quaternions.txt

EXPANDED FORMS OF QUATERNION PRODUCTS
HAMILTON'S

with
s1=w
s2=w'
v1=[x,y,z]
v2=[x',y',z']

qp = [s1,v1][s2,v2] = [s1s2 - v1.v2, s1v2 + s2v1 + v1 x v2]

(w1 * w2) - x1 * x2 - y1 * y2 - z1 * z2
(w1 * x2) + (x1 * w2) + (y1 * z2 - z1 * y2)
(w1 * y2) + (y1 * w2) + (z1 * x2 - x1 * z2)
(w1 * z2) + (z1 * w2) + (x1 * y2 - y1 * x2)

w1 * w2 - x1 * x2 - y1 * y2 - z1 * z2
w1 * x2 + x1 * w2 + y1 * z2 - z1 * y2
w1 * y2 - x1 * z2 + y1 * w2 + z1 * x2
w1 * z2 + x1 * y2 - y1 * x2 + z1 * w2

w" = ww' - xx' - yy` - zz`
x" = wx' + xw' + yz` - zy`
y" = wy' - xz' + yw` + zx`
z" = wz' + xy' - yx` + zw`

SHUSTER'S

qp = [s1,v1][s2,v2] = [s1s2 - v1.v2, s1v2 + s2v1 - v1 x v2]

(w1 * w2) - x1 * x2 - y1 * y2 - z1 * z2
(w1 * x2) + (x1 * w2) - (y1 * z2 - z1 * y2)
(w1 * y2) + (y1 * w2) - (z1 * x2 - x1 * z2)
(w1 * z2) + (z1 * w2) - (x1 * y2 - y1 * x2)

w1 * w2 - x1 * x2 - y1 * y2 - z1 * z2
w1 * x2 + x1 * w2 - y1 * z2 + z1 * y2
w1 * y2 + y1 * w2 - z1 * x2 + x1 * z2
w1 * z2 + z1 * w2 - x1 * y2 + y1 * x2

w1 * w2 - x1 * x2 - y1 * y2 - z1 * z2
w1 * x2 + x1 * w2 - y1 * z2 + z1 * y2
w1 * y2 + x1 * z2 + y1 * w2 - z1 * x2
w1 * z2 - x1 * y2 + y1 * x2 + z1 * w2

w" = ww' - xx' - yy' - zz'
x" = wx' + xw' - yz' + zy'
y" = wy' + xz' + yw' - zx'
z" = wz' - xy' + yx' + zw'

TWO FORMS OF QUATERNION ROTATIONS

since (pq)^-1 = (q^-1)(p^-1)

v' = q^-1 v q -> q1q2  ? (q1q2)^-1 v (q1q2) = q2^-1 q1^-1 v q1 q2 = q1 followed by q2
v' = q v q^-1 -> q2q1  ? (q2q1) v (q2q1)^-1 = q2 q1 v q1^-1 q2^-1 = q1 followed by q2

DERIVING ROTATION MATRICES EQUIVALENT TO QUATERNION ROTATIONS
HAMILTON

[s1,v1][s2,v2] = [s1s2 - v1.v2, s1v2 + s2v1 + v1 X v2]

|s1 -v1x    -v1y    -v1z|   |s2 |   |s1s2-v1.v2 |
|0  0       0       0   |   |v2x| = |   0       | = [s1s2-v1.v2,0]
|0  0       0       0   |   |v2y|   |   0       |
|0  0       0       0   |   |v2z|   |   0       |

|0  0   0   0   |   |s2 |   |   0   |
|0  s1  0   0   |   |v2x| = |s1*v2x | = [0,s1*v2]
|0  0   s1  0   |   |v2y|   |s1*v2y |
|0  0   0   s1  |   |v2z|   |s1*v2z |

|0      0   0   0   |   |s2 |   |   0   |
|v1x    0   0   0   |   |v2x| = |s2*v1x | = [0,s2*v1]
|v1y    0   0   0   |   |v2y|   |s2*v1y |
|v1z    0   0   0   |   |v2z|   |s2*v1z |


|0      0       0       0   |   |s2 |   |       0       |
|0      0       -v1z    v1y |   |v2x| = |v1y*v2z-v1z*v2y| = [0,v1 X v2]
|0      v1z     0       -v1x|   |v2y|   |v1z*v2x-v1x*v2z|
|0      -v1y    v1x     0   |   |v2z|   |v1x*v2y-v1y*v2x|

|s1     -v1x    -v1y    -v1z|   |s2 |   
|v1x    s1      -v1z    v1y |   |v2x| = [s1s2 - v1.v2, s1v2 + s2v1 + v1 X v2] = premult
|v1y    v1z     s1      -v1x|   |v2y|   
|v1z    -v1y    v1x     s1  |   |v2z|   

|s1     v1x     v1y     v1z |   |s2 |   
|-v1x   s1      v1z     -v1y|   |v2x| = [s1s2 + v1.v2, s1v2 - s2v1 - v1 X v2] = conjugated premult
|-v1y   -v1z    s1      v1x |   |v2y|   
|-v1z   v1y     -v1x    s1  |   |v2z|   

|s1     -v1x    -v1y    -v1z|   |s2 |   
|v1x    s1      v1z     -v1y|   |v2x| = [s1s2 - v1.v2, s1v2 + s2v1 + v2 X v1] = postmult
|v1y    -v1z    s1      v1x |   |v2y|   
|v1z    v1y     -v1x    s1  |   |v2z|   

|s1     v1x     v1y     v1z |   |s2 |
|-v1x   s1      -v1z    v1y |   |v2x| = [s1s2 + v1.v2, s1v2 - s2v1 - v2 X v1] = conjugated postmult
|-v1y   v1z     s1      -v1x|   |v2y|
|-v1z   -v1y    v1x     s1  |   |v2z|

q*          =   {{w,-x,-y,-z},{x,w,-z,y},{y,z,w,-x},{z,-y,x,w}}
(q^-1)*     =   {{w,x,y,z},{-x,w,z,-y},{-y,-z,w,x},{-z,y,-x,w}}
*q          =   {{w,-x,-y,-z},{x,w,z,-y},{y,-z,w,x},{z,y,-x,w}}
*(q^-1)     =   {{w,x,y,z},{-x,w,-z,y},{-y,z,w,-x},{-z,-y,x,w}}
q*v*(q^-1)  =   {{w,-x,-y,-z},{x,w,-z,y},{y,z,w,-x},{z,-y,x,w}}*{{w,x,y,z},{-x,w,-z,y},{-y,z,w,-x},{-z,-y,x,w}}
(q^-1)*v*q  =   {{w,x,y,z},{-x,w,z,-y},{-y,-z,w,x},{-z,y,-x,w}}*{{w,-x,-y,-z},{x,w,z,-y},{y,-z,w,x},{z,y,-x,w}}

q*v*(q^-1)  =
= {{w^2 + x^2 + y^2 + z^2, 0, 0, 0}, {0, w^2 + x^2 - y^2 - z^2, 2 x y - 2 w z, 2 w y + 2 x z}, {0, 2 x y + 2 w z, w^2 - x^2 + y^2 - z^2, -2 w x + 2 y z}, {0, -2 w y + 2 x z, 2 w x + 2 y z, w^2 - x^2 - y^2 + z^2}}
-> {{w^2 + x^2 - y^2 - z^2, 2 x y - 2 w z, 2 w y + 2 x z}, {2 x y + 2 w z, w^2 - x^2 + y^2 - z^2, -2 w x + 2 y z}, {-2 w y + 2 x z, 2 w x + 2 y z, w^2 - x^2 - y^2 + z^2}}
=
|w²+x²-y²-z²    2xy-2wz         2wy+2xz     |
|2xy+2wz        w²-x²+y²-z²     2yz-2wx     |
|2xz-2wy        2wx+2yz         w²-x²-y²+z² |
=
|1-2(y²+z²)     2(xy-wz)        2(wy+xz)    |
|2(xy+wz)       1-2(x²+z²)      2(yz-wx)    |
|2(xz-wy)       2(wx+yz)        1-2(x²+y²)  |

Above is one of a few possible simplifications:
M11 = w²+x²-y²-z² = 2w²+2x²-1 = 2(w²+x²)-1 = 1-2y²-2z² = 1-2(y²+z²)
M22 = w²-x²+y²-z² = 2w²+2y²-1 = 2(w²+y²)-1 = 1-2x²-2z² = 1-2(x²+z²)
M33 = w²-x²-y²+z² = 2w²+2z²-1 = 2(w²+z²)-1 = 1-2x²-2y² = 1-2(x²+y²)

(q^-1)*v*q  =
{{w^2 + x^2 + y^2 + z^2, 0, 0, 0}, {0, w^2 + x^2 - y^2 - z^2, 2 x y + 2 w z, -2 w y + 2 x z}, {0, 2 x y - 2 w z, w^2 - x^2 + y^2 - z^2, 2 w x + 2 y z}, {0, 2 w y + 2 x z, -2 w x + 2 y z, w^2 - x^2 - y^2 + z^2}}
-> {{w^2 + x^2 - y^2 - z^2, 2 x y + 2 w z, -2 w y + 2 x z}, {2 x y - 2 w z, w^2 - x^2 + y^2 - z^2, 2 w x + 2 y z}, {2 w y + 2 x z, -2 w x + 2 y z, w^2 - x^2 - y^2 + z^2}}
=
|w²+x²-y²-z²    2xy+2wz         2xz-2wy     |
|2xy-2wz        w²-x²+y²-z²     2wx+2yz     |
|2wy+2xz        2yz-2wx         w²-x²-y²+z² |
=
|1-2(y²+z²)     2(xy+wz)        2(xz-wy)    |
|2(xy-wz)       1-2(x²+z²)      2(wx+yz)    |
|2(wy+xz)       2(yz-wx)        1-2(x²+y²)  |

SHUSTER

|s1     -v1x    -v1y    -v1z|   |s2 |   
|v1x    s1      v1z     -v1y|   |v2x| = [s1s2 - v1.v2, s1v2 + s2v1 - v1 X v2] =  premult
|v1y    -v1z    s1      v1x |   |v2y|   
|v1z    v1y     -v1x    s1  |   |v2z|   

|s1     v1x     v1y     v1z |   |s2 |
|-v1x   s1      -v1z    v1y |   |v2x| = [s1s2 + v1.v2, s1v2 - s2v1 + v1 X v2] = conjugated premult
|-v1y   v1z     s1      -v1x|   |v2y|
|-v1z   -v1y    v1x     s1  |   |v2z|

|s1     -v1x    -v1y    -v1z|   |s2 |   
|v1x    s1      -v1z    v1y |   |v2x| = [s1s2 - v1.v2, s1v2 + s2v1 - v2 X v1] =  postmult
|v1y    v1z     s1      -v1x|   |v2y|   
|v1z    -v1y    v1x     s1  |   |v2z|   

|s1     v1x     v1y     v1z |   |s2 |   
|-v1x   s1      v1z     -v1y|   |v2x| = [s1s2 + v1.v2, s1v2 - s2v1 + v2 X v1] = conjugated postmult
|-v1y   -v1z    s1      v1x |   |v2y|   
|-v1z   v1y     -v1x    s1  |   |v2z|   

q*v*(q^-1)  =
{{w^2 + x^2 + y^2 + z^2, 0, 0, 0}, {0, w^2 + x^2 - y^2 - z^2, 2 x y + 2 w z, -2 w y + 2 x z}, {0, 2 x y - 2 w z, w^2 - x^2 + y^2 - z^2, 2 w x + 2 y z}, {0, 2 w y + 2 x z, -2 w x + 2 y z, w^2 - x^2 - y^2 + z^2}}
-> {{w^2 + x^2 - y^2 - z^2, 2 x y + 2 w z, -2 w y + 2 x z}, {2 x y - 2 w z, w^2 - x^2 + y^2 - z^2, 2 w x + 2 y z}, {2 w y + 2 x z, -2 w x + 2 y z, w^2 - x^2 - y^2 + z^2}}
=
|w²+x²-y²-z²    2xy+2wz         2xz-2wy     |
|2xy-2wz        w²-x²+y²-z²     2wx+2yz     |
|2wy+2xz        2yz-2wx         w²-x²-y²+z² |
=
|1-2(y²+z²)     2(xy+wz)        2(xz-wy)    |
|2(xy-wz)       1-2(x²+z²)      2(wx+yz)    |
|2(wy+xz)       2(yz-wx)        1-2(x²+y²)  |

(q^-1)*v*q  =
= {{w^2 + x^2 + y^2 + z^2, 0, 0, 0}, {0, w^2 + x^2 - y^2 - z^2, 2 x y - 2 w z, 2 w y + 2 x z}, {0, 2 x y + 2 w z, w^2 - x^2 + y^2 - z^2, -2 w x + 2 y z}, {0, -2 w y + 2 x z, 2 w x + 2 y z, w^2 - x^2 - y^2 + z^2}}
-> {{w^2 + x^2 - y^2 - z^2, 2 x y - 2 w z, 2 w y + 2 x z}, {2 x y + 2 w z, w^2 - x^2 + y^2 - z^2, -2 w x + 2 y z}, {-2 w y + 2 x z, 2 w x + 2 y z, w^2 - x^2 - y^2 + z^2}}
=
|w²+x²-y²-z²    2xy-2wz         2wy+2xz     |
|2xy+2wz        w²-x²+y²-z²     2yz-2wx     |
|2xz-2wy        2wx+2yz         w²-x²-y²+z² |
=
|1-2(y²+z²)     2(xy-wz)        2(wy+xz)    |
|2(xy+wz)       1-2(x²+z²)      2(yz-wx)    |
|2(xz-wy)       2(wx+yz)        1-2(x²+y²)  |

DERIVING A ROTATION MATRIX OF P AROUND AXIS R BY ANGLE ALPHA

z = r(r.p)
x = p - z
y = r X x = r X p - r X z = r X p - r X r(r.p) = r X p - 0 = r X p

p' = z + cos(alpha)x + sin(alpha)y

p'= Mz*p + cos(alpha)Mx*p + sin(alpha)My*p = (Mz + cos(alpha)Mx + sin(alpha)My)p = Mp*p

        |   x   0   0   |   |   x   y   z   |   |   x²  xy  xz  |
Mz  =   |   0   y  0    |   |   x   y   z   | = |   xy  y²  yz  |
        |   0   0   z   |   |   x   y   z   |   |   xz  yz  z²  |

{{x^2,x y,x z},{x y,y^2,y z},{x z,y z,z^2}}

        |   1-x²    1-xy    1-xz    |
Mx  =   |   1-xy    1-y²    1-yz    |
        |   1-xz    1-yz    1-z²    |

{{1-x^2,1-x y,1-x z},{1-x y,1-y^2,1-y z},{1-x z,1-y z,1-z^2}}

        |   0   -z  y   |
My  =   |   z   0   -x  |
        |   -y  x   0   |

with C=cos(alpha) and S=sin(alpha)

    |   x²+C-Cx²+S0     xy+C-Cxy-Sz     xz+C-Cxz+Sy |
Mp= |   xy+C-Cxy+Sz     y²+C-Cy²+S0     yz+C-Cyz-Sy |
    |   xz+C-Cxz-Sy     yz+C-Cyz+Sx     z²+C-Cz²+S0 |
    
    |   x²+C-Cx²        xy+C-Cxy-Sz     xz+C-Cxz+Sy |
Mp= |   xy+C-Cxy+Sz     y²+C-Cy²        yz+C-Cyz-Sy |
    |   xz+C-Cxz-Sy     yz+C-Cyz+Sx     z²+C-Cz²    |

    |   C+x²(1-C)       xy(1-C)-Sz      xz(1-C)+Sy  |
Mp= |   xy(1-C)+Sz      C+y²(1-C)       yz(1-C)-Sy  |
    |   xz(1-C)-Sy      yz(1-C)+Sx      C+z²(1-C)   |

DERIVING A QUATERNION FOR A ROTATION AROUND AN AXIS BY AN ANGLE

From the quaternion matrix:

              M11 + M22 + M33 = T = 1-2(y²+z²) + 1-2(x²+z²) + 1-2(x²+y²)
                                T = 3-2(2y²+2z²+2x²)
                                T = 3-4y²-4z²-4x²
by using w²+x²+y²+z²=1:       T+4 = 3+4w²
                              T+1 = 4w²
                          (T+1)/4 = w²

From the rotation matrix
(on the lhs x,y,z are those of the axis of rotation not those of the quaternion):

            (C+x²(1-C) + C+y²(1-C) + C+z²(1-C) + 1)/4 = w²
                         (3C + (x²+y²+z²)(1-C) + 1)/4 = w²
by using x²+y²+z²=1:                 (3C + 1-C + 1)/4 = w²
                                           (2C + 2)/4 = w²
                                            (C + 1)/2 = w²
                                      sqrt((C + 1)/2) = w
cos(alpha/2)=sqrt((cos(alpha)+1)/2):     cos(alpha/2) = w

Finding remaining quaternion components
(lhs x,y,z are those of the quaternion while those in rhs are those of the axis of rotation):

                    M11 = 2w²+2x²-1 = C+x²(1-C)         M22 = 2w²+2y²-1 = C+y²(1-C)         M33 = 2w²+2z²-1 = C+z²(1-C)
                        (C+1)+2x²-1 = C+x²(1-C)             (C+1)+2y²-1 = C+y²(1-C)             (C+1)+2z²-1 = C+z²(1-C)
                              C+2x² = C+x²(1-C)                   C+2y² = C+y²(1-C)                   C+2z² = C+z²(1-C)
                                2x² = x²(1-C)                       2y² = y²(1-C)                       2z² = z²(1-C)
                                  x = sqrt(x²(1-C))/2                 y = sqrt(y²(1-C))/2                 z = sqrt(z²(1-C))
                                  x = x sqrt(1-C)/2                   y = y sqrt(1-C)/2                   z = z sqrt(1-C)/2
sqrt(1-cos alpha)=sin(alpha/2):   x = sin(alpha/2) x                  y = sin(alpha/2) y                  z = sin(alpha/2) z

Therefore, a rotation around r by angle alpha is described by the quaternion
 Q = [cos(alpha/2), r sin(alpha/2)]

PROOF OF TRIG IDENTITIES USED

(cos alpha)² + (sin alpha)² = 1
(cos alpha)² + (sin alpha)² - 1 = 0

(cos alpha)² + (sin alpha)² = 1
1 - (cos alpha)² - (sin alpha)² = 0

cos(alpha + beta) = cos alpha cos beta - sin alpha sin beta

cos(2 alpha) = cos(alpha + alpha)
cos(2 alpha) = cos alpha cos alpha - sin alpha sin alpha
cos(2 alpha) = (cos alpha)² - (sin alpha)²

cos(2 alpha) = (cos alpha)² - (sin alpha)²
cos(2 alpha) = (cos alpha)² - (sin alpha)² + (cos alpha)² + (sin alpha)² - 1
cos(2 alpha) = 2(cos alpha)² - 1
2(cos alpha)² - 1 = cos(2 alpha)
2(cos alpha)² = cos(2 alpha) + 1
(cos alpha)² = (cos(2 alpha) + 1)/2
cos(alpha) = sqrt((cos(2 alpha) + 1)/2)
cos(alpha/2) = sqrt((cos(alpha) + 1)/2)

cos(2 alpha) = (cos alpha)² - (sin alpha)²
cos(2 alpha) = (cos alpha)² - (sin alpha)² - (cos alpha)² - (sin alpha)² + 1
cos(2 alpha) = 1-2(sin alpha)²
cos(alpha) = 1-2(sin(alpha/2))²
2(sin(alpha/2))² = 1 - cos(alpha)
(sin(alpha/2))² = (1 - cos(alpha))/2
sin(alpha/2) = sqrt((1 - cos(alpha))/2)