
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 userangle computation. Depending on the situation, certain userangles 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 userangle in question:
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 2D angles suffice these requirements.
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. Computation of Angular Distance Angular distance can be computed from the scalar product of the two line vectors that form the angle:
The return value of the inverse cosine function ranges
The unit is radian (rad). Computation of Relative Angular Position Any 2D or projected 3D 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 2D analysis or X, Y or Z in the 3D 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:
The projected line vectors (Figure 2) can be obtained from the components of the transformed vectors:
where a & b = the projected line vectors and
p, p_{1} & p_{2} shown in [5] are in cyclic order. The following relationships hold among the angles shown in Figure 2:
or
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:
The relative angular position can be then computed from its sine and cosine values:
The relative angular position obtained from [9] ranges
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. 
© YoungHoo Kwon, 1998 