Let $V$ be a real finite-dimensional vector space with inner product $\langle \cdot , \cdot \rangle$.
Let $x,y \in V$ be linearly independent. I was wondering how a reflection $s_{x,y}$ through the $\text{codim} = 2$ subspace $H_x \cap H_y$ could be defined, where $H_v := \langle v \rangle^\bot$. I noticed that the following construction leads to a reasonable result:
- Project $a \in V$ orthogonally onto $H_x$ $\Rightarrow$ $\pi_x(a) \in H_x$.
- Reflect $\pi_x(a)$ through $H_x \cap H_y < H_x$ $\Rightarrow$ $s_{\pi_x(y)}(\pi_x(a)) \in H_x$.
- Reverse project $s_{\pi_x(y)}(\pi_x(a))$ back from $H_x$
Here, $\pi_x(a)= a-cx$, $s_x(a)= a-2cx$ where $c= \frac{\langle x,a \rangle}{\langle x,x \rangle}$.
In total, we get the neat formula $$s_{x,y}(a)= a - 2 \left(\frac{\langle x,x\rangle \langle y,a\rangle - \langle x,y\rangle\langle x,a\rangle}{\langle x,x\rangle \langle y,y\rangle -\langle x,y\rangle^2}y + \frac{\langle y,y\rangle \langle x,a\rangle - \langle y,x\rangle\langle y,a\rangle}{\langle x,x\rangle \langle y,y\rangle -\langle x,y\rangle^2}x \right).$$ It is symmetric, scale invariant and satisfies $s^2=1$.
Similarly, we can define $s_{x,y,z}$ and so on. The construction is straight forward enough to assume that it is well-known, so I didn't make the effort to figure out the general formula for $s_{x_1, \dots,x_n}$.
So my question would be, is a general formula known and where can I find it?
