Convert from rotation matrix to quaternion

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Quaternion q=(q₀,q₁,q₂,q₃) where |q|=1

Quaternion to rotation matrix.

Rotation matrix to quaternion. r_ij=a_ij is rotation matrix.

Therefore,

By solving this,

Now let's calculate the sign. q and -q represent same rotation, thus,

Other sign will be as follows.

Another choice is to do as follows.

or

or

Source code: Convert rotation matrix to quaternion.

inline float SIGN(float x) {return (x >= 0.0f) ? +1.0f : -1.0f;}
inline float NORM(float a, float b, float c, float d) {return sqrt(a * a + b * b + c * c + d * d);}

q0 = ( r11 + r22 + r33 + 1.0f) / 4.0f;
q1 = ( r11 - r22 - r33 + 1.0f) / 4.0f;
q2 = (-r11 + r22 - r33 + 1.0f) / 4.0f;
q3 = (-r11 - r22 + r33 + 1.0f) / 4.0f;
if(q0 < 0.0f) q0 = 0.0f;
if(q1 < 0.0f) q1 = 0.0f;
if(q2 < 0.0f) q2 = 0.0f;
if(q3 < 0.0f) q3 = 0.0f;
q0 = sqrt(q0);
q1 = sqrt(q1);
q2 = sqrt(q2);
q3 = sqrt(q3);
if(q0 >= q1 && q0 >= q2 && q0 >= q3) {
    q0 *= +1.0f;
    q1 *= SIGN(r32 - r23);
    q2 *= SIGN(r13 - r31);
    q3 *= SIGN(r21 - r12);
} else if(q1 >= q0 && q1 >= q2 && q1 >= q3) {
    q0 *= SIGN(r32 - r23);
    q1 *= +1.0f;
    q2 *= SIGN(r21 + r12);
    q3 *= SIGN(r13 + r31);
} else if(q2 >= q0 && q2 >= q1 && q2 >= q3) {
    q0 *= SIGN(r13 - r31);
    q1 *= SIGN(r21 + r12);
    q2 *= +1.0f;
    q3 *= SIGN(r32 + r23);
} else if(q3 >= q0 && q3 >= q1 && q3 >= q2) {
    q0 *= SIGN(r21 - r12);
    q1 *= SIGN(r31 + r13);
    q2 *= SIGN(r32 + r23);
    q3 *= +1.0f;
} else {
    printf("coding error\n");
}
r = NORM(q0, q1, q2, q3);
q0 /= r;
q1 /= r;
q2 /= r;
q3 /= r;

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