hi all, i am playing with maple at the university, and i have an idea, i was experimenting with defining the triplex algebra
through arbitrary axes, because this is essentially what the difference between various formula variants is, it was an experiment
on defining three base vectors which define the coordinate system ( left hand, right hand, arbitrary ) this was just for me experimenting
with the program and entering formulas, but i saw something ( which i could also have seen by using standard x,y,z rotation matrices, for sure
)
the point that swizzles me is that any combination of 2 rotation matrices leaves out an singularity in the matrix, namely there is one element
in the matrix that is 0
the idea now is to incorporate 3 rotation matrices into the operation, i know this sounds strange, because a point is defined explicit by
only 2 angles, but why not apply the following algorithm to the multiplication:
1. take a point <x,y,z>
2. calculate angles to ANY axis defined, so, xa,ya,za are standing for the actual angles to each axis defined by the vector
3. the multiplication would then go like:
3.1 Multiply Lengths of vectors as usual
3.2 Add the angles of all three values, xa,ya,za, from both input vectors
3.3 create rotation matrix from all three angles
4. multiply vector <length1*length2,0,0> with rotation matrix
et voila
as described above, i had a reason for this method, and the reason is the singularity implied through the 0 in the transformation matrix
first attachment below is the rotation matrix for all 3 axes, in the order of: X,Y,Z , alpha beta and gamma are the angles
second attachment below is the rotation matrix for all 3 axes, in the order of: Z,Y,X , alpha beta and gamma are the angles
so, for containing the complex plane the rotation order must be in that way, that <0,1,0>^2 yields: <-1,0,0>
wich would be a rotation around X Axis by 90 Degree ( up or down ) and an Rotation around Y Axis by 90 Degrees ( up or down ) to land on the point <-1,0,0>
i have not thought about embedding the complex plane, but this should be doable, when thinking of the desired property that <0,1,0>^2 ( complex: (0+1i )) results in <-1,0,0>
but keeping in mind that 3 rotations are done to reach that point
, perhaps the angles have to be shifted for that single purpose that <0,1,0>^2=-1
would be cool if someone could include the method in their rendering process