Bruce Rosario

2022-04-30

The hopf map in terms of quaternions is defined as

$h:r\mapsto {R}_{r}({P}_{0})=ri\overline{r}$

where r is a unit quaternion and ${P}_{0}=(1,0,0)$ is a fixed point. If a point $r\in {S}^{3}$ is sent by the Hopf map to the point $P\in {S}^{2}$ , a formula can be derived for a particular representation for the cosets. In my case, I want to derive a formula for the ${180}^{\circ}$ rotations around an axes through i and other points in ${S}^{3}$ .

$h:r\mapsto {R}_{r}({P}_{0})=ri\overline{r}$

where r is a unit quaternion and ${P}_{0}=(1,0,0)$ is a fixed point. If a point $r\in {S}^{3}$ is sent by the Hopf map to the point $P\in {S}^{2}$ , a formula can be derived for a particular representation for the cosets. In my case, I want to derive a formula for the ${180}^{\circ}$ rotations around an axes through i and other points in ${S}^{3}$ .

gonzakunti2

Beginner2022-05-01Added 16 answers

Step 1

It's the same thing: if $P=({p}_{1},{p}_{2},{p}_{3})\in {S}^{2}$ , then ${p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{2}=1$ so$(1+{p}_{1}{)}^{2}+{p}_{2}^{2}+{p}_{3}^{2}=1+2{p}_{1}+{p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{2}=2+2{p}_{1}.$ .

It's the same thing: if $P=({p}_{1},{p}_{2},{p}_{3})\in {S}^{2}$ , then ${p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{2}=1$ so$(1+{p}_{1}{)}^{2}+{p}_{2}^{2}+{p}_{3}^{2}=1+2{p}_{1}+{p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{2}=2+2{p}_{1}.$ .

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