Talk:Euler–Rodrigues formula

Composition of rotations

I think that there's either an error or some different unspecified convention used. Presented formula doesn't match with either the Hamilton product or the expression presented in the linked Euler's four-square identity article. AnonN10 (talk) 05:08, 10 May 2024 (UTC)

Vector formula incorrect?

I'm finding the vector formula given here gives incorrect results. I'm unable to check if it was misinterpreted in its original text, but none is given, and there is no derivation. Where did it come from? Can someone find an external source?

An AI assistant corrected it to x + a * 2 * cross(ijk, x) + cross(ijk, cross(ijk, x)), which seems to work experimentally, but of course this is not admissible.

Timrb (talk) 04:25, 31 August 2024 (UTC)

I was not able to find the source of the formula given in the article

${\vec {x}}'=a^{2}{\vec {x}}+2a({\vec {\omega }}\times {\vec {x}})+2\left({\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})\right)$

The standard Rodrigues' rotation formula is

${\vec {x}}'={\vec {x}}\cos \theta +({\vec {\omega }}\times {\vec {x}})\sin \theta +{\vec {\omega }}({\vec {\omega }}\cdot {\vec {x}})(1-\cos \theta )$

However, conversion between the two formulas turned out to be exciting because the golden ratio suddenly cropped up. Briefly, we must match the following coefficients:

Coefficient of ${\vec {x}}$ in the standard form is $\cos \theta$ while in the article form is $a^{2}$

Coefficient of ${\vec {\omega }}\times {\vec {x}}$ is $\sin \theta$ in the standart form and $2a$ in the article form

Coefficient of ${\vec {\omega }}({\vec {\omega }}\cdot {\vec {x}})$ is $1-\cos \theta$ in the standard form and 2 in the article form.

From these comparisons, we can derive the following relationships:

1. $a^{2}=\cos \theta$

2. $2a=\sin \theta$

Let's solve these equations:

1. From $a^{2}=\cos \theta$, we get

$a=\pm {\sqrt {\cos \theta }}$

2. From $2a=\sin \theta$, we get

$a={\frac {\sin \theta }{2}}$

To satisfy both equations simultaneously, we equate the two expressions for a

$\pm {\sqrt {\cos \theta }}={\frac {\sin \theta }{2}}$

Squaring both sides to eliminate the square root

$\cos \theta ={\frac {\sin ^{2}\theta }{4}}$

Using the Pythagorean identity $\sin ^{2}\theta +\cos ^{2}\theta =1$

$\cos \theta ={\frac {1-\cos ^{2}\theta }{4}}$

Multiplying through by 4

$4\cos \theta =1-\cos ^{2}\theta$

Rearranging to form a quadratic equation

$\cos ^{2}\theta +4\cos \theta -1=0$

Solving this quadratic equation for $\cos \theta$

$\cos \theta ={\frac {-4\pm {\sqrt {16+4}}}{2}}={\frac {-4\pm {\sqrt {20}}}{2}}={\frac {-4\pm 2{\sqrt {5}}}{2}}=-2\pm {\sqrt {5}}$

Since $\cos \theta$ must be between -1 and 1, we select the valid solution

$\cos \theta =-2+{\sqrt {5}}$

Now, substituting back to find a:

$a=\pm {\sqrt {-2+{\sqrt {5}}}}$

But the golden ratio $\phi ={\frac {1+{\sqrt {5}}}{2}}$, and then

$2\phi =1+{\sqrt {5}}$

Therefore

$a=\pm {\sqrt {2\phi -3}}$

Further, we have the identity

${\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})={\vec {\omega }}({\vec {\omega }}\cdot {\vec {x}})-{\vec {x}}({\vec {\omega }}\cdot {\vec {\omega }})$

If ${\vec {\omega }}$ is a unit vector, ${\vec {\omega }}\cdot {\vec {\omega }}=1$, so the expression simplifies to

${\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})={\vec {\omega }}({\vec {\omega }}\cdot {\vec {x}})-{\vec {x}}$

After some algebraic manipulations, I obtained

${\vec {x}}'={\vec {x}}\pm 2{\sqrt {2\phi -3}}({\vec {\omega }}\times {\vec {x}})+2(2-\phi )\left({\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})\right)$

See also Rodrigues' rotation formula where the following two expressions are derived:

$$\mathbf {v} _{\text{rot}}=\cos(\theta )\,\mathbf {v} +(1-\cos \theta )(\mathbf {k} \cdot \mathbf {v} )\mathbf {k} +\sin(\theta )\mathbf {k} \times \mathbf {v} ,$$$$\mathbf {v} _{\text{rot}}=\mathbf {v} +(1-\cos \theta )\mathbf {k} \times (\mathbf {k} \times \mathbf {v} )+\sin(\theta )\mathbf {k} \times \mathbf {v} .$$

The golden ratio expression that I derived above shows that $\mathbf {v} _{\text{rot}}$ does not depend on $\theta$ or $\cos \theta$. This is not plausible, which means that probably the formula given in the article is erroneous. To express the formula with the Euler-Rodrigues parameter a, the cosine and sine are, respectively

$\cos \theta =2a^{2}-1$

$\sin \theta =2{\sqrt {a^{2}(1-a^{2})}}$

Now, if you express cos through sin, substitute them with a, and equate them as above, you find 0 = 0, which makes a lot more sense.

Lantonov (talk) 09:18, 4 October 2024 (UTC)

I agree with your implicit revulsion at the sloppiness of the article, which should properly be a footnote to the main article's Rodrigues' rotation formula#Derivation, bottom of the paragraph. Indeed, you are comparing with the wrong formula, instead of the bottom one of this section linked, which has a unit-normalized k, instead of the aggressively unnormalized ω here. In your place, I'd normalize the latter to the former, and compare normalizations properly. But the overall point is that this article needs its quality raised. Cuzkatzimhut (talk) 16:45, 5 October 2024 (UTC)

I raise my thumb for merging of articles since I don't see a difference between the terms Euler-Rodrigues formula and Rodrigues' rotation formula. On the other hand, the article Euler-Rodrigues formula has some important material which is missing in Rodrigues' rotation formula such as the matrix-form connections with quaternions, spin matrices, and the Euler four-square identity. Lantonov (talk) 10:47, 10 October 2024 (UTC)

@Lantonov: As User:Cuzkatzimhut has noted, I think you may need to take another look at your comparison above.

It is important to note that $\mathbf {k} \not \equiv {\vec {\omega }}$.

$\mathbf {k}$ is a unit vector along the axis of rotation. But ${\vec {\omega }}$ is defined to be

${\vec {\omega }}\equiv (b,c,d)=\mathbf {k} \sin {\frac {\varphi }{2}}$

The article Rodrigues' rotation formula gives the formula

$\mathbf {v'} =\mathbf {v} +\sin(\theta )\mathbf {k} \times \mathbf {v} +(1-\cos \theta )\mathbf {k} \times (\mathbf {k} \times \mathbf {v} ).$

This article gives

${\vec {x}}'=a^{2}{\vec {x}}+2a({\vec {\omega }}\times {\vec {x}})+2\left({\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})\right)$

Looking at these term-by-term from the right,

${\begin{aligned}2\left({\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})\right)&=2\sin ^{2}{\frac {\varphi }{2}}\;(\mathbf {k} \times (\mathbf {k} \times \mathbf {v} ))\\&=(1-\cos \varphi )\;\mathbf {k} \times (\mathbf {k} \times \mathbf {v} )\end{aligned}}$

${\begin{aligned}2a({\vec {\omega }}\times {\vec {x}})&=2\cos {\frac {\varphi }{2}}\sin {\frac {\varphi }{2}}(\mathbf {k} \times \mathbf {v} )\\&=\sin \varphi \;(\mathbf {k} \times \mathbf {v} )\end{aligned}}$

So that just leaves the factor of $a^{2}=\cos ^{2}{\tfrac {\varphi }{2}}$ applied to ${\vec {x}}$ to explain, which looks to be where the anomaly has occurred.

The formulation using $(a,b,c,d)$ is the point of this article (and pre-dates Rodrigues: 1770 vs 1840). It's quite widely used, so is useful to be able to find. No strong objection in principle to a careful merge into Rodrigues' rotation formula -- the content of the two articles may benefit by being presented together. But it would be a big block of content to add to the end of that other article. Jheald (talk) 14:01, 10 October 2024 (UTC)

Note: have also confirmed that substituting

${\vec {\omega }}\,\times \;\leftrightarrow \;{\begin{pmatrix}0&-d&c\\d&0&-b\\-c&b&0\end{pmatrix}}$

into the corrected formula

${\vec {x}}'={\vec {x}}+2a({\vec {\omega }}\times {\vec {x}})+2\left({\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})\right)$

does indeed lead correctly to the component formula at the top of the article. I think we can remove the {{cn}} tag now? -- Jheald (talk) 15:34, 10 October 2024 (UTC)

This formula seems correct on the basis of the above explanation:

${\vec {x}}'={\vec {x}}+2a({\vec {\omega }}\times {\vec {x}})+2\left({\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})\right)$

It is ${\vec {x}}$ and not $a^{2}{\vec {x}}=\cos ^{2}{\tfrac {\varphi }{2}}{\vec {x}}$ because of the identity:

${\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})={\vec {\omega }}({\vec {\omega }}\cdot {\vec {x}})-{\vec {x}}({\vec {\omega }}\cdot {\vec {\omega }})$

I think that the tag {{cn}} should stand till the source of this formula is found. Otherwise, I like it because it is compact and is based on the Euler parameters, so closer to quaternions.

Lantonov (talk) 20:16, 11 October 2024 (UTC)