There are 8 octants of the computational sphere that correspond to the 8 combinations of
signs +/- of the dot products of an arbitrary axis and the 3 canonical axes X, Y, and Z:
The previous page shows the use of a circuit for axes within the first octant, where all the dot products are positive. The math for the other octants is shown below. There are 8 generic operator types Ai, for i = 1 to 8, where |x| is the magnitude of x, etc..:
dotX dotY dotZ A1(a, b) = ( x * b + y * (-i * b) + z * a, x * a + y * (+i * a) + z * (-b) ) + + + A2(a, b) = ( x * b + y * (-i * b) + |z| * (-a), x * a + y * (+i * a) + |z| * b ) + + - A3(a, b) = ( x * b + |y| * (+i * b) + z * a, x * a + |y| * (-i * a) + z * (-b) ) + - + A4(a, b) = ( x * b + |y| * (+i * b) + |z| * (-a), x * a + |y| * (-i * a) + |z| * b ) + - - A5(a, b) = ( |x| * (-b) + y * (-i * b) + z * a, |x| * (-a) + y * (+i * a) + z * (-b) ) - + + A6(a, b) = ( |x| * (-b) + y * (-i * b) + |z| * (-a), |x| * (-a) + y * (+i * a) + |z| * b ) - + - A7(a, b) = ( |x| * (-b) + |y| * (+i * b) + z * a, |x| * (-a) + |y| * (-i * a) + z * (-b) ) - - + A8(a, b) = ( |x| * (-b) + |y| * (+i * b) + |z| * (-a), |x| * (-a) + |y| * (-i * a) + |z| * b ) - - -
Due to symmetry, octants with opposite signs contain equivalent operators, so that A1 ≈ A8, A2 ≈ A7, A3 ≈ A6, and A4 ≈ A5, and there are only 4 different operator types and circuits: A1 to A4, covering a hemisphere centered on the positive X axis. The 4 circuits use the same components (+/-i phase filters, an inverter, and summation), but differ in which inputs go to which components.
The circuit for operators in the second octant (for z < 0) is shown below:
Circuits for the other two octants are left as exercises for the reader. They involve swapping the +i and -i phase filters.
Below are tests of this circuit on oriented figures (green = input, blue = output) and
random points for the resistor combinations RX = 100 K, RY = 100 K, RZ = 50 K. The left
images show the measured operator axes (yellow), and the right show the registration of
calculated points with actual outputs. Only the average axis (white) is shown in the
The measured axis is within 0.000 degrees of the true axis, and the average point-by-point error is 0.000 degrees. This demonstration suggests that any operator axis in this octant can be produced by this circuit, and that axes in the other two octants can be produced by their corresponding circuits as well. Thus, any arbitrary hermitian operator, with its axis anywhere on the computational sphere, can be calculated. The next section will demonstrate how combinations of hermitian operators can be used to create any rotation, small or large, with its axis anywhere on the computational sphere. These are unitary operators.