An equation for all those points that have the same shortest distance to the same straight line in 3D space. Can you form an equation for a ''pipe'' in 3D space? It means all those points P(x,y,z) that have the same shortest distance for the same straight line l. For example what would the pipe equation be when we know that the line l is x-1=y-1=z-1 and when the distance between all points P(x,y,z) and the line l is 4?

The line $x-1=y-1=z-1$ is the same as $x=y=z$ so it's all the points $(t,t,t)$, $t$ real. The plane perpendicular to this line at $(t,t,t)$ is $$x+y+z=3t$$ The points in this plane at distance 4 from $(t,t,t)$ also satisfy $$(x-t)^2+(y-t)^2+(z-t)^2=16$$ Solve the first equation for $t$ in terms of $x,y,z$, then put that in the second equation, and you should have an equation for the tube.