Duke Rabbit's Game Zone

What is Quaternion

• Invented by Sir William Rowan Hamilton as an extension to the complex numbers in 1843
• Introduced to the field of computer graphics by Shoemake in 1985
• q = (x, y, z, w)

Mathematical Background

• Definition

q = (q_v, q_w) = iq_x + iq_y + iq_z + q_w

i² = j² = k² = -1, jk = -kj = i, ki = -ik = j, ij = -ji = k

• Multiplication

qr = (q_v X r_v+r_wq_v+q_wr_v) - q_v‧r_v

• Addition

q + r = (q_v + r_v) + (q_w+r_w)

• Conjugate

q* = (-q_v, q_w)

• Norm

n(q) = (q_x² + q_y² + q_z² + q_w²)^1/2

• Identity

q_I = (0, 0, 0, 1)

• Inverse

n(q)² = qq* → qq*/n(q)² = 1

q^-1 = q*/n(q)²

Slerp(Spherical Linear Interpolation) of Quaternion Rotation:

Slerp(q₀, q₁, t) = q₀(q₀^-1q₁)^t

                        = q₀^(1-t)q₁^t

                        = q₀^aq₁^b, a+b = 1 

=>Slerp(q₀, q₁, a, b)

所以是否可以推展成:

 Slerp(q₀, q₁, q₂, a, b, c) = q₀^aq₁^bq₂^c, a+b+c = 1 ?

我覺得應該可以，推導如下：

q₀^aq₁^bq₂^c = (q₀^xq₁^y)^zq₂^c; x = a/(a+b), y = b/(a+b), z = a+b

                 = Slerp(Slerp(q₀, q₁, x, y), q₂, z, c); x+y = 1, c+z = a+b+c = 1

如此一來，就可以將一組quaternion的blending拆成各個獨立的運算式。即各自運算qⁿ的結果，最後再乘起來即可：

quat q = Identity;

loop(1 to n) {
    q *= track[i].WeightedQuat();
}

Exponentiation of Quaternion:

q = cosq + vsinq

q^t = costq + vsintq

Implementation Issue

在實作時發現上述的公式有兩個致命的問題：

• When q = (0, 0, 0, -1), q^t is undefined
• 無法產生Shortest Path Slerp

因此必須將公式改成：

Slerp(q₀, q₁, t) = [(sin(1-t)a)/sina]q₀ + [sinta/sina]q₁, sina = (x₀*x₁ + y₀*y₁ + z₀*z₁ + w₀*w₁)

程式必須改成：

QuatWithWeight qw = (q, t);

loop(1 to n) {
    qw = SlerpWithWeight(qw, track[i].Quat(), track[i].Weight());
}

QuatWithWeight SlerpWithWeight(qw, q1, w1)
{
    QuatWithWeight qw1
    weight = qw.w/(qw.w+w1);
    qw1.q = SlerpSP(qw.q, q1, weight);
    qw1.w = weight;
    return qw1;
}

Quat SlerpSP(q1, q2, w)
{
    cos = dot(q1, q2);
    if (cos < 0) {
         q1 = -q1;
    }
    if (cos -> 1) {
        q3 = q1*w + q2*(1-w);
    } else {
        theta = acos(cos);
        q3 = sin(theta*w)*q1 + sin(theta*(1-w))*q2;
        q3 = q3/sin(theta);
    }
    return q3;
}

但無論是哪一個算式，他們的計算順序都會影響最後的結果。換句話說，同樣一組quaternions及weights卻可能得到各種不同的結果，以算式表示就是：

Slerp(q1, q2, q3, a, b, c) ≠ Slerp(q1, q3, q2, a, c, b)

所以如果想要得到一個唯一解也許只有簡化Slerp成Qlerp(In "Spherical blend skinning: A real-time deformation of articulated models" SIGGRAPH 2005)這一途吧