Quaternion¶

Definition¶

A quaternion ${\bf q}$ is represented as a 4-tuple $(a,b,c,d)$ , with basis $\{1,i,j,k\}$ written as

(1) ${\bf q} = (a,b,c,d) = a + b\ i + c\ j + d\ k.$

The basis elements have multiplication property

$i^2 = j^2 = k^2 = ijk = -1, \\ ij = k,\ jk = i,\ ki = j, \\ ji = -k,\ kj = -i,\ ik = -j.$

The Hamilton product of two general quaternion is

(2) $( & a_1, b_1, c_1, d_1)(a_2, b_2, c_2, d_2) \\ = ( & a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2, \\ & a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2, \\ & a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2, \\ & a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2).$

A quaternion can be divided into a scalar part and a vector part

${\bf q} = (r, {\bf v}),\quad \textrm{with}\ r\in\mathbb{R}, {\bf v}\in\mathbb{R}^3.$

We also consider scalar $r$ and 3-vector ${\bf v}$ as special forms of quaternion

${\bf q}_r = (r, {\bf 0}),\quad {\bf q}_{\bf v} = (0, {\bf v}),$

and write ${\bf q}_r$ and $r$ ( ${\bf q}_{\bf v}$ and ${\bf v}$ ) interchangably in this note.

For quaternion ${\bf q}$ defined in (1), its conjugate is

${\bf q}^* = a - b\ i - c\ j - d\ k,$

its norm is

(3) $\|{\bf q}\| = \sqrt{{\bf q}{\bf q}^*} = \sqrt{{\bf q}^*{\bf q}} = \sqrt{a^2 + b^2 + c^2 + d^2},$

and its reciprocal is

(4) ${\bf q}^{-1}=\frac{{\bf q}^*}{\|{\bf q}\|^2},\quad {\bf q}{\bf q}^{-1} = {\bf q}^{-1}{\bf q} = 1.$

Note that the multiplications in (3) and (4) are Hamilton product defined in (2).

Spatial Rotation¶

Given a unit vector $\widehat{\bf u}=(u_x,u_y,u_z)$ with a scalar angle $\theta$ , we define quaternion

${\bf q} = \exp\left(\frac{\theta}{2}(u_x{\bf i}+u_y{\bf j}+u_z{\bf k})\right) = \cos\left(\frac{\theta}{2}\right) + \sin\left(\frac{\theta}{2}(u_x{\bf i}+u_y{\bf j}+u_z{\bf k})\right)$

then for any given vector ${\bf p}$ , its rotation across axis $\widehat{\bf u}$ for angle $\theta$ is

${\bf p}' = {\bf q}{\bf p}{\bf q}^{-1},$

using Hamilton product (2). Note that both ${\bf q}$ and $-{\bf q}$ performs the same rotation.

Conversion between rotation matrics¶

Given a unit quaternion ${\bf q}=(a,b,c,d)$ , it can be converted to a rotation matrix as

$\boldsymbol{R} = \left[\begin{array}{ccc} 1-2c^{2}-2d^{2} & 2bc-2ad & 2bd+2ac \\ 2bc+2ad & 1-2b^{2}-2d^{2} & 2cd-2ab \\ 2bd-2ac & 2cd+2ab & 1-2b^{2}-2c^{2} \end{array}\right]$

To convert from a rotation matrix $\boldsymbol{R}$ to a quaternion,

${\bf q} = \Big( & \frac{1}{2}\sqrt{R_{11}+R_{22}+R_{33}+1}, \\ & \frac{1}{2}\sqrt{R_{11}-R_{22}-R_{33}+1}\ \mbox{sign}(R_{32}-R_{23}), \\ & \frac{1}{2}\sqrt{-R_{11}+R_{22}-R_{33}+1}\ \mbox{sign}(R_{13}-R_{31}), \\ & \frac{1}{2}\sqrt{-R_{11}-R_{22}+R_{33}+1}\ \mbox{sign}(R_{21}-R_{12}) \ \Big).$

This conversion can be implemented with a single square root, but one needs to take special care on numerical stability when doing so.

API References¶

Utility functions for quaternion and spatial rotation.

A quaternion is represented by a 4-vector q as:

q = q[0] + q[1]*i + q[2]*j + q[3]*k.

The validity of input to the utility functions are not explicitly checked for efficiency reasons.

Abbr.	Meaning
quat	Quaternion, 4-vector.
vec	Vector, 3-vector.
ax, axis	Axis, 3- unit vector.
ang	Angle, in unit of radian.
rot	Rotation.
rotMatx	Rotation matrix, 3x3 orthogonal matrix.
HProd	Hamilton product.
conj	Conjugate.
recip	Reciprocal.

quaternion.quatConj(q)[source]¶: Return the conjugate of quaternion q.

quaternion.quatFromAxisAng(ax, theta)[source]¶

Get a quaternion that performs the rotation around axis ax for angle theta, given as:

q = (r, v) = (cos(theta/2), sin(theta/2)*ax).

Note that the input ax needs to be a 3x1 unit vector.

quaternion.quatFromRotMatx(R)[source]¶: Get a quaternion from a given rotation matrix R.

quaternion.quatHProd(p, q)[source]¶: Compute the Hamilton product of quaternions p and q.

quaternion.quatRecip(q)[source]¶: Compute the reciprocal of quaternion q.

quaternion.quatToRotMatx(q)[source]¶: Get a rotation matrix from the given unit quaternion q.

quaternion.rotVecByAxisAng(u, ax, theta)[source]¶: Rotate the 3-vector u around axis ax for angle theta (radians), counter-clockwisely when looking at inverse axis direction. Note that the input ax needs to be a 3x1 unit vector.

quaternion.rotVecByQuat(u, q)[source]¶

Rotate a 3-vector u according to the quaternion q. The output v is also a 3-vector such that:

[0; v] = q * [0; u] * q^{-1}

with Hamilton product.