I am trying to understand this post in terms of differential geometry. Since now I have concluded to the following, please verify my thoughts.
- The normalization function $f(x) = x / ||x||_2$ can be seen as the mapping \begin{equation} f \colon \mathbb{R}^n/\{0\}\to S^{n-1}. \end{equation}
- The Jacobian of $f$ given by \begin{equation} J_f = \frac{1}{\lVert x \rVert}\left(I - x x^T / ||x||_2^2 \right) \end{equation} can be used to define the differential function \begin{equation} f_{*,x} \colon T_x\mathbb{R}^n\to T_xS^{n-1} \end{equation} at point $p$.
My questions are:
- Shall we define a $f_{*,x}$ for all $x \in \mathbb{R^n}$?
- If we apply $x$ on $f_{*,x}$ we obtain the zero vector. Is $x$ thought of as a point, the origin of $T_{x}\mathbb{R}^n$, and thus it is meaningless to apply $x$ in $f_{*,x}$?
- If we want to apply another vector $v$ should bellong to $T_{x}\mathbb{R}^n$?
- If yes, then the output is \begin{equation} f_{*,x}v = \frac{1}{\lVert x \rVert}\left(I - x x^T / ||x||_2^2 \right)v \end{equation} Geometrically I know that $P=I - x x^T / ||x||_2^2$ projects any vector to the left null-space of the single column matrix $x$. How can we interpret in this case $J_f$ that has also a scaling factor $1/||x||$ ?
- How can be apply $f_{*,x}$ in an arbitrary vector $v\in \mathbb{R}^n$? Shall we cover the entire $\mathbb{R}^n$ with tangent planes $T_{x}\mathbb{R}^n$ for all $x\in\mathbb{R}^n$? I am thinking that $T_p\mathbb{R}^n$ is isomorphic to $\mathbb{R}^n$ but I can not figure it out.
Thanks in advance.