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Equation 47 · Part 3 · A Reference Frame Becomes Classical by Publishing Its Orientation

Symbol pi

pblind(ϵ)=ϵ−sin⁡ϵπ.p_{\mathrm{blind}}(\epsilon)=\frac{\epsilon-\sin\epsilon}{\pi}.
π\pi

What this part means

the values from zero to.

Its job in the formula

pi occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Where the article explains it

(Tr⁡(R1TR2)−12)\left(\frac{\operatorname{Tr}(R_1^{\mathsf T}R_2)-1}{2}\right), with values from zero to π\pi .

The passage around this formula

There is a trap. Even an orientation-blind fragment can sometimes satisfy a loose threshold by guessing from the prior. For uniform SO(3) , the probability that a blind estimate lies within geodesic angle ϵ\epsilon of the true rotation is pblind(ϵ)=ϵ−sin⁡ϵπp_{\mathrm{blind}}(\epsilon)=\frac{\epsilon-\sin\epsilon}{\pi}. This expression follows from the Haar distribution of rotation angle, whose density on [0,π\pi] is 2sin⁡2(θ/2)\sin^2(\theta/2)/π\pi . It passes the limiting checks: at ϵ\epsilon=0 , blind success is zero; at ϵ\epsilon=π\pi , it is one. The redundancy definition is informative only when 1-δ\delta>pblind(ϵ)p_{\mathrm{blind}}(\epsilon) . Otherwise an empty box can qualify as a witness.

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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