Creating orientation Quaternion from forward vector

Question

Suppose that I have an orientation Quaternion Q, I can compute its forward vector from V = Q * Vector3.forward easily.

Inversely, suppose that I have its forward vector V, how do I compute Q?

I know that is not possible, please tell me what's needed beside V, in order to compute Q.

Motivation behind the problem: I have a forward direction of a game object, I want to find out its up direction and its right direction. I can find out all these 3 directions if I have the orientation Quaternion.

Answer 1

You can calculate a right and up directly from the forward. however it is not going to be a unique up and right because there is an extra degree of freedom you need to consider.

Code from Building an Orthonormal Basis, Revisited

void branchlessONB(const Vec3f &n, Vec3f &b1, Vec3f &b2){
    float sign = copysignf(1.0f, n.z);
    const float a = -1.0f / (sign + n.z);
    const float b = n.x * n.y * a; 
    b1 = Vec3f(1.0f + sign * n.x * n.x * a, sign * b, -sign * n.x);
    b2 = Vec3f(b, sign + n.y * n.y * a, -n.y);
}

Answer 2

Unless your forward vector is directed exactly up, you can easily find the right and up vectors in Unity:

Vector3 right = Vector3.Cross(Vector3.up, forward);
Vector3 up = Vector3.Cross(forward, right);

Otherwise, right and up will be zero and you will be experiencing gimbal lock, a problem that has plenty of different solutions depending on your case. Choosing an arbitrary right vector may be a solution. The final code could be:

Vector3 right = Vector3.Angle(Vector3.up, forward) < 0.001)
              ? Vector3.right : Vector3.Cross(Vector3.up, forward);
Vector3 up = Vector3.Cross(forward, right);

Answer 3

In order to convert a direction-vector to a quaternion, use Quaternion.LookRotation(v). The (optional) second argument isn't the up-vector of the object. It's the up-vector of the world.