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Published equation contexts

d(R1,R2)=cos⁡−1 ⁣(Tr⁡(R1TR2)−12)d(R_1,R_2)= \cos^{-1}\!\left(\frac{\operatorname{Tr}(R_1^{\mathsf T}R_2)-1}{2}\right)

Why this formula appears here

For SO(3) , a coordinate-free angular distance between two rotations can be defined by d(R1,R2)=cos⁡−1 ⁣(Tr⁡(R1TR2)−12)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 π\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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R1TR_1^{\mathsf T}

Symbol R_1^mathsf T

R1mR_1^mathsf T occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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Tr⁡(R1TR2)−1\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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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.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

d(R1,R2)=cos⁡−1 ⁣(Tr⁡(R1TR2)−12),d(R_1,R_2)= \cos^{-1}\!\left(\frac{\operatorname{Tr}(R_1^{\mathsf T}R_2)-1}{2}\right),

Equation 36 · Evolutionary Physics

A Reference Frame Becomes Classical by Publishing Its Orientation

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

For SO(3) , a coordinate-free angular distance between two rotations can be defined by d(R1,R2)=cos⁡−1 ⁣(Tr⁡(R1TR2)−12)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 π\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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