Equation 36 · A Reference Frame Becomes Classical by Publishing Its Orientation

What does this equation mean?

d(R_1,R_2)= \cos^{-1}\!\left(\frac{\operatorname{Tr}(R_1^{\mathsf T}R_2)-1}{2}\right),

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 withTr(R_1^mathsf TR_2)-1

Divide by2

This relates tod(R_1,R_2)

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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.

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d

Symbol d

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

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R_1

Symbol R_1

$R_1$ occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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R_2

Symbol R_2

$R_2$ occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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R_1^{\mathsf T}

Symbol R_1^mathsf T

$R_1^m$ athsf T occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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=

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

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fraction

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

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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.

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superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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\operatorname{Tr}(R_1^{\mathsf T}R_2)-1

Numerator: Tr(R_1^mathsf TR_2)-1

The complete quantity above the fraction bar.

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2

Denominator: 2

The complete quantity below the fraction bar; it must be nonzero for this division.

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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

For SO(3) , a coordinate-free angular distance between two rotations can be defined by $d(R_1,R_2)= \cos^{-1}\!\left(\frac{\operatorname{Tr}(R_1^{\mathsf T}R_2)-1}{2}\right)$ . with values from zero to $\pi$ . Radians are dimensionless in SI, but d still has the operational meaning of an angular error. A different group requires a declared invariant metric appropriate to that group. A coordinate chart’s Euclidean distance is not an acceptable substitute near its singularities.

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Sources cited in the article section

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