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Equation 47 · Part 1 · A Reference Frame Becomes Classical by Publishing Its Orientation

Symbol p_blind

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

What this part means

pbp_blind is part of the quantity the equation computes from the expression on the right.

Its job in the formula

pbp_blind is part of the quantity the equation computes from the expression on the right.

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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A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the article section

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