- #1

mathmari

Gold Member

MHB

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For $p\in \mathbb{R}^2$ let $\delta_{p,\alpha}=\tau_p\circ \delta_{\alpha}\circ\tau_p^{-1}$.

Let $p,q\in\mathbb{R}^2$ and $\alpha,\beta\in \mathbb{R}$.

(a) Show that $\gamma=\delta_{p,\alpha}\circ\delta_{q,\beta}$ is a rotation of a translation (or both). Give the center of the rotation or the translation vector of $\gamma$ in respect to $p,q,\alpha, \beta$.

(b) Show analytically that the product $\gamma$ of two line reflections is a rotation or a translation. Give the geometric interpretation of the rotation angle/translation vector of $\beta$.

For (a) I have done the following :

\begin{align*}\left(\delta_{p,\alpha}\circ\delta_{q,\beta}\right )(x)&=\left (\tau_p\circ \delta_{\alpha}\circ\tau_p^{-1}\circ\tau_q\circ \delta_{\beta}\circ\tau_q^{-1}\right )(x)\\& =\tau_p\left ( \delta_{\alpha}\left (\tau_p^{-1}\left (\tau_q\left (\delta_{\beta}\left (\tau_q^{-1}(x)\right )\right )\right )\right)\right )\\ & =\tau_p\left ( \delta_{\alpha}\left (\tau_p^{-1}\left (\tau_q\left (\delta_{\beta}\left (x-q\right )\right )\right )\right)\right ) \\ & =\tau_p\left ( \delta_{\alpha}\left (\tau_p^{-1}\left (\tau_q\left (d_{\beta}\left (x-q\right )\right )\right )\right)\right )\\ & =\tau_p\left ( \delta_{\alpha}\left (\tau_p^{-1}\left (d_{\beta}\left (x-q\right )+q\right )\right)\right )\\ & =\tau_p\left ( \delta_{\alpha}\left (d_{\beta}\left (x-q\right )+q-p\right)\right )\\ & =\tau_p\left ( d_{\alpha}\left (d_{\beta}\left (x-q\right )+q-p\right)\right )\\ & = d_{\alpha}\left (d_{\beta}\left (x-q\right )+q-p\right)+p\\ & = d_{\alpha}d_{\beta}\left (x-q\right )+d_{\alpha}(q-p)+p\end{align*}

Is that correct so far? Do we have to substitute the rotation matrices $d_{\alpha}$ and $d_{\beta}$ ? Or is there an other way to continue?

:unsure: