Geometric algebra: longitude and latitude rotor ordering?

In summary, the conversation was about using geometric algebra to solve a problem of locating a satellite using angle measurements from two points. The speaker had a question about the ordering of the longitude and latitude rotations in their solution, but they were able to figure it out themselves and concluded that the rotations only commute when applied to an outwards facing vector. They also mentioned that they wouldn't post the math involved unless asked because they suspect that no one else is interested in it.
  • #1
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[SOLVED] geometric algebra: longitude and latitude rotor ordering?

Was playing around with what is probably traditionally a spherical trig type problem using geometric algebra (locate satellite position using angle measurements from two well separated points). Origin of the problem was just me looking at my Feynman Lectures introduction where there is a diagram illustrating how trianglulation could be used to locate "Sputnik" and thought I'd try such a calculation, but in a way that I thought was more realistic.

I'm pretty sure that I solved the problem, but have a question about a small detail that I glossed over. I'll describe a bit of the context as background.

Part of my solution requires the measured unit vector be rotated back to a reference frame (ie: measure direction cosines in a local frame with frame dir vectors north facing, east facing, and up facing). If I fix a reference frame at (0,0) ( equator/gren. intersection, e1 = east, e2 = north, e3 = up at that point), I can rotate to any given longitude/latitude (and thus inverse rotate my measured direction vector to the satellite location) using the following rotor:

[tex]
R = R_{g} R_{e} = \exp(-e_3 \wedge e_1 \theta/2)\exp(-e_3 \wedge e_1 \alpha/2)
[/tex]

where [itex]\theta[/itex] is the east directed angle measurement, and [itex]\alpha[/itex] is the angle to the north from the equator. This gives a combined rotor equation of:

[tex]
x' = R x R^\dagger
[/tex]

Now, looking at a globe, it seems clear to me that these "perpendicular" (abusing the word) rotations could be applied in either order, but their rotors definitely don't commute, so I assume that together the non-commutive bits of the rotors "cancel out".

Any idea how to demonstrate that the end effect of applying these rotors in either order is the same?

ie: for R_g and R_e above:

[tex]
x' = R_g R_e x R_e^\dagger R_g^\dagger = R_e R_g x R_g^\dagger R_e^\dagger
[/tex]

In general order of rotations should matter, but here (ie: for the rotors above) I don't think it will. I started doing a brute force expansion of sine and cosine terms, but decided I don't want to do that without a symbolic calculator.
 
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  • #2
Peeter said:
Any idea how to demonstrate that the end effect of applying these rotors in either order is the same?

I figured this out (examining the difference of the two rotation varients applied to an arbitrary vector). These longitude and latitude rotations only commute when applied to an outwards facing vector (ie: a point on the sphere represented by a vector). This is what intuition tells you, but that breaks down if you are trying to rotate a frame positioned at that point on the sphere, or any direction vector in that frame that has a component in the east/west or north/south directions.

I suspect that nobody but me is interested in the math involved to prove this so I won't post it unless asked.
 

1. What is geometric algebra?

Geometric algebra is a mathematical framework that extends traditional vector algebra to include not just vectors, but also other geometric objects such as planes, lines, and rotations. It allows for a more intuitive and powerful way to represent and manipulate geometric quantities.

2. How is geometric algebra used in longitude and latitude?

In the context of longitude and latitude, geometric algebra can be used to represent the Earth's surface as a sphere and perform calculations involving rotations and transformations. This can be particularly useful in fields such as geology, geography, and navigation.

3. What is a rotor in geometric algebra?

A rotor is a fundamental concept in geometric algebra that represents a rotation in three-dimensional space. It can be thought of as a combination of a vector and a scalar, and is often used to perform rotations and transformations in geometric calculations.

4. How is the ordering of rotors important in longitude and latitude calculations?

The ordering of rotors is important in geometric algebra because it determines the order in which rotations are applied. In longitude and latitude calculations, the ordering of rotors is crucial in accurately representing and transforming the Earth's surface.

5. How does geometric algebra differ from traditional vector algebra?

Geometric algebra differs from traditional vector algebra in that it includes additional geometric objects and operations, such as reflections, rotations, and translations. This allows for a more comprehensive and intuitive way to perform calculations involving geometric quantities.

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