Geometric Algebra -- Angle Between Two Vectors

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Geometric Algebra -- Angle Between Two Vectors

Postby PatrickPowers » Sun Feb 14, 2021 11:42 pm

In geometric algebra I have a plane defined by a bivector. Given two unit vectors in this plane, how do I find the angle I need to rotate the first vector so that it becomes equal with the second.

The usual method for computing the angle between two vectors gives an answer between zero and pi. That's no good. I need an answer between zero and 2pi. I'd rather not have several cases depending on quadrants. It seems like there has to be a direct way to do it.
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Re: Geometric Algebra -- Angle Between Two Vectors

Postby mr_e_man » Tue Feb 16, 2021 3:32 am

Let's call the vectors u and v, and the (normalized) bivector i. We want a complex number z = cos θ + i sin θ such that u z = v. But this is just z = u⁻¹ v = u v = u∙v + u∧v. Now we have cos θ = u∙v and sin θ = -i(u∧v), and you can use atan2 or something similar to get θ.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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Re: Geometric Algebra -- Angle Between Two Vectors

Postby PatrickPowers » Tue Feb 16, 2021 3:59 pm

mr_e_man wrote:Let's call the vectors u and v, and the (normalized) bivector i. We want a complex number z = cos θ + i sin θ such that u z = v. But this is just z = u⁻¹ v = u v = u∙v + u∧v. Now we have cos θ = u∙v and sin θ = -i(u∧v), and you can use atan2 or something similar to get θ.


Oh that's delightful. This was driving me to exhaustion. I looked around the Internet and the closest to a resource I could find was a physicist refering to GA as a cult. There seems to be nowhere else I can get answers to such questions.

My next question is, how did you learn this so I can do the same? I don't want to have to run for help every time I encounter something like this. I have about five GA books but simple things like this aren't discussed. What they do is give me bunch of operators. I have no idea what to do with them, so I learn zero.
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Re: Geometric Algebra -- Angle Between Two Vectors

Postby mr_e_man » Tue Feb 16, 2021 6:29 pm

I learned some from the websites of Hestenes (section IV of the primer is relevant to your first question), Macdonald, Denker, and maybe others; from wikipedia, and math.stackexchange; and from my own pencil work.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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Re: Geometric Algebra -- Angle Between Two Vectors

Postby PatrickPowers » Wed Feb 17, 2021 10:49 pm

mr_e_man wrote:I learned some from the websites of Hestenes (section IV of the primer is relevant to your first question), Macdonald, Denker, and maybe others; from wikipedia, and math.stackexchange; and from my own pencil work.


That primer is the stuff, terse and down to Earth. I have all of Hestenes' books I could find on line but missed that one.
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Re: Geometric Algebra -- Angle Between Two Vectors

Postby PatrickPowers » Fri Mar 12, 2021 12:33 am

mr_e_man wrote:Let's call the vectors u and v, and the (normalized) bivector i. We want a complex number z = cos θ + i sin θ such that u z = v. But this is just z = u⁻¹ v = u v = u∙v + u∧v. Now we have cos θ = u∙v and sin θ = -i(u∧v), and you can use atan2 or something similar to get θ.


I learned more from these four sentences than from some entire GA books.
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