# Maths - Axis Angle - message from Mike Day

Subject: Axis Angle Page
Date: Wednesday 09 March 2005 17:46
From: "Day, Michael A AMRDEC/UAH" <mike.a.day@us.army.mil>

Howdy

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:

https://www.euclideanspace.com/maths/geometry/rotations/axisAngle/index.htm

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
https://modelexchange.army.mil/

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,

Martin.

Subject: RE: Axis Angle Page
Date: Thursday 10 March 2005 18:46
From: "Day, Michael A AMRDEC/UAH" <mike.a.day@us.army.mil>

Martin

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

Mike Day
Virtual Targets Center
https://modelexchange.army.mil/

 metadata block see also: Correspondence about this page Book Shop - Further reading. Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them. Mathematics for 3D game Programming - Includes introduction to Vectors, Matrices, Transforms and Trigonometry. (But no euler angles or quaternions). Also includes ray tracing and some linear & rotational physics also collision detection (but not collision response). Other Math Books

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