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    "## Quaternion\n",
    "Quaternion is a adding operator Group. It isn't a Field and the missing part is commutative of multiplication.\n",
    "\n",
    "#### Format\n",
    "$$\n",
    "q = a + bi + cj + dk\n",
    "$$\n",
    "\n",
    "#### Algebra\n",
    "$$\n",
    "i^2 = j^2 = k^2 = −1\\\\\n",
    "ij = k, ji = −k\\\\\n",
    "jk = i, kj = − i\\\\\n",
    "ki = j, ik = − j\n",
    "$$\n",
    "\n",
    "#### Conjugate\n",
    "$$q^* = a - bi - cj - dk$$\n",
    "\n",
    "\n",
    "#### Norm\n",
    "$$\\|q\\|=\\sqrt{a^2 + b^2 + c^2 + d^2}$$\n",
    "\n",
    "#### Inverse\n",
    "$$q^{-1} = \\frac {q^*} {\\|q\\|^2} = \\frac {a - bi - cj - dk} {a^2 + b^2 + c^2 + d^2}$$\n",
    "$$ qq^{−1} = q^{−1}q = 1$$\n",
    "\n",
    "#### Identity\n",
    "For adding group, the identity element is real number $0$.\n",
    "For multiplication, the identity element is real number $1$.\n",
    "\n",
    "\n",
    "#### Associativity \n",
    " The associativity holds for every $x, y, z \\in G$ in both adding and multiplication:\n",
    " $$x \\ast (y \\ast z) = (x \\ast y) \\ast z$$\n",
    "\n",
    "#### Distributive\n",
    "$$ p(q + r) = pq + pr$$ \n",
    "\n",
    "#### Commutative\n",
    "Only true for adding\n",
    "$$ p + q = q + p $$\n",
    "\n",
    "Not true for multiplication.\n",
    "$$pq \\neq qp$$\n",
    "\n",
    "DO NOT USE $p/q$ since it isn't clear whether it is $pq^{-1}$ or $q^{-1}q$. \n",
    " \n",
    " ## Quaternions for rotations\n",
    "\n",
    "#### Vector\n",
    " A vector in three-dimensional space can be expressed as a **pure quaternion**, a quaternion with no real part: $q = 0 + bi + cj + dk$. \n",
    "\n",
    "#### Rotation Matrix\n",
    "A rotation can be expressed by a quaternion $q$ if \n",
    "$$\\|q\\| = 1$$\n",
    "\n",
    "#### From  quaternion to a rotation matrix\n",
    "A rotation from one coordinate frame $A$ to another frame $B$ is given by the conjugation operation of transform quaternion $q_R$ :\n",
    "$$q_B = q_Rq_Aq^*_R$$\n",
    "\n",
    "Assuming $q_A$ represent a vector or a point in $R^3$, we have\n",
    "$$\n",
    "q_A = xi + yj + zk \\\\\n",
    "q_R = a + bi + cj + dk\n",
    "$$\n",
    "The transform result is\n",
    "$$\n",
    "\\begin{split} q_B =  q_Rq_Aq^*_R\n",
    "&= (a + bi + cj + dk)( xi + yj + zk )(a − bi − cj − dk)\\\\\n",
    "&= [x(a^2 + b^2 − c^2 − d^2) + 2y(bc − ad) + 2z(ac + bd)]i\\\\\n",
    "&+ [2x(ad + bc) + y(a^2 − b^2 + c^2 − d^2) + 2z(cd − ab)]j\\\\\n",
    "&+ [2x(bd − ac) + 2y(ab + cd) + z(a^2 − b^2 − c^2 + d^2)]k\n",
    "\\end{split}$$\n",
    "\n",
    "If we express this in a matrix formulation\n",
    "$$ q_B = Mq_A$$\n",
    "in which \n",
    "$$\n",
    "M = \n",
    "\\begin{bmatrix}\n",
    "a^2 + b^2 − c^2 − d^2 & 2(bc − ad) & 2(ac + bd) \\\\\n",
    "2(ad + bc) & a^2 − b^2 + c^2 − d^2 &  2(cd − ab) \\\\\n",
    "2(bd - ac) & 2(ab + cd) & a^2 − b^2 − c^2 + d^2\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "#### From a rotation matrix of angles to a quaternion\n",
    "\n",
    "$$\n",
    "\\begin{split} Trace(M)\n",
    "& = M_{11} + M_{22} + M_{33} \\\\\n",
    "& = 2(3a^2 + b^2 + c^2 + d^2 - 1.5) \\\\\n",
    "& = 2(3a^2 + (1 - a^2) - 1.5 \\\\\n",
    "& = 2(2a^2 - 0.5) \\\\\n",
    "& = 4a^2 - 1\n",
    "\\end{split} $$\n",
    "\n",
    "We can solve this equation for $a$:\n",
    "$$|a| = \\sqrt\\frac{Trace(M) + 1} 4$$\n",
    "\n",
    "Once we have $a$, we can obtain $b$ from $M_{11}$\n",
    "\n",
    "$$M_{11} = 2(a^2 + b^2 - 0.5) = 2(\\frac{Trace(M) + 1}4 + b^2 - 0.5)$$\n",
    "\n",
    "$$|b| = \\sqrt{\\frac{M_{11}} 2 + \\frac {1 - Trace(M)} 4}$$\n",
    "\n",
    "Similarly, $c$ and $d$ can be computed from $M_{22}$ and $M_{33}$.\n",
    "$$|c| = \\sqrt{\\frac{M_{22}} 2 + \\frac {1 - Trace(M)} 4}$$\n",
    "$$|d| = \\sqrt{\\frac{M_{33}} 2 + \\frac {1 - Trace(M)} 4}$$\n",
    "\n",
    "#### From Euler angles to a quaternion\n",
    "The general logic is simple \n",
    "1. From Euler angels, construct its rotation matrix.\n",
    "2. Convert rotation matrix to quaternion."
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