# Plotting spaces spanned by parameters

Suppose I have three coordinate functions of a vector of parameters: x[q], y[q], z[q] where q is at least a three-dimensional vector in some domain $Q$. Is it possible to plot in 3D the space spanned by q? By that I mean all the triplets {x',y',z'} such that there is at least one q in $Q$ such that x[q]=x', y[q]=y', z[q]=z'? Of course I would be happy if this worked with intervals (say $Q$ is the unit cube). And of course I would be happy with the surface of the subset.

I need to quickly check that this is a convex subset of $\mathbb{R}^3$. So any other method is welcome.

## 1 Answer

Assuming your three scalar functions x, y, and z have been defined, you'd do something like this:

n = 3;
q = Array[q1, n];
RegionPlot3D[
Length[FindInstance[(x @@ q == x1) && (y @@ q == y1) && (z @@ q == z1), q]] >
0, {x1, -1, 1}, {y1, -1, 1}, {z1, -1, 1}]


Here, n is the dimension of $Q$. This is all I can do without knowing how the unknowns in your question are actually defined.