How to calculate the shortest height of an non regular Tetrahedron

Hello, I need to calculate the shortest height in a non regular tetrahedron. I have a mesh which is formed by tetrahedrons and I need to calculate the Aspect Ratio (max Edge/ short height) for all tetrahedrons that forms the mesh. I was able to find all the lengths of the edges by euclidean distance and now I need to find the shortest height between one vertice and the opposite face to him.

All tetrahedron have different aspects and sizes so I need to know mathematically how to do that. I searched the web for it, but I could just find information about the height in a regular tetrahedron.

Thanks

[Walter Roberson]

We will need you to be clearer as to which distance is the "height". Is it the distance between a vertex and the center of the base opposite?

If you were to put the above tetrahedron on a flat surface with the smallest triangle downwards (the one that is at the bottom in the diagram), then the tetrahedron will lean. Any time one of the interior angles exceeds 90 degrees, there is a placement under which the tetrahedron will lean. When it does lean, "height" can be defined in terms of the maximum z. If you were to fix the points of the base and were to pull the tip further down and away from the base, the distance between the tip and the centroid of the base would keep increasing until the tip was co-planar with the base, but the maximum z component would keep decreasing. It is thus crucial that we know which distance you intend.

[André Cintra]

Well that's a nice question. Since all the tetrahedrons are in a 3d subspace witch should have no "ground", I believe that the calculation should be about the centre of the opposite triangle of the vertex that I want to calculate the height. Since the documentation of Tetgen doesn't say anything about it I will go with that. in the web I found that the height of a regular tetrahedron in relation of it's opposite face is h = sqrt(6)/3*a but I'm not sure if it will work for an non regular Tetrahedron

Answer 1

[Torsten]

It's simply the problem of calculating the distance of a point to a plane in 3d.

Best wishes

Torsten.