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Equation 36 · Part 9 · A Reference Frame Becomes Classical by Publishing Its Orientation

Numerator: Tr(R_1^mathsf TR_2)-1

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),
Tr⁡(R1TR2)−1\operatorname{Tr}(R_1^{\mathsf T}R_2)-1

What this part means

The complete quantity above the fraction bar.

Its job in the formula

Tr(R1mR_1^mathsf TR2R_2)-1 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 fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the article section

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