Planes parallel to axis planes

The following three planes correspond to x = 0.3,  y = 1,  
    and z = (π/2).

p1=ParametricPlot3D[{0.3, y, z},{y, 0, 2},{z, 0, 2},AxesLabel -> {"x", "y", "z"},DisplayFunction -> Identity];
p2=ParametricPlot3D[{x, 1, z},{x, 0, 2},{z, 0, 2},AxesLabel -> {"x", "y", "z"},DisplayFunction -> Identity];
p3 = 
        ParametricPlot3D[{x, y, (π/2)}, {x, 0, 2}, {y, 0, 2}, 
          AxesLabel -> {"x", "y", "z"}, 
          DisplayFunction -> Identity];
Show[{p1, p2, p3}, 
        DisplayFunction -> $DisplayFunction];

Spheres

\(The\ following\ equation\ in\ cartesian\ coordinates\ is\ for\ a\ sphere\ of\ radius\ r, \ 
    centred\ at\ \((x\_0, y\_0, z\_0)\) . \ 
        In\ this\ form\ it\ is\ impossible\ to\ \(\(plot\)\(.\)\)\)

r[x_, y_, 
        z_] := (Sqrt[(((x - (Subscript[x, 0])))^2) + (((y - (Subscript[y, 0])))^2) + (((z - (Subscript[z, 0])))^2)])

We can plot the sphere though if we use parametric equations. The following plot is for a sphere of radius 2 centred at the origin, whilst the plot after that is for the same radius sphere centred at ((Subscript[x, 0]), (Subscript[y, 0]), (Subscript[z, 0])).

r = 2; ParametricPlot3D[{r Cos[ϕ] Sin[θ], r Sin[ϕ] Sin[θ], r Cos[θ]}, {ϕ, 0, 2π}, {θ, 0, π}];

(Subscript[x, 0]) = -1; (Subscript[y, 0]) = -2; (Subscript[z, 0]) = π;
r = 2;  
    ParametricPlot3D[{r Sin[θ] Cos[ϕ] + (Subscript[x, 0]), 
        r Sin[θ] Sin[ϕ] + (Subscript[y, 0]), 
        r Cos[θ] + (Subscript[z, 0])}, {ϕ, 0, 2  π}, {θ, 
        0, π}];

Surfaces whose equation has a variable missing

If our equation has a variable missing, that variable can take any value.

Clear[f];
\(f[x_, y_] := x\^2
\)
p1 = 
        Plot3D[f[x, y], {x, -2, 2}, {y, -π, π}, 
          AxesLabel -> {"x", "y", "z"}, 
          DisplayFunction -> Identity];
Show[p1, DisplayFunction -> $DisplayFunction];

Axi-symmetric surfaces

Axi-symmetric surfaces are best plotted in spherical or cylindrical coordinates to take advantage of the angular symmetry. This is best illustrated by the following axi-symmetric equation first being plotted in cartesian coordinates, and then in cylindrical polar coordinates as a parametric plot.

\(\(\(f[x_, y_] := x\^2 + y\^2 + 5\)\(
\)
    \)\)
p1 = 
        Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}, 
          AxesLabel -> {"x", "y", "z"}, 
          DisplayFunction -> Identity];
p2 = 
        ParametricPlot3D[{R Cos[ϕ], R Sin[ϕ], (R^2) + 5}, {ϕ, 
            0, 2  π}, {R, 0, 1}, 
          DisplayFunction -> Identity];
Show[GraphicsArray[{p1, p2}]];

Both plots are then superimposed to prove that they are the same surface.

Show[{p1, p2}, 
        DisplayFunction -> $DisplayFunction];

Cross-Sections

A cross-section results when a plane intersects with the surface f[x,y]. The plane corresponds to x = constant, or y = constant. The example below gives three cross-sections for x = c = 0, 1, and 2; and then a single cross-section for y = k = -1. In this example, because our function is axi-symmetric, we expect the cross-sections in x to be the same as the cross-sections in y.

p1=ParametricPlot3D[{0, y, z},{y, -2.1, 2.1},{z, 4.5, 10},AxesLabel -> {"x", "y", "z"},DisplayFunction -> Identity];
p2=ParametricPlot3D[{1, y, z},{y, -2.1, 2.1},{z, 4.5, 10},AxesLabel -> {"x", "y", "z"},DisplayFunction -> Identity];
p3=ParametricPlot3D[{2, y, z},{y, -2.1, 2.1},{z, 4.5, 10},AxesLabel -> {"x", "y", "z"},DisplayFunction -> Identity];
p4=ParametricPlot3D[{x, -1, z},{x, -2.1, 2.1},{z, 4.5, 10},AxesLabel -> {"x", "y", "z"},DisplayFunction -> Identity];
p5 = 
        ParametricPlot3D[{R Cos[ϕ], R Sin[ϕ], (R^2) + 5}, {ϕ, 
            0, 2  π}, {R, 0, 2.1}, 
          DisplayFunction -> Identity];
p6  =  
        Show[{p1, p2, p3, p5}];
p7  =  
        Show[{p4, p5}];
Show[
        GraphicsArray[{p6, p7}]];

These are plots of the cross-sections for x = c = 0, 1,and 2.

For[c = 0, c < 3, 
        (Subscript[p, c]) = Plot[f[c, y], {y, -3, 3}, PlotRange -> {0, 12}, 
            DisplayFunction -> Identity, 
            AxesLabel -> {"y", "z"}]; c++];
Show[{(Subscript[p, 0]), (Subscript[p, 1]), (Subscript[p, 2])}, 
        DisplayFunction -> $DisplayFunction];