# Maths - Rotations - Correspondance 3

From: William Lupton
To: Martin Baker
Subject: Euler parameter typo?
Date: 01 March 2002 09:41

Martin,

I believe that there is a typo in the Euler parameters for the NASA
standard airplane case
(https://www.euclideanspace.com/threed/scenegraph/rotations/euler.htm). Writing
c1 for cos(psi/2) etc, the stated formulae are:

e0 = c1c2c3 + s1s2s3
e1 = c1c2s3 + s1s2s3
e2 = c1s2c3 + s1c2s3
e3 = s1c2c3 - c1s2s3

However, I think that the e1 formula should be:

e1 = c1c2s3 - s1s2c3

Apart from anything else, this fits the pattern of the others
(alternating sign, and c in one term being s in the other).

Another thing that worried me was the complexity of the Euler to
Quaternion conversion
parameters were already the desired Quaternion, i.e. (w, x, y, z) = (e0,
e1, e2, e3). Incredibly, the (rather complex) formulae presented are
indeed analytically the same as the Euler parameters (I checked...
that's how I found the error!). IMHO, the Euler parameter method is
better because it requires the same number of trig operations, no square
root, no worry about dividing by zero, uses familiar formulae, and is
fairly clearly normalised.

It's amazing how hard it is to find good material on this subject on the
web. Thanks for your amazing site.

William Lupton

From: "Martin Baker"
To: "William Lupton"
Subject: Re: Euler parameter typo?
Date: 03 March 2002 18:18

William,

Thank you very much for this, I have corrected the e1 formula.

On the Euler to Quaternion conversion, would it be possible to send me the
working to show that the formulae presented are analytically the same as the
Euler parameters. I tried, but the trigonometry beat me.

I think I need to standardise on the quaternion terminology, on the website,
because programmers will be using quaternion terminology to write 3D
programs. However I agree that it would be best to use the formulae from the
Euler parameters, for the reasons you state.

Do you have any views about what system of euler angles that I should
concentrate on? NASA standard airplane? Or do you think that I need to
include the formulas for each type? By the way, do you know, are the NASA
standards available on the web?

Is it OK if I include your messages in my web pages?

Martin

Martin,

> On the Euler to Quaternion conversion, would it be possible to send me the
> working to show that the formulae presented are analytically the same as the
> Euler parameters. I tried, but the trigonometry beat me.

I attach a file euler.abw (abiword) and also euler.doc (MS Word, written by
abiword). Please let me know if these are not satisfactory (abiword can also
generated HTML but it lost the subscripts, at least for me with my rather old
Netscape version).

> Do you have any views about what system of euler angles that I should
> concentrate on? NASA standard airplane? Or do you think that I need to
> include the formulas for each type? By the way, do you know, are the NASA
> standards available on the web?

I hope I didn't give the impression that I am an expert! I didn't know anything

You must have seen the pages at mathworld.wolfram.com? They seem quite good and
seem to feel that they need to give the formulae for several different Euler
angle conventions. I don't know if the NASA standards are on the web... you
will know that a Google search for "nasa standard airplane" comes up with only

> Is it OK if I include your messages in my web pages?

Of course, if you would like to.

My pleasure.

William

From: "Martin Baker"
To: "William Lupton"
Subject: Re: Euler parameter typo?
Date: 09 March 2002 10:38

William,

Thank you very much for this, I have put it here:
https://www.euclideanspace.com/threed/scenegraph/rotations/william.htm
and I have cross-linked this from the other web pages. I hope this is
alright as I am just about to go on a weeks holiday. Ill check it again when
I get back.

I used the word version that you sent me, but word does generate very large
HTML files, so I'll try abiword, does it have an equation editor?

Martin

From: "William Lupton"
To: "Martin Baker"
Subject: Re: Euler parameter typo?
Date: 11 March 2002 08:37

Martin,

The web page looks good.

You wrote:

> I used the word version that you sent me, but word does generate very large
> HTML files, so I'll try abiword, does it have an equation editor?

The HTML file size is probably mostly due to the subscripts!

Abiword has no equation editor. StarOffice may do (I'm not sure) but it is so
resource hungry as to be impracticable to use (IMHO). Lyx (which generates
LaTeX) is surprisingly good and has a GUI interface to TeX's math mode.
Actually, I probably should have used straight HTML!

William