3 Bounding Rotations

Lemma 12

✓

For any \(\alpha , \theta ,\varphi \in \operatorname{\mathbb {R}}\) and \(a \in \{ x,y,z\} \) one has \(\| R(\alpha )\| = \| R_a(\alpha )\| =\| M(\theta , \varphi )\| = 1\).

Proof ▶

See [ SY25 ] , Lemma 9.

Lemma 13

Let \(\varepsilon {\gt}0\), \(|\alpha -\overline{\alpha }|\leq \varepsilon \) and \(a \in \{ x,y,z\} \) then \(\| R_a(\alpha )-R_a({\overline{\alpha }})\| =\| R(\alpha )-R(\overline{\alpha })\| {\lt} \varepsilon \).

Proof ▶

See [ SY25 ] , Lemma 10.

Lemma 14

✓

For all \(a,b \in \operatorname{\mathbb {R}}\) with \(|a|,|b|\leq 2\) the following inequality holds:

\[ (1+\cos (a))(1+\cos (b))\geq 2+2\cos \Big(\sqrt{a^2+b^2}\Big), \]

with equality only for \(a=0\) or \(b=0\).

Proof ▶

Use the Jensen inequality. See [ SY25 ] , Lemma 11.

Lemma 15

For any \(\alpha ,\beta \in \mathbb {R}\) one has

\[ \| R_x(\alpha )R_y(\beta )-\operatorname{\mathrm{Id}}\| \leq \sqrt{\alpha ^2+\beta ^2} \]

with equality only for \(\alpha = \beta = 0\).

Proof ▶

See [ SY25 ] , Lemma 12.

Lemma 16

Let \(\varepsilon {\gt}0\) and \(|\theta -\overline{\theta }|,|\varphi -\overline{\varphi }| \leq \varepsilon \) then \(\| M(\theta , \varphi )-M(\overline{\theta },\overline{\varphi })\| , \| X({\theta , \varphi })-X(\overline{\theta },\overline{\varphi })\| {\lt} \sqrt{2}\varepsilon .\)

Proof ▶

See [ SY25 ] , Lemma 13.

Lemma 17

Let \(P \in \operatorname{\mathbb {R}}^3\) with \(\| P\| \leq 1\). Further, let \(\varepsilon {\gt}0\) and \(\overline{\theta },\overline{\varphi }, \theta , \varphi \in \operatorname{\mathbb {R}}\) such that \(|\overline{\theta }-\theta |, |\overline{\varphi }- \varphi | \leq \varepsilon \). If \( \langle X(\overline{\theta },\overline{\varphi }),P \rangle {\gt}\sqrt{2}\varepsilon \) then \( \langle X(\theta , \varphi ),P \rangle {\gt}0. \)

Proof ▶

See [ SY25 ] , Lemma 14.

Lemma 18

Let \(P \in \operatorname{\mathbb {R}}^3\) with \(\| P\| \leq 1\). Further, let \(\varepsilon , r{\gt}0\) and \(\overline{\theta },\overline{\varphi }, \theta , \varphi \in \operatorname{\mathbb {R}}\) such that \(|\overline{\theta }-\theta |, |\overline{\varphi }- \varphi | \leq \varepsilon \). If \( \| M(\overline{\theta },\overline{\varphi }) P \| {\gt} r + \sqrt{2}\varepsilon \) then \( \| M(\theta ,\varphi ) P \| {\gt} r. \)

Proof ▶

See [ SY25 ] , Lemma 15.

Lemma 19

Let \(\varepsilon {\gt}0\) and \(|\theta -\overline{\theta }|,|\varphi -\overline{\varphi }|,|\alpha -\overline{\alpha }|\leq \varepsilon \) then \(\| R(\alpha ) M(\theta , \varphi )-R(\overline{\alpha })M(\overline{\theta },\overline{\varphi })\| {\lt} \sqrt{5} \varepsilon .\)

Proof ▶

See [ SY25 ] , Lemma 16.