Mesh Integrity & Explanation

The polyhedral gravity model by Tsoulis et al. is defined using a polyhedron with OUTWARDS pointing plane unit normals. The results are negated if the plane unit normals are INWARDS pointing.

This property correlates to the sorting order of the vertices in the faces. In the context of most programs, one uses here the terminology clockwise and anti-clockwise ordering of vertices in the definition of the faces. The classification also depends on the coordinate system’s definition (right- or left-handed).

To be more illustrative and independent of the concrete coordinate system, we stick to the orientation of the normals: OUTWARDS and INWARDS pointing.

For example, given you have the following cube:

import numpy as np

CUBE_VERTICES = np.array(
  [[-1, -1, -1], [1, -1, -1], [1, 1, -1], [-1, 1, -1],
   [-1, -1, 1], [1, -1, 1], [1, 1, 1], [-1, 1, 1]]
)
CUBE_FACES_OUTWARDS = np.array(
  [[1, 3, 2], [0, 3, 1], [0, 1, 5], [0, 5, 4], [0, 7, 3], [0, 4, 7],
   [1, 2, 6], [1, 6, 5], [2, 3, 6], [3, 7, 6], [4, 5, 6], [4, 6, 7]]
)
cube_inwards = Polyhedron(
  polyhedral_source=(CUBE_VERTICES, CUBE_FACES_OUTWARDS),
  normal_orientation=NormalOrientation.OUTWARDS,
)

Its plane unit normals are pointing OUTWARDS. However, if you simply invert the order like in the following, they will point INWARDS:

CUBE_FACES_INWARDS = CUBE_FACES_OUTWARDS[:, [1, 0, 2]]
cube_outwards = Polyhedron(
  polyhedral_source=(CUBE_VERTICES, CUBE_FACES_OUTWARDS),
  normal_orientation=NormalOrientation.INWARDS,
)

And then, there is still the possibility that one can define a polyhedron by hand with mixed orientations! This would lead to entirely different (wrong) results.

So, what can we do?

The Polyhedron class checks the pointing direction of every single normal. This way, the Polyhedron ensures correct results even if a mistake occurs during the definition.

Since we are in 3D and might have concave and convex polyhedrons, the most viable option is the Möller–Trumbore intersection algorithm. It checks the amount of intersections each plane unit normal has with the polyhedron. If this is an even number, the normal is OUTWARDS pointing, otherwise INWARDS. In the current implementation, we implement a naiv version which takes \(O(n^2)\) operations - which can get quite expensive for polyhedrons with many faces.

To make this as straightforward to use, we provide four options for the construction of a polyhedron in the form of the enum PolyhedronIntegrity:

PolyhedronIntegrity	Meaning	Invalid Polyhedron Possible	Overhead
AUTOMATIC (default)	Throws an Exception if the NormalOrientation is different or inconsistent than specified AND Prints a short-version of the above explanation to stdout informing the user about \(O(n^2)\) runtime	NO	\(O(n^2)\)
VERIFY	Throws an Exception if the NormalOrientation is different or inconsistent than specified	NO	\(O(n^2)\)
HEAL	Modifies the specified NormalOrientation and the faces array to a consistent polyhedron if they are inconsistent	NO	\(O(n^2)\)
DISABLE	Disables all checks	YES	None