Any geometrical meaning of multiplication of quaternions?

[Kumar8434]

Let's just talk about unit quaternions.
I know that $$\left(\cos{\frac{\theta}{2}}+v\sin{\frac{\theta}{2}}\right)\cdot p \cdot \left(\cos{\frac{\theta}{2}}-v\sin{\frac{\theta}{2}}\right)$$
where $p$ and $v$ are purely imaginary quaternions, gives another purely imaginary quaternion which corresponds to $p$ rotated by an angle $\theta$ about the axis specified by $v$. So the product $q\cdot p \cdot q'$ has a geometrical meaning.

But what about any arbitrary unit quaternion multiplication $q_1\cdot q_2$? What does it mean geometrically (just like unit complex number multiplication means adding their angles)?

If $z_1\cdot z_2=z_3$, then $z_3$ is the point we end up at when we rotate point $z_1$ by the argument of $z_2$ or vice-versa. Now, if $q_1\cdot q_2=q_3$, then $q_1,q_2,q_3$ are points in four dimensions. What is the relation between these three points?

 

[FactChecker]

It gets complicated. In my opinion, the best geometric treatment of 3D and higher is with Geometric Algebra. It systematically treats many geometric concepts with algebraic operations.
(see )

PS. A word of warning. Although Geometric Algebra consolidates and replaces a lot of specialized mathematical "gimic" algebras, it is not well known, the learning curve is not trivial, and translating the more popular algebras to it is not always easy.

 

[fresh_42]

For the unit quaternions we have $U(1,\mathbb{H}) \cong SU(2,\mathbb{C}) \cong \mathbb{S}^3$ (cp. section 3 in https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/).
Thus the multiplication is the same as in the special unitary group, or on the 3-sphere. Both, $SU(2,\mathbb{C})$ and $\mathbb{S}^3$ are geometric objects, so the multiplication directly translates into geometry.

 

[StoneTemplePython]

I tried a book on geometric algebra a while ago but it didn't speak that much to me.

For quarternions, thinking about them as part of special unitary group, and in particular using a matrix representation, as done in Stillwell's Naive Lie Theory, is how I'd do it.