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Formula Values

Once the ambiguous case for $ \alpha $ has been eliminated by the definition of the $ \arcsin$ function, there might be zero or one values $ \alpha $ (and corresponding points $ I$) for a given $ v_{T}$, $ v_{M}$, and $ \theta $.

(If the distance between $ M$ and $ T$ 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 $ v_{M} = v_{T}$. These special cases are excluded from all the considerations below.)

Equation (1) is undefined when $ v_{M}$ is zero. If $ T$ happens to be headed directly for $ M$ with a non-zero velocity (that is, $ a_{T} = a_{S} - 180$, which means $ \theta $ is zero, and $ v_{T}$ is non-zero), then a point of interception $ I$ will exist at point $ M$. Otherwise, no point of interception will exist. In either case, the value of $ \alpha $ is irrelevant to the physical problem since point $ M$ does not move.

Equation (1) is undefined when the parameter of the $ \arcsin$ function is less than $ -1$ or greater than $ 1$. Consider the parameter of the $ \arcsin$ function: $ \sin \theta$ is a number between $ -1$ and $ 1$ which will modify the speed ratio to determine how far $ M$ has to turn from its line of sight in order to try to intercept $ T$: a faster $ T$ and a $ \theta $ close to $ 90^{\circ}$ lead to a more radical turn. If, for instance, $ T$ is headed directly toward or directly away from $ M$, then $ \theta = 0^{\circ}$ or $ 180^{\circ}$, $ \sin \theta = 0$, and the relative speeds of $ M$ and $ T$ are irrelevant to $ \alpha $: the intercept angle $ \alpha $ will be 0 (i.e. $ M$ should run directly towards $ T$ to try to intercept). If, on the other hand, $ T$ is running directly broadside to $ M$'s line of sight, $ \theta = 90^{\circ}$, then $ M$ is going to have to turn quite a ways from its line of sight to try and intercept $ T$: how far, exactly, is determined only by the ratio of their speeds, since $ \sin \theta$ will be $ 1$.

So, when the ratio of the velocities modified by the relative angle of their vectors is less than $ -1$ or greater than $ 1$, $ T$ will escape from $ M$ off to one side or the other, and no point $ I$ is defined. This doesn't describe all possible escapes of $ T$; rather, it describes the escapes where $ T$ is running obliquely--at something like a right angle--across $ M$'s line of sight. Such an escape has an understandable representation when $ \theta $ is less than $ 90^{\circ}$, as shown by $ \theta_{1}$ in Figure 4: $ T$ is moving so fast (relative to $ M$) that even though $ T$ is headed in the general direction of $ M$, $ M$ still can't close the gap, even with a maximum $ \alpha $ of $ 90^{\circ}$. (Because of the ambiguity expressed by $ \sin \theta = \sin (180 - \theta)$, a similar escape where $ \theta $ is greater than $ 90^{\circ}$ is not easily pictured--see $ \theta_{2} = (180 - \theta_{1})$ in the figure.

Figure 4: An oblique escape; $ \alpha $ undefined. The black lines show an example acute $ \theta $; the gray lines the reflected obtuse $ \theta $.
Image oblique-escape
The mathematical representation begins to have little correspondence with the physical sitatution, and it seems strange that $ M_{2}$ would flip around to the other side of $ T$. 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 $ \alpha $ 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 ( $ \theta_{2} >
90^{\circ}$) of the previous paragaph begins to illustrate the failure of the representation (that is, the failure of the formulaic solution for $ \alpha $) when $ v_{M}$ 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 $ \alpha $ equation) in red.

Figure 5: A strange pursuit by $ M$; $ \alpha $ defined but useless. The red lines show the mathematical representation; the gray lines show the calculated $ \alpha $ in the physical representation.
Image strange-escape
If the black $ M$ and $ T$ are points in a plane, then $ \theta $ is the angle greater than $ 90^{\circ}$ between $ M$'s line of sight to $ T$ and $ T$'s velocity vector $ v_{T}$. The formula treats $ \theta $ the same as $ 180 - \theta$, and since $ v_{M}$ is so small, the red triangle on which the formula is based must be drawn with the acute $ 180 - \theta$. An acute $ \alpha $ is then chosen by the $ \arcsin$ function in the formula (for a height dropped from $ I$ to $ MT$), rather than the formulaically equivalent $ 180 - \alpha$ pictured as angle $ TMI$). If this $ \alpha $ were translated back into the physical situation (as shown by the gray angle and vector at point $ M$), $ M$ 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 $ \theta $ is greater than $ 90^{\circ}$ but $ v_{M}$ is at least as large as $ v_{T}$, as shown in Figure 6.

Figure 6: A sensible $ \alpha $ with an obtuse $ \theta $. The red lines show the mathematical representation; the gray lines show the calculated $ \alpha $ in the physical representation.
Image largetheta-interception
Now when the red mathematical solution is translated back into the physical situation, the calculated $ \alpha $ makes sense--and it appears $ M$ will intercept $ T$ sometime in a little less than two ticks. Formula (1) yields a sensible $ \alpha $ for $ \theta > 90^{\circ}$ (or $ \theta < -90^{\circ}$) all the way up to $ v_{M} = v_{T}$. (When $ v_{M} = v_{T}$, then $ \alpha = 180^{\circ} -
\theta$, and $ T$ is instructed to run parallel to $ T$. Sure, $ M$ 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.

Figure 7: Two cases representing all possible escapes. On the left, $ \theta <
90^{\circ}$; on the right, $ \theta >= 90^{\circ}$.
Image escape-cases
When $ \theta <
90^{\circ}$, then formula (1) will be undefined. When $ \theta >= 90^{\circ}$, either formula (1) will be undefined, or $ v_{T}$ will be greater than $ v_{M}$, 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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