Problem

The monster $M$ wants to catch the elusive tofubeast $T$.

$M$ and $T$ are two points on a plane. $T$ is moving in direction $a_{T}$ (an angle clockwise relative to $0^{\circ}$ north) at a constant velocity $v_{T}$. $a_{S}$ is the direction from $M$ to $T$ (angle of sight, relative to the same $0^{\circ}$ north), $d$ is the distance between them, and $v_{M}$ is the constant velocity of point $M$. All these are given, and shown in Figure 1. Angles $\theta$ and $\alpha$ are derived (from givens, and from the unknown direction $a_{M}$ in which $M$ will travel):

$$\theta = a_{T} - (a_{S} - 180)$$

$$\alpha = a_{S} - a_{M}$$

Figure 1: The physical problem.

Suppose a point $I$ such that if $T$ and $M$ continue at their constant speeds and directions they will intercept at time $t_{0}$. In other words, at any point in time $t$, with $\Delta t = t_{0} - t$ seconds until interception, the points can be represented as shown in Figure 2, where $r_{XY}$ is the rate at which points $X$ and $Y$ are approaching. (Notice in particular the way assumptions of the problem are captured: an extant $I$, and $\theta$ and $\alpha$ constant over time.)

Figure 2: A mathematical representation of the problem.

The problem:

1. Make an expression for $\alpha$ in terms of $v_{T}$, $v_{M}$, and $\theta$.

2. How many points $I$ are there for a given $v_{T}$, $v_{M}$, and $\theta$?