1. Chapter 2. Geometrical Methods
- Quaternions -
Mathematical summary of quaternion algebra & calculus
Expalins how quaternions relate to rotations
전자공학/ Networked Virtual Computing Lab/ 조성업

2. Networked Virtual Computing Lab.
Quaternion Algebra - 1
Quaternion definition
q = w + xi + yj + zk
Where w, x, y, z are real numbers
i2 = j2 = k2 = -1
ij = -ji = k / jk = -kj = i / ki = -ik = j
Algebra
q0 = w0 + x0i + y0j + z0k / q1 = w1 + x1i + y1j + z1k
Addition
q0 + q1 = (w0 + w1) + ( x0 + x1 )i + ( y0 + y1 ) j + ( z0 + z1 )k
Subtraction
q0 - q1 = (w0 - w1) + ( x0 - x1 )i + ( y0 - y1 ) j + ( z0 - z1 )k
Multiplication
q0 * q1 = ( w0w1 - x0 x1 - y0y1 - z0z1 ) (w0 x1 + x0 w1 + y0z1 + z0y1) i (w0 y1 - x0 z1 + y0w1 + z0 x1) j (w0 z1 + x0 y1 - y0 x1 + z0 w1) k
교환법칙 성립 안함
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3. Networked Virtual Computing Lab.
Quaternion Algebra - 2
Conjugate of Quaternion
q*= (w + xi + yj + zk ) * = w - xi - yj – zk
( q* )*= q
( pq)*= q*p*
Norm of Quaternion
Real value
N(q) = N(w + xi + yj + zk) = w2 + x2 + y2 + z2
N(q) = N(q*)
N(pq) = N(p)N(q)
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4. Networked Virtual Computing Lab.
Quaternion Algebra - 3
Multiplicate Inverse of Quaternion
Notation: q  q-1
Property: q q-1 = q-1 q = 1
Inverse: q-1 = q* / N(q)
Quaternion을 real-value scalar로 나눌때에는 w, x, y, z 각각을 나눈다.
( q-1 )-1 = q / ( pq ) -1 = q-1 p-1
Selection Function
Select real part of Quaternion
W(q) = W(w + xi + yj + zk) = w
W(q) = ( q + q* ) / 2
Networked Virtual Computing Lab.

5. Networked Virtual Computing Lab.
Quaternion Algebra - 4
3D-Vector view of Quaternion
let v = xi + yj + zk = ( x, y, z )
q = w + v
Multiplication q0 * q1 = ( w0 + v0 )( w1 + v1 ) = ( w0 w1 - v0  v1 ) + w0 v1 + w1 v0 + v0 v1
if v0 v1 = 0 ( parallel ) then q0 * q1 = q1 * q0
4D-Vector view of Quaternion
4D-Vector = (w, x, y, z)
q0  q1 = w0w1 + x0x1 + y0y1 + z0z1 = W(q0 q1*)
Unit Quaternion
Quaternion q for which N(q) = 1
q = q-1 / qq = q
q = cos + u sin 
u = u0i + u1j +u2k = (u0 ,u1 ,u2 )
u has length 1
u u = -1
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6. Relationship of Quaternions to Rotations - 1
Unit Quaternion & Rotation
let unit quaternion q = cos + u sin 
then in R (v) = q v q* …
R (v) 는 3D vector v 를 3차원 축 u 에 대해서 2 만큼 회전시키는 함수가 된다.
R (v)가 rotation function 이라는 것을 증명하려면 다음 4가지를 증명해야만 한다.
R (v) is a 3D-Vector
R (v) is a length-preserving function
R (v) is a linear transformation
R (v) does not have an reflection component
Networked Virtual Computing Lab.

7. Relationship of Quaternions to Rotations - 2
Proof 1: R (v) is a 3D-Vector
W(R(v)) = W(q v q* ) = [(q v q* ) + (q v q* ) *] / = [(q v q* ) + q v *q* ] / = q[ (v + v *) / 2 ]q* = qW(v )q* = q  0  q* = 0
Proof 2: R (v) is a length-preserving function
N(R(v)) = N(q v q* ) = N(q)N(v )N(q*) = N(q)N(v )N(q) = 1  N(v )  1 = N(v )
Networked Virtual Computing Lab.

8. Relationship of Quaternions to Rotations - 2
Proof 3: R (v) is a linear transformation
R(av + w) = q (av + w) q* = q av q* + qw q* = a(q v q*) + qw q* = aR(v) + R(w)
Proof 4: R (v) does not have an reflection component
Let v = M(v) & v = M-1(v)
w = M-1(w) = M-1(R(v)) = M-1(R(M(v)))
w = P v
P is dependent on unit quaternion q
P(q) is an orthonormal matrix
| det(P(q)) | = +1 or –1
For all unit quaternion q, | det(P(q)) | = +1
No refelection v w
Rotation R(v)
Quaternion계 M
실수계
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9. Relationship of Quaternions to Rotations - 3
Rotation axis & Rotation angle in Quaternion- 1
q = cos + u sin  & R (v) = q v q*
Prove that in R …
3D-Vector u is the unit rotation axis
the rotation angle is 2
Proof: 3D-Vector u is the unit rotation axis
R(u) = q u q* = (cos + u sin ) u (cos - u sin ) = (cos) 2 u – (sin ) 2 u = (cos) 2 u – (sin ) 2 (-u ) = u
u 는 회전 후에도 그대로 유지되므로 회전 축이다.
u 2 = -1
Networked Virtual Computing Lab.

10. Relationship of Quaternions to Rotations - 4
Rotation axis & Rotation angle in Quaternion- 2
Proof: the rotation angle is 2
Let u, v, w be a right-handed set of orthonormal vectors All 3 vectors are unit length u  v = u  w = v  w = u  v = w / v  w = u / w  u = v
v  rotation()  q v q* : v  (q v q* ) = cos ()
For unit quaternions with zero real part p2 = -1 / v * = - v
cos () = v  (q v q* ) = W(v* q v q* ) = W[- v (cos + u sin) v (cos - u sin)] = W[(-v cos -v u sin)(v cos - v u sin)] = W[- v2(cos)2 + v2u sin cos – vu v sin cos + (uv)2(sin)2] = W[(cos)2 - (sin)2 – (u + vu v )sin cos] (uv = - v  u + v u = v u = -w / vu v = -w v = wv -w  v = u )
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11. Networked Virtual Computing Lab.
Continue…
Rotation axis & Rotation angle in Quaternion- 3
Proof: the rotation angle is 2
cos () = W[(cos)2 - (sin)2 – (u + vu v )sin cos] = W[(cos)2 - (sin)2 – u (2 sin cos) ] = (cos)2 - (sin) = cos(2)
Rotation angle is “ 2 ”
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12. Conversion Between Angle-Axis & Rotation Matrix - 1
from Angle & Axis to Rotation Matrix
 = angle of rotation / U = unit-length axis of rotation
I = Identity matrix
S =
R = I + sin S + (1 – cos)S2
0 -u2 u1
u u0
-u1 u0 0
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13. Conversion Between Angle-Axis & Rotation Matrix - 2
from Rotation Matrix to Angle & Axis
Extract rotation angle  & unit-length rotation axis U from rotation matrix R
trace(M) = sum of diagonal terms of matrix M
Some Algebra
cos = (trace(R) – 1)/2
R – RT = 2(sin)S
 = cos-1( (trace(R) – 1)/2 )  [0, ]
 = 0 : no rotation / any axis is valid
  (0, ) : 두 번째 식을 이용해 S를 구하고 U 를 구한다.
 = : R – RT = 0  U 를 구할 수 없다. 다음 식을 이용한다.
R = I + S2 =
1-2(u12 + u22) (u0u1) (u0u2)
2(u0u1) (u02 + u22) 2(u1u2)
2(u0u2) (u1u2) (u02 + u12)
Networked Virtual Computing Lab.

14. Conversion between Quaternion & Angle/Axis
from Angle & Axis to Quaternion
q = w + xi + yj + jk = cos(/2) + sin(/2)(u0i + u1j + u2k)
U = (u0, u1, u2) 축을 기준으로
(radian)만큼 회전시키는 quaternion
Rotation axis U 와 rotation angle  이 주어졌을 때의 quaternion은…
w = cos(/2) / x = u0 sin(/2) y = u1 sin(/2) / z = u2 sin(/2)
from Quaternion to Angle & Axis
|w| = 1 :  = 0 & any axis will do
|w|  1 :  = 2cos-1(w) & U = (x, y, z) /  1-w2
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15. Conversion between Quaternion & Rotation Matrix - 1
from Quaternion to Rotation Matrix
Solution: w, x, y, z  find  & U  R = I + sin S + (1 – cos)S2
Some identities
2sin2(/2) = 1- cos 
sin() = 2sin(/2)cos(/2)
From above two identities…
2wx = (sin)u0 / 2wy = (sin)u1 / 2wz = (sin)u2
2x2 = ( 1 – cos )u02 / 2xy = (1 - cos )u0u1 / 2xz = (1 - cos )u0u2
2y2 = ( 1 – cos )u12 / 2yz = (1 - cos )u1u2
2z2 = ( 1 – cos )u22
Right-hand side of all above equations are terms in R = I + sin S + (1 – cos)S2  replacing them yields…
R =
1-2y2 – 2z xy + 2wz 2xz – 2wy
2xy – 2wz x2 – 2z yz + 2wx
2xz + 2wy yz – 2wx x2 – 2y2
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16. Conversion between Quaternion & Rotation Matrix - 2
from Rotation Matrix to Quaternion
given R find w, x, y, z, & (u0, u1, u2)
Some algebra
cos = (trace(R) – 1)/2
R – RT = 2(sin)S
cos2(/2) = ( 1 + cos  )/2
Solution: from w = cos(/2) w2 = cos2(/2) = ( 1 + cos) /2 = ( 1+trace(R) )/  |w| =  trace(R)+1 / 2
Trace(R) > 0
|w|>1/2 & choose positive w =  trace(R)+1 / 2
R – RT = 2(sin)S 를 이용해 U 계산
2wx = (sin)u0 / 2wy = (sin)u1 / 2wz = (sin)u2 를 이용해 x, y, z 계산
Trace(R)  0
|w|  1/2
R =
1-2y2 – 2z xy + 2wz 2xz – 2wy
2xy – 2wz x2 – 2z yz + 2wx
2xz + 2wy yz – 2wx x2 – 2y2
Networked Virtual Computing Lab.