Difference between the two quaternions

Question

Solved

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I'm making a 3D portal system in my engine (like Portal game). Each of the portals has its own orientation saved in a quaternion. To render the virtual scene in one of the portals I need to calculate the difference between the two quaternions, and the result use to rotate the virtual scene.

When creating the first portal on the left wall, and second one on the right wall, the rotation from one to another will take place in only one axis, but for example when the first portal will be created on the floor, and the second one on the right wall, the rotation from one to another could be in two axis, and that's the problem, because the rotation goes wrong.

I think the problem exists because the orientation for example X axis and Z axis are stored together in one quaternion and I need it separately to manualy multiply X * Z (or Z * X), but how to do it with only one quaternion, (the difference quaternion)? Or is there other way to correct rotate the scene?

EDIT:

Here on this picture are two portals P1 and P2, the arrows show how are they rotated. As I am looking into P1 I will see what sees P2. To find the rotation which I need to rotate the main scene to be like the virtual scene in this picture I'm doing following:

1. Getting difference from quaternion P2 to quaternion P1
2. Rotating result by 180 degrees in Y axis (portal's UP)
3. Using the result to rotate the virtual scene

This method above works only when the difference takes place in only one axis. When one portal will be on the floor, or on te ceiling, this will not work because the difference quaternion is build in more than one axis. As suggested I tried to multiply P1's quaternion to P2's quaternion, and inversely but this isn't working.

EDIT 2:

To find the difference from P2 to P1 I'm doing following:

Quat q1 = P1->getOrientation();
Quat q2 = P2->getOrientation();

Quat diff = Quat::diff(q2, q1);  // q2 * diff = q1 //

Here's the Quat::diff function:

GE::Quat GE::Quat::diff(const Quat &a, const Quat &b)
{
    Quat inv = a;
    inv.inverse();
    return inv * b;
}

Inverse:

void GE::Quat::inverse()
{
    Quat q = (*this);
    q.conjugate();
    (*this) = q / Quat::dot((*this), (*this));
}

Conjugate:

void GE::Quat::conjugate()
{
    Quat q;
    q.x = -this->x;
    q.y = -this->y;
    q.z = -this->z;
    q.w = this->w;

    (*this) = q;
}

Dot product:

float GE::Quat::dot(const Quat &q1, const Quat &q2)
{
    return q1.x*q2.x + q1.y*q2.y + q1.z*q2.z + q1.w*q2.w;
}

Operator*:

const GE::Quat GE::Quat::operator* ( const Quat &q) const
{
    Quat qu;
    qu.x = this->w*q.x + this->x*q.w + this->y*q.z - this->z*q.y;
    qu.y = this->w*q.y + this->y*q.w + this->z*q.x - this->x*q.z;
    qu.z = this->w*q.z + this->z*q.w + this->x*q.y - this->y*q.x;
    qu.w = this->w*q.w - this->x*q.x - this->y*q.y - this->z*q.z;
    return qu;
}

Operator/:

const GE::Quat GE::Quat::operator/ (float s) const
{
    Quat q = (*this);
    return Quat(q.x / s, q.y / s, q.z / s, q.w / s);
}

All this stuff works, because I have tested it with GLM library

Answer 1

I solved my problem. As it turned out I don't need any difference between two rotations. Just multiply one rotation by rotation in 180 degrees, and then multiply by inverse of second rotation that way (using matrices):

Matrix m1 = p1->getOrientation().toMatrix();
Matrix m2 = p2->getOrientation().toMatrix();
Matrix model = m1 * Matrix::rotation(180, Vector3(0,1,0)) * Matrix::inverse(m2);

and translation calculating this way:

Vector3 position = -p2->getPosition();
position = model * position + p1->getPosition();
model = Matrix::translation(position) * model;