Study Coordinates

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Recall that a pair $(e,g)\in \bH\times\bH$ lying on the Study quadric $S\subset \bR\P^7$ define the pair $(p,C) \in SE(3) = \bR^3\times SO(3)$:

(1)
\begin{align} p &= ge'/ee' \\ Cv &= eve'/ee' \end{align}
  1. Find the matrix $C$ in terms of $e_0,e_1,e_2,e_3$.
  2. Verify that the above construction gives an isomorphism of $S\setminus\V(ee')$ and $SE(3)$. (Express $(e,v)$ in terms of $(p,C)$.)

Hints

  1. Several lines of code in M2 using the quaternion type in platforms.m2 should do this: see what $Cv$ is for $v=i,j,k$.
  2. Note that $g$ can be found as long as $e$ is known.

Solution

  1. Set $e = e_{0} + e_{1} i + e_{2} j + e_{3} k$ and $g = g_{0} + g_{1} i + g_{2} j + g_{3} k$ and $v = v_{1} i + v_{2} j + v_{3} k$.

Compute the RHS of the equation $Cv = eve'/ ee'$ using Mathematica. They have a Quaternions package that works well with symbolic computation.

The LHS is matrix multiplication where we view $v \in \mathbb{R}^{3}$ as a column vector. Denote $C \in \mathbb{R}^{3 \times 3}$ as:

(2)
\begin{eqnarray} \nonumber C = \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{bmatrix} \ \end{eqnarray}

We will have three linear equations in the variables $v_{1}, v_{2}, v_{3}$. The defining equations for $C$ in terms of $e_{0},e_{1},e_{2},e_{3}$ are found by comparing coefficients. Thus,

(3)
\begin{eqnarray} \nonumber C = \frac{1}{e_{0}^{2} + e_{1}^{2} + e_{2}^{2} + e_{3}^{2}} \begin{bmatrix} e_{0}^{2} + e_{1}^{2} - (e_{2}^{2} + e_{3}^{2}) & 2(e_{1}e_{2}-e_{0}e_{3}) & 2(e_{0}e_{2} + e_{1}e_{3})\\ 2(e_{1}e_{2} + e_{0}e_{3}) & e_{0}^{2}-e_{1}^{2} + e_{2}^{2} -e_{3}^{2} & -2(e_{0}e_{1} + e_{2}e_{3})\\ -2(e_{0}e_{2} + e_{1}e_{3}) & 2(e_{0}e_{1} + e_{2}e_{3}) & e_{0}^{2} - e_{1}^{2} - e_{2}^{2} + e_{3}^{2} \end{bmatrix} \end{eqnarray}

Discussion

  • What is a Study quadric?

$e_0g_0+e_1g_1+e_2g_2+e_3g_3 = 0$

  • What does apostrophe mean (as in $e'$)?

It means conjugation of quaternions, i.e. given $(q_0,q_1,q_2,q_3)\in\mathbb{R}^4$, then

(4)
\begin{equation} (q_0+q_1 i+q_2 j + q_3 k)'=q_0-q_1 i-q_2 j - q_3 k \end{equation}
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