This is a problem that a teacher told me about that's been bothering me for a while. I'm positive that this has been explored before because it seems way too useful for physicists to not have come up with a way to describe it. I just don't think I have the right name. Anyway, onto the problem.
Let's say I have a parametric position function in $\mathbb{R}^2$ defined by $(x(t),\,y(t))$ for $t\geq 0$ and $t$ is a real number. I want to define a "homing" function of sorts that we'll call $(h_x(t),\,h_y(t))$ where $h_x(0)= a$ and $h_y(0)=b$, or, that the starting point will be $(a,\,b)$. At any time $t_0$, the homing function will directly approach the point $(x(t_0),\,y(t_0))$ at a fixed speed $s$. Can $(h_x(t),\,h_y(t))$ be written in terms of $x(t)$, $y(t)$, $a$, $b$, and $s$, and if so (I'm thinking that it has to be so) then how?
You could put it more colorfully by introducing things to represent the points. For example, "A squirrel is running in a field such that at any time $t$, his position is $(x(t),\,y(t))$. A dog is chasing this squirrel at a constant speed of $s$ and is on the point $(a,\,b)$ when $t=0$. Describe the parametric position function of the dog as it chases the squirrel." I've been thinking about it for a while now but I've only gotten toes deep in it with any approach I take.