Plotting a manifold within an existing surface

I have a dynamical system $\textbf{x}'=\textbf{f(x)}$ where $x\in \mathbb{R}^4$. However, $x_1 +x_2 +x_3 + x_4 =1$ and $0\leq x_i \leq 1$ for $i = 1,2,3,4$, meaning the system lives on the Tetrahedron.

I am trying to plot solutions to this system and the manifold implicitly given by $x_1 x_4 - x_2 x_3 =0$. This is because solutions to my system lay close to this manifold and this is a result of interest.

So far, I have been plotting on the Tetrahedron using a crude projection matrix:

$B = \left( \begin{array}{cccc} 0 & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{\sqrt{\frac{3}{2}}}{4}+\frac{1}{4 \sqrt{6}} & \frac{\sqrt{\frac{3}{2}}}{4}+\frac{1}{4 \sqrt{6}} & -\frac{\sqrt{\frac{3}{2}}}{2}-\frac{1}{2 \sqrt{6}} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2 \sqrt{3}} & -\frac{1}{2 \sqrt{3}} & -\frac{1}{2 \sqrt{3}} \\ \end{array} \right)$

I have been numerically solving my system to find a single orbit, $x(t)$. Then, by performing $B.x(t)$, I am able to visualise the solution on the simplex using:

Show[Line[B.x[t]],Line[B.{1,0,0,0},B.{0,1,0,0},etc]]

to plot both the solution and the edges of the tetrahedron.

However, I have recently discovered

ImplicitRegion[], DiscretizeRegion[]

and other similar functions. This has led me to believe there is a far better and more proper way of getting these plots. So far I have been able to see the $x_1 x_4 - x_2 x_3 =0$ manifold by using

R = DiscretizeRegion[ImplicitRegion[x1 (1-x1-x2-x3) - x2 x3 ==0, {{x1,0,1},{x2,0,1},{x3,0,1}}]]

However, when I attempt to combine this with the regular tetrahedron, by using

Show[Graphics3D[Tetrahedron[],R]

The result is a god awful mess. This is clearly because the exact projections being used are different, but I am unable to figure out how to fix this.

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• Related: (23565) – Silvia May 13 '16 at 6:04

You can combine Graphics3D and the MeshRegion using their MeshPrimitives. For example,

Show[Graphics3D[{Blue, Tetrahedron[]}], Graphics3D[{Red, MeshPrimitives[R, 1]}]] or

Show[Graphics3D[{Opacity[0.3], Tetrahedron[]}],
Graphics3D[{Red, MeshPrimitives[R, 1]}] /. Line -> Tube,
Boxed -> False] 