I know that octant in a circle looks like this image:
Now how can I move a circle from one octant to another octant? I'm picturing it something like this:
I don't know if the circle will be moved on the octants of the coordinate plane (given point X,Y,Z), or the octants in the circle itself. Will it be possible to move a circle from one octant to another octant?
Paying attention just to the circle of equation $x^2+y^2=1$, all rotation of angle $\theta\ne 0$ of $\mathbb R^2$ move the eight octants. When you want each octant move to another one, you take $\theta= \frac{k\pi}{4}$ where $k\in \mathbb Z$.
For instance to move the octant $0$ to the octant $1$ (hence the octant 1 to the 2, the 2 to the 3,etc) you can take $\theta=\frac {\pi}{4}$ in other words, you take the rotation of $\mathbb R^2$ defined by the matrix (a rotation is a linear function!) $$R=\begin{bmatrix}\frac{\sqrt2}{2} &-\frac{\sqrt2}{2}\\ \frac{\sqrt2}{2} & \frac{\sqrt2}{2} \end{bmatrix}$$