The equation you gave, $ax+by+c=0$, is the equation for a line in two dimentions. In three dimensions, you can define a line by a point and a vector. This is not in contradiction with the idea that two points determine a line, as obviously there is a vector between two points and therefore by specifying two points you have also specified a point and a vector. This is probably all too pedantic to be worthwhile discussing further, so let's move on to the specific useful example.
Let one point be defined as $P=(x_p,y_p,z_p)$, and another point be defined as $Q=(x_q,y_q,z_q)$. Then we can define the line $L$ as follows:
where $t$ is any real number. To dissect this a little bit, this shows us that $P$ is in the set (for $t=0$), and $Q$ is in the set (for $t=1$). By allowing $t$ to be any real number, we are scaling the vector between $P$ and $Q$, and this is what gives us the whole line.
Now, we consider any arbitrary point on the line, which we will define as $(x,y,z)$. If this point is on the line, then it is in the set $L$ and so we must have
These equations are parametric, that is they depend on a parameter $t$, but such equations are necessary in order to describe a line in three dimensions. Obviously, if the line happened to be in one of the planes, say the $xy$ plane, then $z_p=z_q=0$, and so the equations could be solved for $t$. Substituting, you would get the familiar two-dimensional equation for a line, however in general the best you can do is solve for $t$ in the above equations and reduce the three equations to two.
It is also common to write the so-called "symmetric" equations for a line in three dimensions by solving for $t$ and setting all three equal to each other, like so:
Unfortunately, I don't think these sorts of equations will help with your robot project, but perhaps the discussion will engender some good ideas. You can take a look at Paul's Online Notes for more discussion about how lines are represented in three dimensions. He doesn't follow exactly the same approach as I have outlined here, but it is very similar.
You mentioned that the robot arm is on a circular base, and as such it might be advantageous to look into using Spherical Coordinates (which may simplify many of the computational aspects of movement).