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 $\theta$.

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.

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.

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, $\theta >= 90^{\circ}$.

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.