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Formula Values

Once the ambiguous case for has been eliminated by the definition of the function, there might be zero or one values (and corresponding points ) for a given , , and .

(If the distance between and is zero, then a single point of interception exists at the shared point--or an infinite number of points of interception over time if in addition . These special cases are excluded from all the considerations below.)

Equation (1) is undefined when is zero. If happens to be headed directly for with a non-zero velocity (that is, , which means is zero, and is non-zero), then a point of interception will exist at point . Otherwise, no point of interception will exist. In either case, the value of is irrelevant to the physical problem since point does not move.

Equation (1) is undefined when the parameter of the function is less than or greater than . Consider the parameter of the function: is a number between and which will modify the speed ratio to determine how far has to turn from its line of sight in order to try to intercept : a faster and a close to lead to a more radical turn. If, for instance, is headed directly toward or directly away from , then or , , and the relative speeds of and are irrelevant to : the intercept angle will be 0 (i.e. should run directly towards to try to intercept). If, on the other hand, is running directly broadside to 's line of sight, , then is going to have to turn quite a ways from its line of sight to try and intercept : how far, exactly, is determined only by the ratio of their speeds, since will be .

So, when the ratio of the velocities modified by the relative angle of their vectors is less than or greater than , will escape from off to one side or the other, and no point is defined. This doesn't describe all possible escapes of ; rather, it describes the escapes where is running obliquely--at something like a right angle--across 's line of sight. Such an escape has an understandable representation when is less than , as shown by in Figure 4: is moving so fast (relative to ) that even though is headed in the general direction of , still can't close the gap, even with a maximum of . (Because of the ambiguity expressed by , a similar escape where is greater than is not easily pictured--see in the figure.

The mathematical representation begins to have little correspondence with the physical sitatution, and it seems strange that would flip around to the other side of . This difficulty of representation is discussed more fully in the next paragraphs.)

A description of all possible escapes cannot use equation (1) as its sole criterion, since in many cases the equation will happily yield an even though an interception is impossible (and even though the reprentation expressed by the formula has long since stopped having any correspondence with the physical situation). The reflection case ( ) of the previous paragaph begins to illustrate the failure of the representation (that is, the failure of the formulaic solution for ) when is not large enough for an interception.

Figure 5 further clarifies the potential difficulty, and combines a physical representation (at a certain point in time) in black, and the corresponding mathematical representation (for the equation) in red.

If the black and are points in a plane, then is the angle greater than between 's line of sight to and 's velocity vector . The formula treats the same as , and since is so small, the red triangle on which the formula is based must be drawn with the acute . An acute is then chosen by the function in the formula (for a height dropped from to ), rather than the formulaically equivalent pictured as angle ). If this were translated back into the physical situation (as shown by the gray angle and vector at point ), would be heading in an apparently arbitrary direction. Magic formula (1)'s result is not to be trusted.

Consider, on the other hand, a representation of the problem when is greater than but is at least as large as , as shown in Figure 6.

Now when the red mathematical solution is translated back into the physical situation, the calculated makes sense--and it appears will intercept sometime in a little less than two ticks. Formula (1) yields a sensible for (or ) all the way up to . (When , then , and is instructed to run parallel to . Sure, won't make the interception by running parallel--but at least it's a sensible direction in which to run.)

The two illustrations in Figure 7, then, are the physical representations of the general case for non-interception.

When , then formula (1) will be undefined. When , either formula (1) will be undefined, or will be greater than , or both. In either situation, an interception is impossible according to the assumptions of the problem; but otherwise, formula (1) will give a correct interception value.

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