An oversight which I made, in an earlier posting: Matrices with Negative Determinants.

One of the subjects which I have written about a number of times, especially in This Posting, is the use of ‘rotation matrices’, and what their determinant is. This subject actually requires some understanding of Linear Algebra to be understood in turn. But it also requires just a bit more insight, into what the equations stand for.

A matrix can exist, the columns of which are mutually perpendicular – i.e., orthogonal – in addition to being unit vectors each. What I wrote was that, in such a case, the determinant of the matrix would equal (+1), and that its transpose can be used, in place of computing its inverse.

Such a matrix can be used to rotate objects that are distinctly not rectangular in appearance, but rotate them nonetheless, in computer games, CGI, etc.

A situation which I had overlooked was, that the determinant of such a matrix could also be (-1). And if it is, then to apply this matrix to a 3D system of coordinates has as effect:

  • To convert between a right-handed coordinate system and a left-handed coordinate system accurately, or
  • To derive a model that is the mirror-image of the original model.

What tends to happen in Scientific Computing, as well as in certain other areas, is that right-handed coordinate systems are often used, and left-handed coordinates less-frequently so. Yet, left-handed coordinate systems are still used. And so, if that is the case, this conversion will need to take place eventually, and no longer counts as a rotation. I.e., it has been observed that, if a right-handed helix is rotated whichever way, it stays a right-handed helix. Well, if such an orthonormal matrix with a determinant of (-1) is applied to its model coordinates, then it will become a left-handed helix…