# Consistent normals or boundary orientation for a simplex

I’ve got four points in 3D PGA, P_1, P_2, P_3, P_4. They are neither in one plane nor on a line nor concentrated in a single point. They can be used to describe a 3-simplex. Now, one can use these points to get a description for the boundary planes. If I know nothing, I just use a cyclic permutation of three of the four points B_1 = P_1 \vee P_2 \vee P_3, B_2 = P_2 \vee P_3 \vee P_4, B_3 = P_3 \vee P_4 \vee P_1, B_4 = P_4 \vee P_1 \vee P_2.

The problem now is, that I want to have consistent normals of those planes, but B_1,\dots, B_4 do not have consistent normals. I used the test X \wedge B_i = c_i I, where X should be an inner point of the 3-simplex and c_i is a measure for the perpendicular signed distance of X to B_i. (In fact, later I want to test, if some given point lies inside the 3-simplex.)

So with the B_i given above, the c_i have different signs (more precise: two have one sign, two have another sign, the signs alternate). Consistent outward or inward pointing normals would lead to a consistent sign. I observed that by changing the order of two points in the two offending B_i, the signs are consistent afterwards and therefore the normals (which is no surprise, since the \vee is like \wedge and changes sign, when factors are interchanged).

My questions are: Why is that, that a cyclic permutation of the points does not lead to consistent normals? How do I avoid to change factors in the B_i afterwards? Is there a general rule to get consistent oriented boundarys for a pointset describing a simplex or more general a convex polygon in arbitrary dimensions?