If I understand it right, the equation f(x,y)=0 represents the cross-section of the cylinder by the plane z=0. Think of that as being some sort of curve (maybe a circle) centred at the origin. Then the cross-section of the cylinder for other values of z ought to look like the same curve, but centred at a point on the line in the direction of the vector (a,b,c).

The point on this line having its third coordinate equal to z is . That makes me think that the equation of the cylinder should be . But I don't know how to give a convincing explanation of that.