For any P \in \mathbb{R}^3 one has \|M(\theta, \phi) P\|^2=\|P\|^2-\langle X(\theta,\varphi),P\rangle^2.

Let V_1,V_2,V_3,Y,Z \in \mathbb{R}^3 with \|Y \|=\| Z \| and Y,Z \in \mathrm{span}^+(V_1,V_2,V_3). Then at least one of the following inequalities fails:

• \langle V_1, Y \rangle > \langle V_1, Z \rangle

• \langle V_2, Y \rangle > \langle V_2, Z \rangle

• \langle V_3, Y \rangle > \langle V_3, Z \rangle

For A,\overline{A},B,\overline{B}\in \mathbb{R}^{n\times n} and P_1,P_2\in \mathbb{R}^n it holds that

|\langle AP_1,BP_2\rangle-\langle \overline{A}P_1,\overline{B}P_2\rangle| \leq \|P_1\|\,\|P_2\|\,\big( \|A-\overline{A}\|\,\|\overline{B}\| + \|\overline{A}\|\,\|B-\overline{B}\|+\|A-\overline{A}\|\,\|B-\overline{B}\|\big).

For A,B\in \mathbb{R}^{n\times n} and P_1,P_2\in \mathbb{R}^n one has

|\langle AP_1,AP_2\rangle-\langle BP_1,BP_2\rangle| \leq \|P_1\|\,\|P_2\|\,\|A-B\|\,(\|A\|+\|B\| + \|A-B\|).

Let A,B,C\in \mathbb{R}^2 be such that \langle R(\pi/2) A,B\rangle, \langle R(\pi/2) B,C\rangle, \langle R(\pi/2) C,A\rangle > 0. Then the origin lies strictly in triangle ABC.

Let P_1, P_2, P_3 \in \mathbb{R}^3 with \|P_1\|,\|P_2\|,\|P_3\| \leq 1 be \epsilon-spanning for (\bar\theta, \bar\phi) and let \theta, \phi \in \mathbb{R} satisfy |\theta - \bar{\theta}|, |\phi - \bar{\phi}| \leq \epsilon. If \langle X(\theta, \phi), P_i \rangle > 0 for i=1,2,3, then X(\theta, \phi) \in \spanp(P_1, P_2, P_3).

Let P, Q \in \mathbb{R}^3 with \|P\|, \|Q\| \leq 1, and let \epsilon>0, \bar\theta_1,\bar\phi_1,\bar\theta_2,\bar\phi_2,\bar\alpha \in \mathbb{R}. Set

T := \left(R(\bar\alpha) M(\bar\theta_1, \bar\phi_1) P + M(\bar\theta_2, \bar\phi_2) Q\right)/2 \in \mathbb{R}^2,

and assume \delta \geq \|T - M(\bar\theta_2, \bar\phi_2) Q\|. If |\bar\theta_1-\theta_1|, |\bar\phi_1-\phi_1|, |\bar\theta_2-\theta_2|, |\bar\phi_2-\phi_2|, |\bar\alpha - \alpha| \leq \epsilon, then R(\alpha)M(\theta_1, \phi_1) P and M(\theta_2, \phi_2) Q lie in \mathrm{Circ}_{\delta + \sqrt{5} \epsilon}(T).

Let \mathbf{P} be a convex polygon and Q \in \mathbf{P} a vertex. Let \overline{Q} \in \mathbb{R}^2 with \|Q - \overline{Q}\| < \delta for some \delta>0. Assume there exists r > 0 with \|Q\| > r such that

\frac{\langle Q, Q - P_j \rangle}{\|Q\|\|Q - P_j\|} \geq \delta/r

for all other vertices P_j \in \mathbf{P} \setminus Q. Then Q is \delta-locally maximally distant with respect to \overline{Q}.

Let \epsilon>0 and \theta,\bar\theta, \phi, \bar\phi \in \mathbb{R} with |\theta - \bar{\theta}|, |\phi - \bar{\phi}| \leq \epsilon. Define M = M(\theta, \phi) and \overline{M} = M(\bar\theta, \bar\phi), and let P, Q \in \mathbb{R}^3 with \|P\|, \|Q\| \leq 1. Assume that

\frac{\langle \overline{M} P,\overline{M} (P-Q)\rangle - 2 \epsilon \|P-Q\| \cdot (\sqrt{2}+\varepsilon)}{ \big(\|\overline{M} P\|+\sqrt{2} \varepsilon \big) \cdot \big(\|\overline{M}(P-Q)\|+2 \sqrt{2} \varepsilon\big)} > 0. $$

Then:

\frac{\langle MP,M(P-Q)\rangle}{\|MP\|\,\|M(P-Q)\|} \geq \frac{\langle \overline{M} P,\overline{M} (P-Q)\rangle - 2 \epsilon \|P-Q\| \cdot (\sqrt{2}+\varepsilon)}{ (\|\overline{M} P\|+\sqrt{2} \varepsilon ) \cdot (\|\overline{M}(P-Q)\|+2 \sqrt{2} \varepsilon)}.

Let P_1,P_2,P_3, Q_1,Q_2,Q_3 \in \mathbb{R}^3. Define P := (P_1|P_2|P_3) and Q := (Q_1|Q_2|Q_3) and assume Q is invertible. Then P_1, P_2, P_3 and Q_1, Q_2, Q_3 are congruent iff P^t P = Q^t Q.