Manuscript VIII-A@3 · VIII-A-v3.tex · lemma · 2398a747

Lemma: Exact rotation residuals

A4: two independently rejected partial updates › Changed deficit with a stale rotation · lines 2232–2268 · proof included or linked

VIII-A@3#A:lem:rotation-residual, VIII-A@3#A:eq:rotation-residual, VIII-A@3#A:eq:row-residual

Back to manuscript

For every $θ, Δ \in R$ ,

∥ R (θ) - R (θ + Δ) ∥_{2} = 2 ∣ sin (Δ/2)∣ .

At $θ = π /2$ ,

∥ R (π /2) - R (π /2 + Δ) ∥_{row} = ∣ sin Δ ∣ + ∣ 1 - cos Δ ∣ .

Proof. The time block of the difference is zero. Its spatial block is $Rot (θ) (I_{2} - Rot (Δ))$ . Orthogonal multiplication preserves the spectral norm, and

(I_{2} - Rot (Δ))^{T} (I_{2} - Rot (Δ)) = 4 sin^{2} (Δ/2) I_{2} .

This proves A:eq:rotation-residual. At $θ = π /2$ , the spatial difference is

(sin Δ 1 - cos Δ cos Δ - 1 sin Δ) .

Both absolute row sums give A:eq:row-residual.

Displayed formulas in this passage

VIII-A@3#A:eq:rotation-residual · line 2235
$∥ R (θ) - R (θ + Δ) ∥_{2} = 2 ∣ sin (Δ/2)∣ .$
VIII-A@3#A:eq:row-residual · line 2241
$∥ R (π /2) - R (π /2 + Δ) ∥_{row} = ∣ sin Δ ∣ + ∣ 1 - cos Δ ∣ .$
unlabeled · line 2254
$(I_{2} - Rot (Δ))^{T} (I_{2} - Rot (Δ)) = 4 sin^{2} (Δ/2) I_{2} .$
unlabeled · line 2261
$(sin Δ 1 - cos Δ cos Δ - 1 sin Δ) .$

Lemma: Exact rotation residuals

Displayed formulas in this passage

Nearby in this section