Application of Quaternion in Industry 四元数在工业界的应用-优快云博客

Application of Quaternion in Industry

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0. Quaternion & Vec3

quaternion Q

class Q(object):
    def __init__(self, x=0.0, y=0.0, z=0.0, w=1.0) -> None:
        self.x = x
        self.y = y
        self.z = z
        self.w = w
    
    ...

    def print(self):
        print(self.x, self.y, self.z, self.w)
    
    # operator override
    def __mul__(self, num):
        return Q(self.x * num, self.y * num, self.z * num, self.w * num)

    def __truediv__(self, num):
        assert np.abs(num) > 1e-10
        return Q(self.x / num, self.y / num, self.z / num, self.w / num)

Vec3

class Vec3(object):
    def __init__(self, x=0.0, y=0.0, z=0.0) -> None:
        self.x = x
        self.y = y
        self.z = z

    def from_array(self, array):
        self.x = array[0]
        self.y = array[1]
        self.z = array[2]
    
    def to_array(self):
        return np.array([self.x, self.y, self.z], dtype=np.float32)
    
    def print(self):
        print(self.x, self.y, self.z)
    
    # operator override
    def __add__(self, vec3):
        return Vec3(self.x + vec3.x, self.y + vec3.y, self.z + vec3.z)

    def __sub__(self, vec3):
        return Vec3(self.x - vec3.x, self.y - vec3.y, self.z - vec3.z)

    def __mul__(self, num):
        return Vec3(self.x * num, self.y * num, self.z * num)

    def __truediv(self, num):
        assert np.abs(num) > 1e-10
        return Vec3(self.x / num, self.y / num, self.z / num)

1. Definition

$q = x i + y j + z k + w$

其中 $a, b, c$ 是虚部， $w$ 是实部。 $i^2=j^2=k^2=ijk=-1$ .

class Q(object):
    def __init__(self, x=0.0, y=0.0, z=0.0, w=1.0) -> None:
        self.x = x
        self.y = y
        self.z = z
        self.w = w

2. Property

2.1 Norm

四元数 $q = x i + y j + z k + w$ 的模长：

$\sqrt{x^2 + y^2 + z^2 + w^2}.$

class Q(object):
    def __init__(self, x=0.0, y=0.0, z=0.0, w=1.0) -> None:
        self.x = x
        self.y = y
        self.z = z
        self.w = w
    
    def get_norm(self) -> float:
        return np.sqrt(self.x**2 + self.y**2 + self.z**2 + self.w**2)

2.2 Conjugation of quaternion

四元数 $q = x i + y j + z k + w$ 的共轭四元数 $q^*$

$q^* = -xi-yj-zk+w$

共轭四元数满足条件：

$qq^*=q^*q=||q||^2$

class Q(object):
    def __init__(self, x=0.0, y=0.0, z=0.0, w=1.0) -> None:
        self.x = x
        self.y = y
        self.z = z
        self.w = w

    def get_conjugation(self):
        return Q(-self.x, -self.y, -self.z, self.w)

2.3 Inverse of quaternion

四元数 $q = x i + y j + z k + w$ 的逆四元数 $q^{-1}$

$q^{-1}=\frac{q^*}{||q||^2}$

class Q(object):
    def __init__(self, x=0.0, y=0.0, z=0.0, w=1.0) -> None:
        self.x = x
        self.y = y
        self.z = z
        self.w = w
    
    def get_norm(self) -> float:
        return np.sqrt(self.x**2 + self.y**2 + self.z**2 + self.w**2)

    def get_conjugation(self):
        return Q(-self.x, -self.y, -self.z, self.w)

    def get_inverse(self):
        q_conjugation = self.get_conjugation()
        q_lengthSqr = self.get_norm()**2
        return Q(q_conjugation.x/q_lengthSqr, q_conjugation.y/q_lengthSqr,\
                 q_conjugation.z/q_lengthSqr, q_conjugation.w/q_lengthSqr)

3. Operation

3.1 Make quaternion with axis and angle

绕轴 $\mathbf{u}$ 逆时针旋转 $\theta$ 角度的四元数表示：

$\sin{\frac{\theta}{2}}\mathbf{u}.x~i + \sin{\frac{\theta}{2}}\mathbf{u}.y~j + \sin{\frac{\theta}{2}}\mathbf{u}.z~k + \cos{\frac{\theta}{2}}$

def MakeQuaternion(angle_radian: float, u: Vec3) -> Q:
    q = Q()
    halfAngle = 0.5 * angle_radian
    sinHalf = np.sin(halfAngle)

    q.w = np.cos(halfAngle)
    q.x = sinHalf * u.x
    q.y = sinHalf * u.y
    q.z = sinHalf * u.z

    return q

3.2 Multiplication with two quaternion

满足分配律和结合律，但不满足交换律， $q_Aq_B \neq q_Aq_B$
$q_Aq_B$ :

$\begin{aligned} q_Aq_B &= (q_A.w * q_B.x + q_A.x * q_B.w + q_A.y * q_B.z - q_A.z * q_B.y)i + \\ &= (q_A.w * q_B.y + q_A.y * q_B.w + q_A.z * q_B.x - q_A.x * q_B.z)j + \\ &= (q_A.w * q_B.z + q_A.z * q_B.w + q_A.x * q_B.y - q_A.y * q_B.x)k + \\ &= (q_A.w * q_B.w - q_A.x * q_B.x - q_A.y * q_B.y - q_A.z * q_B.z) \end{aligned}$

def MultQuaternionAndQuaternion(qA: Q, qB: Q) -> Q:
    q = Q()

    q.w = qA.w * qB.w - qA.x * qB.x - qA.y * qB.y - qA.z * qB.z
    q.x = qA.w * qB.x + qA.x * qB.w + qA.y * qB.z - qA.z * qB.y
    q.y = qA.w * qB.y + qA.y * qB.w + qA.z * qB.x - qA.x * qB.z
    q.z = qA.w * qB.z + qA.z * qB.w + qA.x * qB.y - qA.y * qB.x

    return q

3.3 Multiplication with quaternion and vec3

对三维向量 $\mathbf{v}$ 应用四元数 $q$ ：

$\mathbf{v}' = \mathbf{v} + 2.0 * q.w * q.xyz \times \mathbf{v} + 2.0 * q.xyz \times (q.xyz \times \mathbf{v})$

def MultQuaternionAndVector(q: Q, v: Vec3) -> Vec3:
    uv, uuv = Vec3(), Vec3()
    qvec = Vec3(q.x, q.y, q.z)
    uv.from_array(np.cross(qvec.to_array(), v.to_array()))
    uuv.from_array(np.cross(qvec.to_array(), uv.to_array()) * 2.0)
    uv *= (2.0 * q.w)

    return v + uv + uuv

3.4 Normalize quaternion

四元数 $q = x i + y j + z k + w$ 的归一化结果为：

$\hat{q} = \frac{q}{||q||}$

def NormalizeQuaternion(q: Q) -> Q:
    qq = Q(q.x, q.y, q.z, q.w)
    lengthSqr = q.x*q.x + q.y*q.y + q.z*q.z + q.w*q.w

    if lengthSqr < 1e-10:
        qq.w = 1.0
        return qq
    
    qq /= np.sqrt(lengthSqr)

    return qq

3.5 Quaternion from u to v

向量 $\mathbf{u}$ 到向量 $\mathbf{v}$ 的四元数 $q$ 为：

$q = N or ma l i ze (cross (u, v), 1.0 + d o t (u, v))$

def QuatFromTwoUnitVectors(u: Vec3, v: Vec3) -> Q:
    r = 1.0 + np.dot(u.to_array(), v.to_array())
    n = Vec3()

    # if u and v are parallel
    if r < 1e-7:
        r = 0.0
        n = Vec3(-u.y, u.x, 0.0) if np.abs(u.x) > np.abs(u.z) else Vec3(0.0, -u.z, u.y)
    else:
        n.from_array(np.cross(u.to_array(), v.to_array()))

    q = Q(n.x, n.y, n.z, r)
    return NormalizeQuaternion(q)