Understanding how to use Quaternion to rotate object

Question

I m having hard time to understand Quaternions and to use them in my engine for rotating object. I m looking for step by step explanation, actually a correction of my view of Quaternions.

Here is how I understand they work.

q = cos(40*) + sin(40)(i+j+k)

->what I understand is that i+j+k represents the axis of rotation. Basically if i = 1 that means we rotate object around X-axis.

Formula q * point * p-1, is what I dont quite understand. Lets for the sake say I understand why forumla goes like this. What I dont seems to get how would I apply this to my engine to rotate object.

What I tried is to create 3dVector with object position, example (0,0,-5). I would use quaternion that would rotate this vector around y axis, so q * ThisVector * q(conjugate), and then result I would put in translate matrix. But no work. I cant seem to find a way to implement them. I saw some articles on quaternion to matrices, But what do I put in that matrix, a result from qThisVectorq(conjugate) or it is just q that is transfromed in matrix?

EDIT What I want is to rotate object on Y-axis using quaternion. So far,because of lack of my understanding, I m failing to rotate. I created a class 3dVector ->three dimensional vector, and instantiate a new position (0,0,-5). This position I sent as translate matrix to opengl and multiplied with glPosition.

When I tried to rotate that object using quaternion. I just took that position (0,0,-5) and used it in formula q = q * p * q(conjugate). I manually calculated that but result I get didnt make object rotate on Y-axis around itself. Now I m not sure, but I think it actually rotated object but not around him self,he was moved to the right, so he was not in the same position anymore

EDIT_2:

I have cube that is on position [0,0,-5] and I want to rotate it around y-axis by some angle.

I tried to write code in python

poinToRotate = Quaternion(w=0,x=0,y=0,z=-5)

angle =45

sinus = math.sin(math.radians(angle));
q=Quaternion(
w=math.cos(math.radians(angle)),
x=sinus*0,
y=sinus*1,
z=sinus*)

newPoint = q.rotate(poinToRotate)

Rotation is somewhere good but it cube does not rotate around it self but around some point. But when

   p = Quaternion(w=0,x=0,y=0,z=0)

to make it to the center, result is all zeros.

Answer 1

Explaining why quaternions work the way they do and their relation to mainstream linear transformation rotations would require some more advanced mathematics knowledge regarding group theory and how it applies to complex numbers and matrices. For your needs however what you need to know is the following:

• Quaternions are 4-dimensional (w,x,y,z) objects with some interesting properties which we can abuse in order to do rotations in 3-dimensional space (x,y,z).
• A 3-dimensional vector or point in space can be represented as a quaternion if we create a quaternion where \$w=0\$ and we set the rest of \$x,y,z\$ to the point/vector coordinates
• The above hacky quaternions have the following property: if a quaternion \$p\$, that represents a point in 3D space, is multiplied by another quaternion and its conjugate in a sandwich \$q*p*q^{*}\$ a new quaternion results from this operation that also represents a point in 3D space, specifically one that has been rotated by using the following linear transformation matrix $$R=\begin{pmatrix} 1-2y^2-2z^2 & 2xy-2zw & 2xz+2yw \\ 2xy+2zw & 1-2x^2-2z^2 & 2yz-2xw \\ 2xz-2yw & 2yz+2xw & 1-2x^2-2y^2 \end{pmatrix}$$ where the \$w,x,y,z\$ values represent the values of the \$q\$ quaternion.
• The above rotation transformation matrix represents a rotation around the axis represented by \$x,y,z\$ with angle \$2*cos^{-1}(w)\$

To summarize, if you have a 3D point \$x,y,z\$ that you want to rotate by angle \$θ\$ around the axis represented by vector \$[a,b,c]\$ using quaternions, what you need to do is:

• Create a quaternion \$p=0 + xi + yj + zk\$
• Create a quaternion \$q=cos(θ/2) + ai + bj + ck\$ where \$cos(θ/2)^2 + a^2 + b^2 + c^2 = 1\$ i.e. a unit quaternion. If your axis vector \$[a,b,c]\$ is a unit vector then it is easier to do \$q=cos(θ/2) + sin(θ/2) * (ai + bj + ck)\$
• Do the sandwich multiplication \$p'=q*p*q^{*}\$
• Extract \$x',y'\$ and \$z'\$ from \$p'\$ and you now have the rotated 3D point \$(x', y', z')\$