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Equation 36 · Part 4 · A Reference Frame Becomes Classical by Publishing Its Orientation

Symbol R_1^mathsf T

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),
R1TR_1^{\mathsf T}

What this part means

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

Its job in the formula

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

The passage around this formula

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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Learn the underlying idea

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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