On a 2D plane, a transform containing a mirror reflection will change a directed acyclic graph from clockwise to anti-clockwise. In math this can be done that given a triangle ABC, we can check which direction the cross product $\vec{AB} \times \vec{BC}$ is pointing to, e.g. if the plane ABC is on $x-y$ space, sign of $\vec{AB} \times \vec{BC}$ on $z$ direction can serve as the criterion.

How about in the 3D space? Given a transform $\tau$ which transformed cube $ABCDEFGH$ into $A'B'C'D'E'F'G'H'$, or a regular tetrahedron $ABCD$ into $A'B'C'D'$, is there a way to check if it's gone through a mirror reflection?

How is the case in 4D or even higher space?


I prefer thinking about tetrahedron in 3d as the counterpart to the 2d triangle. Doesn't have to be a regular one. This post discusses how to compute the oriented volume of such a tetrahedron. If applying your transformation changes the sign of that volume, you can say it has changed orientation.

For linear transformations, you can detect that the transformation reverses orientation by finding that the determinant of the transformation matrix has negative sign. The same is true for an affine transformation, if you compute the determinant only for the linear part of the transformation, ignoring the translation part.

Some other classes, projective transformations for example, might change the orientation for parts of the space but not everywhere. But I guess that is not the class of transformations you had in mind.


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