Quaternary element and Matrix

The Quaternary element is the simplest super-plural element. For example, a + bi + cj + dk.

Make the p = w1 + x1 * I + y1 * j + z1 * k = w1 + v1 (Real + virtual ).

Set the QQ = w2 + x2 * I + y2 * j + z2 * k = w2 + v2 (real part + virtual part ).

Addition: p + q = (w1 + w2) + (x1 + x2) * I + (y1 + y2) * j + (z1 + z2) * k

Multiplication: p * q = (w1 + (x1 * I + y1 * j + z1 * k )) * (w2 + (x2 * I + y2 * j + z2 * k ))

= W1 * w2-v1. v2 (DOT multiplication) + v1 X v2 (Cross multiplication) + w1 * v2 + w2 * v1

 

Purpose: Replace the rotation matrix.

For example, in 3D ry, if the θ degrees of a particle wound vector (a, B, c) are required to be rotated, the following can be represented by a four-element (x, y, z, w:

Make s = sin (θ/2), c = cos (θ/2)

X = s *,

Y = s * B,

Z = s * c,

W = c

 

Conversion Between the rotation matrix and the Quaternary element:

[W2 + x2-y2-z2, 2xy-2wz, 2xz + 2wy]
[2xy + 2wz, w2-x2-y2-z2, 2yz-2wx]
[2xz-2wy, 2yz + 2wx, w2-x2-y2-z2]