Parametrisation of a line in a plane

Consider the plane x + y + z = 0.

If we wanted to draw a line that is totally embedded within this plane, we could do it by defining a position vector that points exclusively to that line.

For example, suppose we wanted to draw a circle in the plane x + y + z = 0. We could do this by defining a position vector \(r\+~\)(t) = \(TraditionalForm\`\(x(t)\)\ \(i\+ ∼ \%⋀\)\)+  \(TraditionalForm\`\(y(t)\)\ \(i\+ ∼ \%⋀\)\) +  \(TraditionalForm\`\(z(t)\)\ \(i\+ ∼ \%⋀\)\) where
x(t) = Cos (t), 
y(t) = Sin(t), and 
z(t) = -(x(t) + y(t) ) = - ( Cos(t) + Sin(t) ).

x[t_] := Cos[t];
y[t_] := Sin[t];
z[t_] := -(x[t] + y[t]);
ParametricPlot3D[{x[t], y[t], z[t]}, {t, 0, 3  π/2}, 
        ViewPoint -> {10, 20, 50}];

Change the limit on t from ((3  π)/2) to 2 π to complete the circle.

Notice that to produce this line we have only used a single parameter → t.

We know that we can parameterise a 3 dimensional surface by reducing its 3 variables to 2. So to parameterise a 3 dimensional line we reduce its 3 variables to 1.

With this in mind, to produce the surface corresponding to x + y + z = 0, we need to introduce another variable:

x1[u_, t_] := u Cos[t];
y1[u_, t_] := u Sin[t];
z1[u_, t_] :=  -(x1[u, t] + y1[u, t]);
p1  =  
        ParametricPlot3D[{x1[u, t], y1[u, t], z1[u, t]}, {u, 0.5, 1.5}, {t, 
            0, 2  π}, ViewPoint -> {10, 20, 50}];

In this parameterisation, u corresponds to the radius of the circle. So as we change the value of u we are simply drawing a new circle of a different radius. 

In the axample above, if we change the minimum value of u from 0.5 to 0, this will fill in the empty section in the centre.

 To confirm that the disk above is indeed the same surface as  x + y + z = 0, the next example superimposes the figure above with a different parameterisation for the same surface. The last two figures show the superposition from two different angles.

x2[u_, t_] := u t;
y2[u_, t_] := (u^2) t - u (t^(1/2));
z2[u_, t_] := -(x2[u, t] + y2[u, t]);
p2  =  
      ParametricPlot3D[{x2[u, t], y2[u, t], z2[u, t]}, {u, -2, 2}, {t, 0, 
          1}, ViewPoint -> {10, 20, 50}]; 
    p3 = Show[{p2, p1}, ViewPoint -> {10, 20, 50}, 
        DisplayFunction -> Identity];
p4 = 
        Show[{p2, p1}, ViewPoint -> {30, -30, 0}, 
          DisplayFunction -> Identity];
Show[GraphicsArray[{p3, p4}]];

Try some different parameterisations yourself.