Maths - AxisAngle to Matrix

Minor Corrections: AxisAngle to Matrix

By: Hauke (haukeheibel) - 2007-04-26 00:41

Hi there,

First of all thanks for the nice web-site. Optically it could be nicer but the content rocks!

So here are my minor remarks regarding the conversion between axis angle representation and matrix representation.

a) When substituting the basis vectors of the plane perpendicular to the rotation vector for the first time, you change the order of the cross product between basis2 and the rotation axis without changing the sign. You are writing:

------------- begin cite -------------

P2 = P1 + (cos(angle) - 1) * basis1 + sin(angle) * basis2

substituting the basis values above gives:

P2 = P1 + (cos(angle) - 1) * (axis × axis × P1) + sin(angle) * axis × P1

-------------- end cite --------------

whereas the last line should in my humblest opinion be:

P2 = P1 + (cos(angle) - 1) * ((axis × P1) × axis) + sin(angle) * axis × P1

As you can see I also added brackets around the substitution of basis2, since the cross product is not associative.

b) In the diagram where you are visualizing the plane being perpendicular to the rotation vector, the rotation vector and the point to be rotated, basis2 should actually point in the other direction when sticking to the right hand side rule.

As said before. These are minor remarks and I appreciate the work that has been done a lot.

Regards,
Hauke Heibel

RE: Minor Corrections: AxisAngle to Matrix

By: Martin Baker (martinbaker

) - 2007-04-26 02:58

Hello Hauke,

Good to hear from you, thanks very much for letting me know about this, I think its important to get the details right.

I have modified this page as you suggested:
http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToMatrix/

I have changes the proof so that the equation is converted into matrix form in two stages to avoid applying associative rule. I've also changed the diagram.

I am pleased you mentioned the 'look' of the site as I would welcome your, and anyone else's, views on that.
Over the past year I have tried to improve the look of the site by changing the buttons at the top by not having a background of primary colour. I have also limited the width of the main text to 600 pixels to make the text easier to read. I then added 'breakout' boxes on the right.

But I don't know how to improve it further:
* What do you think of the multilevel navigation buttons at the top? I know its a bit strange but I wanted a way that users can instantly see where they are in the hierarchy of the site, can you think of a better way to do that?
* I have not set a background colour for the pages as I thought this would override the browser defaults and remove the choice from users?
* I've started using .png diagrams instead of .gif diagrams. Unfortunately older versions of ie render this with a grey background instead of a transparent background.

Do you think these thinks need changing? Or is it just my lack of artistic ability? Don't worry about being diplomatic, I can take it!
I think it is important, what the site looks like, I don't want to frighten off potential readers, any ideas you have will be appreciated.

Thanks,

Martin

Design (was "Minor Corrections: ...")

By: Hauke (haukeheibel) - 2007-04-26 04:49

Hi Martin,

(referring to: http://sourceforge.net/forum/message.php?msg_id=4281274)

Uh well, what can I say. Design is definitely a matter of personal taste. The idea of the multi-level navigation is nice. On the other side there are always hundreds of solutions to a design issue and each one of them has its advantages and disadvantages. You'll probably never be able to make everybody happy.

There is one single thing I would try to prevent in the future. I would suggest using more saturated color tones. For instance on your main site, I assume the colors in your table indicate the section to which the content is referring. May be it is a little bit too much of information. The pattern the differently colored cells are creating does not look very nice.

That you did not go for a background image is something I would stick to. Again, too much information as color and images will distract the users from the actual content.

Regarding the problem with png vs. gif and old versions IE not properly handling the transparency settings of .pngs.... forget about it. The creators of web-browsers seem to care very little about standards and are coming up every other day with a new "standard" which only their lab knows about. I think its virtually impossible to make your web-site look the same on every browser and especially not if you are going into so much detail as taking care even of different browser versions.

As said before - I personally like web-sites using little color combined with plain and simple design. Sourceforge.net itself is a good example.

You should take care and definitely think twice before putting a lot of effort into re-designing your web-site. It became quite huge over time and will for sure require a lot of resources, many people would prefer seeing you in spending on the actual content as you are doing at the moment.

I will for sure stick to reading euclideanspace.com no matter what the design will be like. In the end the content is more important.

I wish you all the best,
Hauke

RE: Design (was "Minor Corrections: ...")

By: Martin Baker (martinbaker

) - 2007-04-26 10:26

Thanks again Hauke,

I agree with all you say and I have taken the background colours off the home page.

Its good to get feedback on this as I find I get certain ideas and perhaps loose the ability to see the site as users (especially new users) would see it.

I think I agree about content over presentation, having said that, if the presentation puts people off then there will be less feedback and the content will suffer.

I'm gradually trying to introduce the use of css (cascading style sheet) to define some of the presentation so there are now a few things I can change quite quickly across the site.

Cheers,

Martin

RE: Minor Corrections: AxisAngle to Matrix

By: Hauke (haukeheibel) - 2007-04-27 01:31

Hi Martin,

I just stumbled over another small error within the axis angle conversion.

At the top of the page where you are writing the matrices, right above the code sample, the scaling c in front of the identity matrix has to go away. And on the diagonal elements of the squared antisymmetric matrix you need to add -1, so x*x-1 and so on.

Finally I am a little bit confused about the "Issues" section. What's the issue? ;)

Cheers,
Hauke

RE: Minor Corrections: AxisAngle to Matrix

By: Martin Baker (martinbaker

) - 2007-04-27 02:19

Hauke,

Can you explain a bit more? Most of the matrices on the page have a 'c' term in each of the leading diagonal elements, so I hope that's right or the page will need a big rewrite!

As an example, if we take the top left element of the matrix, this can be expressed as:
1 + (1-c) * (x*x - 1)
= 1 - (1-c) + (1-c)*x*x
= c + t*x*x (because t = 1-c)
Is this correct?

> Finally I am a little bit confused about the "Issues" section. What's the issue?
Yes, I take your point, I think I was just trying to define the notation and index numbering of the matrix. That's not really necessary on that page, I'll just link to the 'standards' page instead.

Thanks,
Martin

RE: Minor Corrections: AxisAngle to Matrix

By: Hauke (haukeheibel) - 2007-04-27 02:45

Hi again,

You are right. It can be written, as you did. The only point one could argue about is that the matrix being multiplied by t does not correspond to [~axis]^2. I was simply confused by the fact that the forumla does not correspond to:

[R] = [I] + sin(angle)[~axis] + (1-cos(angle))[~axis]^2

which is (as you have written at the bottom of the page)

[R] = I + s*[~axis] + t*[~axis]^2
= I + t*[~axis]^2 + s*[~axis];

and can be formulated as you did:

= c*I + t*([~axis]^2 + I) + s*[~axis]
= c*I + t*[~axis]^2 + t*I + s*[~axis]
= (c+t)*I + t*[~axis]^2 + s*[~axis]
= (c+1-c)*I + t*[~axis]^2 + s*[~axis]
= I + t*[~axis]^2 + s*[~axis]

Regards,
Hauke

RE: Minor Corrections: AxisAngle to Matrix

By: Martin Baker (martinbaker

) - 2007-04-27 03:50

Hauke,

That's interesting,
I + t*[~axis]^2 + s*[~axis]
does look simpler than:
c*I + t*([~axis]^2 + I) + s*[~axis]
but when we write out the individual elements of the matrix, then the second form looks simpler.

I'll therefore write this equivalence more explicitly on the web page to try to reduce the confusion.

Thanks,

Martin

Maths - Forum discusion with Hauke