ParametricPlot3D with three variables

Question

ParametricPlot3D can be used with one or two variables to plot paths and surfaces in 3-D such as

ParametricPlot3D[{x + y, x y, Sin[x] + Cos[y]}, {x, 0, 1}, {y, 0, 1}]

However, say I wanted to plot a 3-D solid where the x, y and z coordinates are defined in terms of 3 variables. Is there some way to use ParametricPlot3D or other Mathematica functions to perform this task?

A simple test case could be

ParametricPlot3D[{x, y, z}, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]

which should output a cube.

Answer 1

This will be easy to break as there's a bunch of things it doesn't check for but when I wrote it I was not after good code but rather a quick solution. It should definitely get you started. It plots a ParametricPlot3D for ever two variables using the endpoints of the third (assuming it a parameter) and you can specify whether to include all of them or not.

ClearAll[myParametricPlot3D]
Options[myParametricPlot3D] = {Exclude -> {}};

myParametricPlot3D[vField_, 
  param1Range_?VectorQ /; Length[param1Range] === 3, 
  param2Range_?VectorQ /; Length[param2Range] === 3, 
  param3Range_?VectorQ /; Length[param3Range] === 3, 
  opts : OptionsPattern[{ParametricPlot3D, myParametricPlot3D}]] := 
 Module[{p1, p2, p3, goodOpts, newOpts},
  goodOpts = FilterRules[{opts}, Options[ParametricPlot3D]];
  newOpts = FilterRules[{opts}, Options[myParametricPlot3D]];
  p1 = If[! (OptionValue[Exclude]~Intersection~{3} === {}),
    {},
    Block[{a, b, c, p11, p12},
     {a, b, c} = Evaluate@param3Range;
     p11 = 
      ParametricPlot3D[Evaluate@vField /. a -> b, 
       Evaluate@param1Range, Evaluate@param2Range, Evaluate[goodOpts]];
     p12 = 
      ParametricPlot3D[Evaluate@vField /. a -> c, 
       Evaluate@param1Range, Evaluate@param2Range, Evaluate[goodOpts]];
     Show[p11, p12]
     ]];
  p2 = If[! (OptionValue[Exclude]~Intersection~{1} === {}),
    {},
    Block[{a, b, c, p11, p12},
     {a, b, c} = Evaluate@param1Range;
     p11 = 
      ParametricPlot3D[Evaluate@vField /. a -> b, 
       Evaluate@param2Range, Evaluate@param3Range, Evaluate[goodOpts]];
     p12 = 
      ParametricPlot3D[Evaluate@vField /. a -> c, 
       Evaluate@param2Range, Evaluate@param3Range, Evaluate[goodOpts]];
     Show[p11, p12]
     ]];
  p3 = If[! (OptionValue[Exclude]~Intersection~{2} === {}),
    {},
    Block[{a, b, c, p11, p12},
     {a, b, c} = Evaluate@param2Range;
     p11 = 
      ParametricPlot3D[Evaluate@vField /. a -> b, 
       Evaluate@param3Range, Evaluate@param1Range, Evaluate[goodOpts]];
     p12 = 
      ParametricPlot3D[Evaluate@vField /. a -> c, 
       Evaluate@param3Range, Evaluate@param1Range, Evaluate[goodOpts]];
     Show[p11, p12]
     ]
    ];
  If[p1 === {} && p2 === {} && p3 === {},
   $Failed,
   Show[{p1, p2, p3}]
   ]
  ]

You can use it (in your example):

myParametricPlot3D[{x, y, z}, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]

And there's an option to exclude a plot with a particular parametric dependence. So the following, doesn't plot the parametric dependence from z at 0, 1:

myParametricPlot3D[{x, Sin[y z], x z}, {x, 0, 1}, {y, -1, 2}, {z, 0, 
  1}, PlotRange -> All, MeshFunctions -> {#3 &}, 
 PlotStyle -> Opacity[0.9], Exclude -> {3}]

USE "HEAVY" OPTIONS LIKE PlotPoints WITH CAUTION! (you are applying them to six different plots)

Answer 2

Generally the plot of 2 variables define the surface in 3D space. Similarly, in the general case the plot of 3 variables define 3D space in 4D space. So, to visualize the plot of 3 variables you can use a vector field. In case of your example it looks like

 VectorPlot3D[{x, y, z}, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]