3 Bounding Rotations

Lemma 9

✓

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

Proof ▶

See [ SY25 ] , Lemma 9.

Lemma 10

✓

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 11

✓

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 ▶

The composition \(R_d(\alpha )R_{d'}(\beta )\) is a rotation, i.e. conjugate to some \(R_z(\gamma )\) by an isometry, which gives \(\| R_d(\alpha )R_{d'}(\beta )-\operatorname{\mathrm{Id}}\| ^2 = 2(1-\cos \gamma ) = 3 - \operatorname{\mathrm{tr}}(R_d(\alpha )R_{d'}(\beta )) = 3 - (\cos \alpha + \cos \beta + \cos \alpha \cos \beta )\). Writing the right hand side as \(2(1-\cos \alpha ) + 2(1-\cos \beta ) - (1-\cos \alpha )(1-\cos \beta )\) and using \(2(1-\cos x) \leq x^2\) (with equality only at \(x = 0\)) yields the bound and the equality condition. This is a direct route to [ SY25 ] , Lemma 12, that avoids the Jensen-inequality argument of Lemma 11 and the doubling induction.

Lemma 12

✓

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 13

✓

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 14

✓

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. Corrigendum: the triangle inquality only implies greater than *or equal to*.

Lemma 15

✓

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.