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Function Repository Resource:

Monge

Source Notebook

Return the Monge point and six midplanes of a tetrahedron

Contributed by: Ed Pegg Jr

ResourceFunction["Monge"][vertices]

returns the Monge point and six midplanes of the tetrahedron defined by vertices.

Details and Options

In a tetrahedron, a midplane is perpendicular to one edge and concurrent with the midpoint of the opposing edge. Each tetrahedron has six midplanes, which intersect at the Monge point.

Examples

Basic Examples

Find the Monge point and six midplanes for a tetrahedron:

In[1]:=
ResourceFunction[
 "Monge"][{{-1, 0, 3}, {1, -1, 1}, {2, 0, -2}, {-2, -2, 1}}]
Out[1]=

A graphic of a tetrahedron, the Monge point and the six midplanes:

In[2]:=
tet = {{0, 1, -2}, {1, 3, 3}, {3, -1, 0}, {-1, 0, 0}};
monge = ResourceFunction["Monge"][tet];
Graphics3D[{Tube[#] & /@ Subsets[tet, {2}],
  Green, Sphere[monge[[1]], .2], Opacity[.6],
  Red, Polygon /@ Subsets[tet, {3}], Yellow, Opacity[.2], monge[[2]]}]
Out[4]=

Scope

The Monge point can be outside of the tetrahedron:

In[5]:=
tet = {{2, -2, 3}, {1, 0, 0}, {2, 2, 1}, {3, -1, -2}};
monge = ResourceFunction["Monge"][tet];
Graphics3D[{Tube[#] & /@ Subsets[tet, {2}],
  Green, Sphere[monge[[1]], .2], Opacity[.6],
  Red, Polygon /@ Subsets[tet, {3}], Yellow, Opacity[.2], monge[[2]]}]
Out[7]=

Requirements

Wolfram Language 11.3 (March 2018) or above

Resource History

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