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…

Dirk

 

Understanding the 2×2 Rotation Matrix

When Students have taken their first Linear Algebra course, they should have been taught, that a column vector can be multiplied by a matrix, to result in a column vector. They should also have been taught, that when matrices are used to multiply a column vector more than once, to result in a final column vector, the operation proceeds from right to left, and that the matrices which do so can themselves be multiplied, as the operation is associative. This multiplication can result in one matrix, as long as the number of rows of the first (right-hand) matrix is always equal to the number of columns in the second (left-hand) matrix.

One subject which does not usually get taught in beginning Linear Algebra courses, is that when the vectors are part of the same coordinate system, the matrix is equally capable of defining a rotation. What tends to get taught first, is transformations that appear linear and parallel.

The worksheet below is intended to show, that the correct choice of elements in a matrix, can also define a rotation:

Work-Sheet

(Edit 12/30/2018, 10h30 : )

The work-sheet has been updated, also to give a hint as to how 3D, Euler Angles may be translated into a 3×3 matrix.

(Edit 1/4/2019, 8h05 : )

I have now created a version of the same work-sheet, which can be viewed on a smart-phone. Please excuse the formatting errors which result:

Work-Sheet for Phones

Dirk