← Back to article

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

What does this equation mean?

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

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start withepsilon-sinepsilon
Divide bypi
This relates top_blind(epsilon)
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

Read it piece by piece

pblindp_{\mathrm{blind}}

Symbol p_blind

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

Understand this part →

ϵ\epsilon

Symbol epsilon

epsilon occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Understand this part →

π\pi

Symbol pi

the values from zero to.

Understand this part →

=

=

The expressions on both sides represent the same quantity under the stated assumptions.

Understand this part →

See an illustrated explanation →
fraction

fraction

Divide the expression above the line by the one below it.

Understand this part →

See an illustrated explanation →
subtraction

subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

Understand this part →

subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Understand this part →

ϵ−sin⁡ϵ\epsilon-\sin\epsilon

Numerator: epsilon-sinepsilon

The complete quantity above the fraction bar.

Understand this part →

How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

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.

Read the equation in its article →

Sources cited in the article section

These citations give research context. Read each source to check which claims it supports.

Return to A Reference Frame Becomes Classical by Publishing Its Orientation

See this formula across 1 published context →

Browse the mathematical compendium →