Maths - Axis Angle - message from Mike Day

Subject: Axis Angle Page
Date: Wednesday 09 March 2005 17:46
From: "Day, Michael A AMRDEC/UAH" <>


Let me first say that I have been using your euclideanspace site for a number of moths now and have found it to be the most instructive site on the Internet for 3D graphics coding. I am not very adept at math, and the intuitive examples have helped me tremendously. I would like to try to give a little back.

This page:

has this paragraph:

Any rotation can be represented in this way, in other words, given a solid object with orientation 1 and the same object with a different orientation 2. Then we can always find an axis and angle which will rotate from orientation 1 to orientation 2. Can anyone help me prove this?

IF a local coordinate system was assigned to each object, can you not do it this way?

  1. Consider object1's x axis and object2's x-axis as vectors.
  2. Take the cross product of object 1's x-axis and object 2's x-axis. This gives you an axis of rotation.
  3. Find the angle between object 1's x-axis and object 2's x-axis.

This gives you an angle of rotation.

Mike Day
Virtual Targets Center

Sent: Thursday, March 10, 2005 11:29 AM
To: Day, Michael A AMRDEC/UAH
Subject: Re: Axis Angle Page

Hallo Mike,

Yes, thanks I didn't realise it before but I see what you mean, if we can always generate a axis-angle that is proof that a single rotation always exists.

I will update the web page. Would it be alright with you if I included a copy of your message on the site, I like to do this where I can as it acknowledges your contribution and hopefully encourages others to contribute, if you wish I can remove your e-mail address or remove anything you don't want published.

Thanks again,


Subject: RE: Axis Angle Page
Date: Thursday 10 March 2005 18:46
From: "Day, Michael A AMRDEC/UAH" <>


Yes, you can include whatever you'd like. I am not above shameless

Mike Day
Virtual Targets Center

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