and the projection axis, respectively. P can be any global or local frame. p can be any axis (X or Y in the 2-D analysis or X, Y or Z in the 3-D analysis). Let the line vectors involved in the angle computation be a (line 1) and b (line 2). To compute the projected line vectors, vectors a and b must be first transformed to the projection reference frame:

.    [3]

The projected line vectors (Figure 2) can be obtained from the components of the transformed vectors:

    Figure 2

,    [4]

where a & b = the projected line vectors and

.    [5]

p, p1 & p2 shown in [5] are in cyclic order. The following relationships hold among the angles shown in Figure 2:

,    [6]

or

.    [7]

In other words, the relative angular position of projected line 2 to 1 is the same to the difference between their angular positions. The sine and cosine values of the angular positions of the projected lines are the components of the unit vectors:

.    [8]

The relative angular position can be then computed from its sine and cosine values:

.    [9]

The relative angular position obtained from [9] ranges

    [10]

Again, the unit is radian (rad). If both the sine and cosine values of the relative angular position in [6] and [7] are 0, the system is not solvable.

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Young-Hoo Kwon, 1998-